yqyxwvuts jsrsorwj~}t|zyrowxswjvutsrqpnlkjigfhdd8e&xqvGI rqphgf8db` { { om h e i e aca
X V U S Q PHFD C A 97 YWRTR&IGE0B@86
3 1 ) %# 504( 20('&$"!
8# 5 3 "sC1)s! #1T1Brn11B`5EWqTpICB`5ED%o8%u)`7n`@T0%X0%UB1I@k!%GDUET8E%B`Q11y3j@igf@1(%1(%D@B%C(1B%8 3 S # 3 5 !Y 7 7 48 7 V 4& ! 7 48 3 48! 4 ! S 4&# V# v ml 4# 9 c Y 4!# 4 7 x h 3 9 '&# '&# c Y e# 9 '& 3 2 8 7 9 7 3 9 Y 7 7 9 # 4 V# 9 V 8 7 58& 4# x S F # 3 5Y! 7 5 V 8 7 9# 3 c 8 3 5 A F !Y 7 V Y8Y 8 7 5 ! S 4# 9 0d$dX0)1XETXE1iUI$CB%I0)T7%gus rq@)BuuEeXg(1DDCU10XWE0@@`TSI`!%GXDc Y 4!# 4 7 x v t S F Y 38& h 89#58& 5 c 7 5 V c 7 a S !#Q Y& 7 V V 5& 8 75 49Q9#8 & F 48 3 5 A9 8 7 5 4 3 2 ! '&# '&# ! ET8E%B`Q11y3dwus rqpi1%fIg(B%fDDeXPDd)b`9$%BWI)XWIUB6TSBRR(EPIHGEDCB@6)$1((0)(%$"
Bfx 5By1HBEi0``41Ik&EE)H0i0yiXsk!11%BXW1 Sq' e 3 9 9 5 7 7 5! # S 3 48# 9 7 8 x V 4 ' # 5 V 5 V 5 # 4 c 3 Y 49 c 35 7 e ! 8 48& 5# Y8 4 9Q 0`D)%8IWo8B%#DCBBQ$dBEEk!`4SX0pTQ%fuBE)pE0TY%#qp 1713 o85 fRW%0y E%y#x sB1i`7BrsX 4 Y V# 1 ! 8# Q 3 5 8 4 5 7 5! E%y#x ! 85 7 e S 3 3 48 4#Y8 9#9& S e c 8 & 4 7 489 S9Q c 39 7 79 7 c #8& ! E)H7$B`TS17g1`2%uk!%0EI&B(@`90 qY UE0E#0Ek!Dc B5`ECT3BRU1HB11Bw1 q3 UE%frETS T8B 45 4 5 x 489Q ! 7 5& 8 ! S 48 3#8! 5& 8 5 V u 3 8 4#8! x ! '& 5# 48 5 V # 38& 7 1 S 3 I TE 5#TBQ21E%#0E)B0iu!" (`1`AEI&)E%uC7BI!v14%XpB9 `70%Es%y#@)(%1 !21EBXi10%f)y5TU12 # #8 7 5 c ! ! 3 38& 3 4 ! 489 78 ! V 4 3! # V#e Y& 7 V 5 9 ! '& 5# 485 V ! S 8 4 4# 4#8Y V5Y e #8 7 0%EWVPDgE)01%f)%y#1`7i"B`QE(80EiXbFWET81E0X%0BbF I)XVWI&i1(D%1$!v1%X"TgI1`cTe%UT%E0XB0ik9S %E0 7 7 c Y ! 79 4 3 5#8 ! & 5# 48 V # 38 7 1 3 3 S 498Y 49Q9#8 & S 4 4 4 7 489 485 V 0``4SW) q3 iDkc q4 BrbTTE ")B1c o8)DcDE%`74 )'(%1G!21EB5XG10%f&)y51cDc)%8I qY GT8%0DTSBR%%nIiDc `k9ET81T7XT5%TQ%Xg9 P E Y E G E 6C ygr CUg@BgrG a CE a Y I Y
3 5 #35 3 1CU12 c !Y V# 1 !Y8 4# 9 Qp! 3 5! c ! V#8 7 D@0X%0yE0TY%quCBEWD@X%0%w0 ! 35#e ! c S 4! 7 48 4#&! c 7 c !9 ! 5 4 c 7 7 5 ) 4 18 3 "C)B%BgDX`1XTVEdB`5E1T%I%Di1) q3 gBCPErsk!DX1dk!By5ED5 w x 8 79! 5 7 BEC 0 Dc x 5 c ! 3 5 # e ! S #4 7 a Y c S # 3 8 9 ! # 5S 9 8 ! s y1' W1ir1B%B``1)0Dc D60%1E)7 (B`$sgS ) F 4& 5!! 485 V `YIDEE9 v1%X`S 8 I c Y 4 7 3 8 A DWET8110uDB Sq5 w "BR9 &Q w mx # 9 4 7 aY c !# S 8 7 5 ! S 4 4# Y8 4#9 B) k!1)0Dn%T5B9 s! s%`!`4@DkcB`eE@E0TY%Q p 11 q3 i1`2`QEBWPTB`5BXB)BIb1g"C)B%B@ 0"DkcB`eEg%I`YEBrQp "D%vuX@!21EXU"DP`%BBE0bET8Dc ) 7 c 3 4 485 V 4 e S5 V5 '5& c ! 35#e ! ! 4 4# !Y8 4#9 ! c !8 4 V 485 V ! c 4#5eY89& 4 S 3 e 7 4#8 4# 5 eY89 7 7 4 cY 4 '&# 9 7 5 8 S V 5& ! # 5& 7 3 # 3 5 4 3#8! 7 5& 4 9Q9#8 & # 4 9 '& c 7 5& S ! 7 9 TY11k9%EG`EBE& 113 0`DX`V1(%BW17 I`1XBI6)%B0W1G)Br k!1%Es)B0nuE0)TSBRR(E0n1c %ET81)(P)B0E! T6)B$( c S 3e 7 4# 1@TY)B1B`9%E8 T%B0%0& s)BEg)10 4# 5 eY89 p! 3 5! 7 3 8 #8 7 ! 4# 5 eY89& c & 7 S 9 4 3 2Y 7 3 4 7# 3 1 4 3 2 # 4 9 '& # 4 V# 3 c S 9 4& 4# 89 S 3!Y E%0"T%B0%0@DgI)TBR`Q1g11nT81E)By5H1$%ET81)(n0`X0%1p1rB)T0)`7E1n(o8)s"E# 3 c & 7 3 2Y! 7 5& 7 8! 4& c ! 7 48& 3 cY# ! S ! 79 c 8 79! 4 9# 9 S 3 4!Y# ! # 5& S # 3! ! 9 VY '&! ! 498# Di0)01"s)Ip)13 s0 6 D)T5%f)D"E`U))@%%`!B%B)19 Dc `!1Dkcs"E@)%B0p`1EUBi01(%)`7(sI& c V 4 c# 5 A 5& 3 4# 5Y ' c ! 3 5# e ! c ! V#8 7 ! S 4 4# Y8 4#9 7 c V 4 c# 5 A 5& c 8 4# 5Y ' 1WXE`!D(BrB0iDc 5 `E01% p8 D@r1B%B@"DUX%0Ee0 s%`!`4WD`cTe%WEI`Y%rQp )1 q3 WXE`!D(BrB0iDIWT%B1Er8p c ! 3 5# ! S 4#&Y c c S 4!! 5 8! ) 8#& 4 8! 7 489 S 9 9! 3 2 8 S 3 7 3 48 4#Y8& 9# 9& S 9 e 8! 3 2!# 1Ur1B%Be `iE`#E0DDbT)TAEECPu" `S4 S%I%I%`!1c u"dT5%)`3Q %p1P0bT`S1X1`2%uk!%0EIB(bTB0 qY c s 1%%5 Btx !# S 8 7 5 ! S 4 4# Y8 4#9Q 7 3 !Y '& 9889 ! V 4 c# 5 A 5& c ! 3 5# e ! %T5B9 s! s%`!`4@DkcB`eEgE0TY%qp 11WF )(B%EnXE`!1%CIDnr1En C Dc 8 A 5 7 3 8 7 48 A 5 7 5) 9 7! c 5 Q 5 4 #9 8 7 V S9 eY 4 7 aY c Iu1BW16T%DBPB"S@8)0G$)60X0`0D1)0D) 3 4# AY S V 4 c# 5 A 5 c #8& ! 3 1`2%10TeB9 XE`!D(BrB0& D%EIrsW9 # 38& 7 1EI)By51 `S )k2T4)1GWsq DkcB`eEG%I`YEBRqp k9D"S`1(u1k2TQ%XGT)BEBXB1 q' c riGTgf1T31@0i21EX`S s 8 7 9 3 S 9 7 Vr V ( 4 4# Y8 4# 9Q S c) S A 98! 3 4 48 5 V 4 58 5 V 5 98 S 8 p8 S 3 5) 48 5 V 8! h S5 l Rux 879!5 %r 70 s!%`!`4@DkcB`e4EgE0TY%qp 11 q3 @T%B1Er8p Dn"C1Ee `%T5Bo8))IDc $T S 4 # Y8 4#9Q 7 c 4#5Y ' c ! 3 5# ! S !# S9 4 7 aY 7 e Dc 8 5 7 3 8 7 8 A 5 7 Iu1ABw1g0`4EDfB5 7 489& 48 S9 7 c # 38& 7 1 S 8 79 3 S 9 7 !! !9 VY '& ! T5%0CTaEg9)B q9 g1EI)By55`Ck2`411e)EEk!`4S Bi0)(E! p Dc 3 4# Y S9 4# Y ' 1k2E)A0B`e@`E501% p8 `9@Es!"E1E# 1)) S 8 7 Y# 4 3 2 ! 9 VY '& ! c S A 98! 3 4 58 5 V 5 ' 4# 5 eY89& S c 8 A 5 Bi1(E! p "DUT)%sU1`2)BEBXB1gT%B0%0ik9UD0uDB1) #8& %EIrs! p 18T1%Es)B0&P@f1cDBGR)30r0b7Xs!kD(BrABI$Dc I `%B0)Er8p D"C)B%B"DiEEk!T4D`cTe%# E0TY%qp 1)UF 0)(B%E9 7 3 4 3#8! 7 5 9 9 7! 5 Q 5 4 8 h V 4 c# 5 5& 8 4# 5Y ' c ! 3 5# e ! c ! S 4 4 Y8 4#9Q 7 3 # '& 988 # 3 5 4# eY89& 8 & # 48 3 8 3 7 e 4 4 !Y8 4#9Q ! c S A 98! 3 4 585 V 5 ' 4# 5 eY89& 4 1Br`E5B0%W%s8I0gsk!4`TS%Dg10rd$!1`cTe%#"E0YT%qp "DP`1B%sP1k21B%XB1U`%BBE0XET8Dc ) S 3e 7 4# TY)B1B`9%E8 `4EB0%& )13 0`4DX`V1'((9 B17iI`1XB50X)E#I& 1i1CXk!1%Es)B0& udE`#BXWV 0& 1H%ET81C(b0`WE)H`X)B$( # 5 eY89 7 7 cY 4 &# 7 5 8 S V & ! 5 7 3 # 3 5 4 3#8! 7 5 ) 4 5 VY c # 4 9 '& # 4 V# S ! 7 9 Vc W W3Q b `gT5gG BX TVETUrTSW CR3WBI@ HFD Q a I GE Y W IE E PE EQ P I GE
! 7 3 ! 3 S 3 2 4#&Y c 7 c 7 48& 3 c 5#8 4 88 ! 79 c 49# 99!! q v !! 3# A V5 8 75! 4#5Y ' 0"11n)k20)@%T#%I"Df0 q HT5%f)1%E`7UEEII& C1(0CE% )DE)0%1PB17 BEUT%B1E8 `TSB11iEEIIb1U%"T`S0`EEryiCB`5E`&`411B 0`D"W`V1((9 B)I`1XBIUCEI& 1U1Eb1`2%29B)iDkcB`eEX`E%IXBBe SQ 3 5 7 88& c ! S 48 785 x ! 7 489 S 9 ! 7 4 cY 4 '&# 75 7 8 S V5& ! # 5 7 3 # 3! 3 48 S9 79 4 4# 4#8Y V5Y k9W1W%DB& B9 B)Eft &If@8)0G6 5B)W0r1B`500DiXXBI& 3!21EX@"DW`%B0)E`9@))0%IY I0)`SR%%WIU)B$( S c # c 9 3 S# 5 A 8 9 7! c 5 Q 5 4 4 # 9 Y S QY c V V 5 ) 485 V ! c 4# 5Y '8 S ! c 78 q 49Q9#8 & ! 7 9
'% # &$"!
RigDM(k)
K
k BGLan
M(X) Rig RigSH(k)
X
S = Spec(A) = Spec(k)
RigSH(k) k BGL
k
d=0
RigSH(k)
SH(Gmk ) ~
X
d = dim(X)
CHd (X) = homDM(K) (Z(d)[2d], M(X)) RigDM(k) k M(X) ~ k k MGL k K K X k
i,j (X) = homRigSH(k) (M(X), MGLan [i](j)).
K i (X) = homRigSH(k) (M(X), BGLan [i])
Hp (X, q) = homRigDM(k) (M(X), [p](q))
CHd (X) = H2d (X, (d)).
X
k
k
X
SH(k) Rig : SH(k) k
/ homDM(k) (Z(d)[2d], M(X)) K CHd (X) k X K X RigSH(k) / RigSH(k) X X k
K
k
= Spec(K) MGLan P1 X k
Y! V 5 s5CXI&X9 ) # 3 3 4 48 V # 38& 3#8 S c 7 48 4#8! 1E!1`2`QEB5Wg%1EI)E%u! p k9g1HT5%fk&E%u"E# `9SpF & 7 S9 4 3 2 4 Q # c 7 5 ## 5& 9# Q c # 3 & 3#8! 88 7 7 4 9 7 798 4# 5 eY89 0)0``Q1 qY Sb`9U%D)BrE!%EI$T84(0D%1E8I)%Esqp EEI 11T7 C1(@T%B0E0& 7 8! 1)3pu" #3 S 87 1U &eTp5Dc # 3 3 4 85 V 38& 3#8 7 3 # 8 4 u 45 c )!# 3 49 #9 1E!d1k2TQ4%XfE#)EfCE%u! p 110%uk!Dp8TBDgTS4 (1T`STSBpG tl BCux ! 3 # AY S ! 3 5 ep 5# "1`24%1B`e9r1B%#Bq%1UDc 3 4 5 79 75 7 V 4 # 5 V 5 7 5 4 c 5 7 4 9 7 798 S9 V 1 S # )k2171&wB1UXE`!1')EWsk!4f1 q3 c E!rsk!Dw7)) T7 C1(@WVEk!T4BX%#By5&T$9GC Y 5 7 9# # A a 38& 1 & E8B1X%s!PE1Ci#1EfC7By550PDc ! V 4 #5 V 8 39 c 4# AY S 3 5# p5# "WEk!1'1%BXB5%Di!"DU13`2%1B`e9@C1EerBE)U C 7 7 4 9 7 7 8 4# 5 Y89& S # A 38& 7 7 7 8 4# ! # 110`7 ))9%g`EBeEi`9U1cE1Ca #10%f)5y1 13 0i`70%E8s%Us! )!# 5 3%TS 8WB1T)biT0&E%BEs0 Di7B`EEk4`%kS 7 V e 7 5 S 3 9 Y 4 5 ! ! 9 4 8 8 c 5 4 8 9 ! S 9 Y # 9 I 8 c Y 7 3 7 48& S 8 7 4 3! 9 A S 3 1g%o841)iB`5Ef"s! `9B9TQ1EX7 &B)qop`S11$TS &"BR9 Q ll "ux
1 fv,~ : DM(K, Q) ^ K ^ RigDM(K, Q)
Rig
Y! 5 V 5& S ! 7 5#Y 4! 7 sCXIX`9)EDkcs)5 C Dc `EB0%qs)BEU11i0 4# 5 eY8 9&p! 3 5! 7 3 8 #8 7 ! 4#5 eY89 %E0"T%B0E0& c & 7 S9 4 3 2Y 7 c ! 5 4 c 7 5 ) # 4 9 '& c 7 5&! # DpIC0`TQ)")1 q3 psCEk!DBgS%ET81C()B0E`S B9 6 8 7 Ve 7 S 7 3 u 7 5 8 S 3 7 3 48 4#Y8& 9#9 0XB1T511pDba W0p`TS)i)k2Esk!EEf%B0& c 8! S 3 4!Y# ! # 5& S 3 2 9#! 5 3! 7 )# S 4! # 3 1Us 011`cE%g)EIi`W)%sC)1sdB5 q0Coa`41XTVEU1B5 T5%)`3`9U1C DiE0TY1XIiTS %E)e$e 7 489 S9Q S # 35 c Y8 S V 5& 9#85 7 7 8! ! V 4 ' # 5 V 58 3 c 3 4# AY S 9 3 5# p 5# S 8 7 5 u" "Xsk!11%BXB%D q9 @1`2%1B`e@C1EreBE)g`iBDc # 3! # A a # 38& 7 7 3 !# S 4 3 c 8 7 V e 7 S 7 7 4 7 9 7 798 4# 5 eY89& 7 3 8! )si%1)g10%f)y51 1(B`59 T81DCT74 WB1T5)W B11TS)1(`EBEb11Us # 3 & S 8 75 c S 7 7 489 )U iTUB1gk!0)1B`5E%# !8 7 4& 5& F ! 3 4#8Y V 5Y e # 5& 7 3 x ! S 48& 3#8! 7 5& 485 ! (0`Ib I$%"1`2%E0XB0%BIC0y5`1TA%f)%Es)I6!v1%XV "Dc S 3 e 7 4# 4# 5eY89& # 3!) 3 4 485 V 4 TY11k9%E8 `EB0%k9S 1E"d1`2`QEBWET8Dc ) # 38& 3#8 S%1EI)E%u! p 11n0%ukDnoB%01`30XskI"E1nCTD 7 3 # 8 !4 u 84 9 # Q c S4 ) 8 7 V Y !4 & Y # ! 3S 8 7 V e 7 S 8 3 5 F 4& 5!! 8 7 V 3 4 7 5 79& 8! 0X01B`51 DE8 $`YIDs%B9 0X0)k211ns 3 4 48 5 4 S 9 h 3 5# e 7 3 2 Y# 38& 7 5 8! s S 3 7 3 48 4#Y8& 9# 9& 8! 3 2 4 9#! 5 3! )# S 4! # 3 1k2`QEBXV `!T5B@$Dc C)B%BX1 q3 n%1%fuBI& s)S B`TS)1k2Es`!%0%fB(Dc s 11(0Er11Eiufq0)Ta`41XTVE1B5 4# 5Y '8 S Q 3 5 7 88& c & 7 48# S c # 38& S S #& 4 9Q 7 5 4! 49 8 79#Y ! )! 3 4 485 V 4 S9 T%B1Eg`TSB11UEEI0DUI)1T7%sCWk9UDi1Ef"TU`gE0)T7BI& E)`7i(0rs"D"1`2`QEXnk!B`5@h c ! 3 5# e u 3 9 ! 3 4!! S& ! 7 48 5 7 !& # c 78 c S 9# 38&7 5 7 5 1 7 8 4& 4# 7 x 8 V# ! 4 4# 48 5 V ! DU"C)B%BebU)k2EE`90U)B`5EB1U0bE1)0%I qY @"`(1EIuI& B0 9)1 q3 c IbrT0)`7E1dy0W%0r@1`cTe%!21EBWDc `E01%ik9gWXI& Ik&gB%1`2`41D )v 1c 4# 5Y '8 S 8 7 V V 5 4 4 49# 3 S u q T%s1T7 q4 c r1EP`fy1P1c 48# 3 5# e S 39 95 7 7 48& # 3! 7 4 7 T5%fu 1EW)1 q3 c )13 V 8! s c 7 4 4! 5 V 5&Y c c 3 5# e S 8! q !# 7 5 4 9 e 37 5 4 7 aY srS DB`5ET8sCXB0DgD$r1E`s $%1iBEk!)uB0& F 1)0Dc 3 5# ep! 3 5! 7 F # 4& 5!! 8 3 r1B%BquCBEp13 0`IDs%B9 DC 7 e c 4 S9 h c 3 5# e $H Dnk!B`5@DgC1E@TS 85 7 7 e S 3 4!Y ! # 5& 8#& 4 c 7 489 S9Q 7 4 7 3 ! #5& 7 %1$H 011`cE%# CEIUDc %I%I%`!1B`5E)`3@11 q3 c 11PV )%B0H)3 T8B 45 6G VX CU R6 c Y Ty)srE CUCdQ c c TCC VEI a c Hy E GEW E Y E Y GEW YGE E0Df0g`BB`5BX1 q' "C)B%Be "TS 1EnB uCio811)T7@XE`!1)EWXB1 q' WF Y# e c 7 4 e S 5 V 5 c ! 3 5# ! # 3! lx # 9 4 3 c 4 V 4 ' # 5 V 5 V 5 S 48 7 q )!# S )TaED`c4 ! Brx 3%T51 8 I !8 7 4 QY ! 7 4! S& 4 ! 8 7 489& 48 S9 7 c # 38& 7 1 S #9 4 3 c ED`cn)B`5sCT30)T7n"TS 0pB`5E0CTaEg9)B q9 i10%f)y5`n)iT81DCT74 l ux ! u S V 5& S# 389 7 V 4 ' # 5 7 c ! 5 4 7 e F 4& 5!! DT1XIDc E1%)@XE`!1)EWV 1 q3 UErsk!Dc $H @TY0Ds%B9 4 4# 3 48 S9 79 Y8 4#9Q DkcB`eE$1`2%29B)@E0TY%P`9S 8 I ) 89 c S 3 5 A S c c# 5 A E6 Dg`1CP`9gD%CgTS &Q "BR9 x u S V5 T)WI& `S u" $rI bDg1`2%29B)9 IDkcB`e%$1B`5)B9 rqS 0X%By5!DCH$T"I`1XBIs 1i)ET7Bin)BEr11) 8! q x c 3 48 S 9 7 p 4 4# 3 e S 9 7 # V# 1 8 3 7 e 8 S V 5& 8! 3 2 8 7 9 78 4 9 V ! 7 5! 5 3 #35 )Br !& 1# 3!# 9 ' ! S # 3 8 7 S 3 7 7 q 4 3 ! 79 0B9f1s(0r@n`)s! 0`119 0! )2 )1c ! 4 7 !& 79 c 7 5 ! Dc `1)a 0)1CBrs"E# pEB0g"Dr1B%#BquCBEW`S 1TeE1c # 5& ! c 3 5 ep! 3 5! 7 4!Y 32 !7 S 9 1g)B`50r1Bi &Q "BR9 x ! 4 7 a !& 79 c 7 5 !## 5 "T))"I)B1)CE%EI& u S V 5 3 4 e S V5 c 35#e V 4 S V V5& Y S9Y# #8r 8 3 1cXDT1XI& DciTB`5B5XB) q' ir1B%BiXTqp `iXXBIXE`!`4"Ei%E0D0r )1U"T0D& p "Dc y)' DiC1Ei ! 79 c ! S& 9 ! x5 c 3 5# e 7 9 4 3! 8! q ! !! c 4& V 4 ' # 5 V 5 V 5 S c 3 4#8Y V 5Y p 4 4# 7 48 4#&! c 7 3 ! 7 48 A q 4 9Q9#8 & ! 7 9 B%8BTQ1Eik9Sns $C)3s%1qpTI@Xsk!11%BXBX1' rqUDg1`2%E0W0re0D`cTe%HB`5ED`E0E"D11nC0`ED5 IR0)TSBR%%$0i)B$( Brx V ' # 5 V V 5 S c 7# V 5 & 5 ! # s 4 5 x 4 V 5# c 5 7 5 c S e 7 4#8 S 8 79 48 3 7 4#&Y c F 4& 8! Xsk!411%BXB5X1' rqnE#1)0%1XB0&UF )7BDci%8u"E)S BfybHE !T#ByQU`XB%DD1BXV Di`B1B`9%En`%`!`4`SEDX0 %`#E0D@i`TS0B91 s x V 4 ' # 5 V 5 V5 S 8 XE`!1)EWXB1 q' i0 T`S0%1E)pT5%i%By58)(E8 S# 389 7 7 489 V# 1! 79# c ! 7 4!# V 4 ! 8 ! 7585 c 9 7! c 5Q 5 c 485 DX)T5E(0XXTVW`S I )B%iG D@f1DBG4 D@v1%XV TS &Q "BR9 7 c !# S9 ! 5 4 7 e ! 79 11 q3 P(B`5fErsk!Dc $d ))c
S
8 0
G(K, )
n = Spec(K[t0 , . . . , tn ]/(1 - Gmd Y i+j =d-1
B1
M(X)
K
X
v
CHd (X) = homDM(K) (Z(d)[2d], M(X))
D(v)
homDM(k) (Z(d)[2d], i j M(X))
i j
Cor((Gm, 1)d , X)
d
D(v) G(K) / / G(k)
/
K
B1 n = Spm(K{t0 , . . . , tn }/(1 - rig
n i=0 ti ))
X
X
DMct (K, Q)
MM(K)
MM(K)
G(K, v, ) ~
/ RigDM(K, Q) ^
K
G(K, v, ) Q Q ~
K
X
Cor((B1 , 1)d rig , X an )
Cor((Gm, 1)d , X)
Gmk
v : k C ~ QUDM(k, Q)
v
QUDM(k, Q) DM(Gmk , Q)
i
H ~
:K C
j
/ Cor((B1 , 1)d , X an ) rig
MM(K)
G(K, ) Q Q.
Cor((Gm, 1)d Y, X) Gmi {1} Gmj Y Gmd Y
/ homDM(k) (Z(d)[2d], M(X))
G(K, )
^ ^ RigDMct (K, Q) RigMM(K) ^ RigDM(K, Q) QUDM(k, Q)
^ f : RigDM(K, Q) ~ t t t
QUDM(k, Q) DM(Gmk , Q) DMct (Gmk , Q)
/ homDM(k) (Z(d)[2d], i j M(X)) .
B1 Gman d
k
I(v)
DM(Gmk , Q)
1
d
n i=0 ti ))
/ DM(k, Q) v
G(K)
n (n )an rig
1
X an
/ DM(k, Q)
H
/ D(Q)
H
ml "ux
~
/ D(Q) t k
K v
fv,~
3 4 485 48# c c 3 5# S 8! q )k2TQ%XV `EE0)T7 q4 P1C1Ee `$s $
c 8 1U0
Dci`S(#13EI&)3%E8sbB`5Ef"u%1Uk9S `Q1s! 9 8 # ! 7 48& 5# 8 7 9 4 3 3 4# AY S9 3 5# p5# 1k2E)0B`ePC1EreBE)W`S
Dc Y%517b%swT5%fu 1EP%sI0& Dfy17 C T5%f"u1EP)1%EIo8B%01612 8 9#! 7 48& # 3! 88 c 39 95 7 48& # 3! 7 3 #8r 4 9# Q c 4 3 fl BCux ! 3 4# AY S 9 ! 3 5# ep 5# c V 4 ' # 5 7 3 4 3 c "1`2%1B`enr1B%Bq%1@DUXsk!11%BXV 1iT81D)`74 #3 )s! !#Q 4 e9 E(0B`7iBBiTV q4 c )0EI)By51 TB(10``STSB9 G # 38& 7 S)!# 3 4 #9 D1`2`QEBWHB`5ET8ErWI"DUDUC1EUTS TC) Sq9 c 3 4 485 V 7 4 4! 5 V 5&Y c c 3 5# e # S #8 5 7 c S# 389 7 !# S 9 8! s V 4 ' # 5 V 5 7 5 V 7 3 #8r 49# Q c 4 3 0EB1XDG0%1E)$%T5BUsGS WEk!11%BXB1Xe1X%E0T8(0D12
v Spec(k) ^ ~ D(v, ) = Grig (K, ) ~ v DM(k, Q) ^ ~ Grig (K, )
3 12
)
# 38& 7 1 1EfCBy5
l 3ux ! 3 4# AY S 9 ! 3 5# p 5# c V 4 ' # 5 V 7 3 4 3 c 4 )8& 9 u #8 # 38&7 5& 7 1k2E)0B`er1EreBE1U1@XE`!)1EX1$o811)T7DqfD@E%Ir E)Ef"uB0fB Sq5 Y8 5 7 8! ! V 4 ' # 5 V 58 3 c 3 4# AY S 3 5# ep 5# S 8 7 5 c # A a # 38& 7 7 3 4 3 c EB1gs$XE`!)1EXBE1 q9 W)k2%10TeB9 C)B%Bq%1P`UBDXE))g10%f)y51 1T81DCT74
/ G(K, ) ~
/ QUDM(k, Q)
D(v, ) ~
/ G(k, ) ~ ^ ~ Grig (K, ) ^ ~ Grig (K, ) Rig Gmk
I(v, ) ~
f
l l
Cl f mf Cl mf Cl 3l tR3l t R3l
l1"l m t Cl mCl t l l l l
l
l1 m B f l f Sf m f
m t R m
tdm fSt f St
m t l l Bl
f
l l
m
! 3 4# AY S9 485 V ! V#8 7 7 48 4#&! @UU@U@U@UgUU@U@U@UU@UgU@UU@U1k2E1B`ei!21EBXUDc X%0E0 T5%DT%I%$( m Df Dt @UU@U@U@UgUU@U@U@UU@UgUg)T5%`&`411B9 01k2E1B`e9 "EI`Y%nDc "DkcB`eE# !21EBft t Df Dt ! 7 489 S 8 ! 3 4# AY S !Y8 4#9Q ! ! 4 4 485 ! 498 7 V SY !Y8 4# 5# 8 @UU@U@U@UgUU@U@U@UU@Ugb%T#B%0XTYnE0TY%1E1 0 7 48& 3#8! 7 Dc B`5EI)E%uCB5 `lDf Dt ! 4 4# !Y8 4#9 ! 485 ! @UU@U@U@UgUU@U@U@UU@UgU@UUW"D`cTe%n%I`YEBQ Dc !21EBXV Dc `EBEX9 4# 5eY89& !8 7 4#9Q 4 8 !8 7 e# Q 7 5&# 3! )!8# 1! 7 9#8 & Q9 u 3 9& 4 1Y# c ! S V @UU@U@U@UgU %k9%T7qp 0%B%0BI(1s13%E0y8)B%%i"BRpBI%k!9%1UD`1X0Du m ) Dt ! 7 489 S 9 8 4e S5 V5 '5& S c 4 585 V5 #9 & 7 4#9Q @UU@U@U@UgUU@U@U@UUj)T5%0k&`4110`BT5W1B0X`9UDU`1B%XB)' )gI)B`9%7 s t ) Dt 89 c S 3 5 A S c 4! 48 3 9 7 48 4#8!# S c 38 @UU@U@U@UgUU@U@U@UU@U %6&DUT1CXk9@DET8Eo8%IrHBT5%fk&E%u"EWk9UDc D)%G `l) Dt 8 7 4#9Q 4 8 8 7 e# Q 7 5&# 3! )!8# 1! 79#8 & Q9 3 9& 4 1Y# 7 ' 4 Q 7 4 4 e S 5 V 5 ' @UU@U@U@ B`9%T7qp I B%0BI(1sD3%E0y8)(ERHB0Ek!9E1 1 q3 c (`&1sk!igTB`5BXB)B5 !# 4 58 5 V 5 ' !& 7 9 c 7 5 !## 5& 7 48& 3#8! 7 @UU@U@U@UgUU@U@U@UU@UgU@ "E)g`1EX1WF "I)B1)BrE%EIDc B`5EI)E%uCB5 t 1m Dt !# 4 585 V 5 ' ! 4 7 !& 79 c 7 5 !##5& 8 48 S# c#9 4 c 35# @UU@U@U@UgUU@U "E)g`1EX1WF `1)a IC1)CE%%BI0621E`9E(0k&@Dr1B%gh `l1m Dt 48 S# c#9 4 c 35#e 8 4 585 V5 #9 & 7 4#9Q @UU@U@U@UgUU@U@U@UU@UgU@UU@21E`9E%B0k&UDUr1B%Bi0$`1B%XB)' )gI)B`9%7 s 8 7 4# 9Q 4 !8# 1! 7 9#8 & Q9 3 9& 4 1Y# 7 !# 4 @UU@U@U@UgUU@U@U@UU@UgU@Uk9EB`7qp %E0y8)B%%i"BRB0Ek!9E1f) q3 c "E)TA6A Dt Dt ! 498 7 V S !Y8 4# 5# !8 7 4#9Q @UU@U@U@UgUU@U@ %T#B%0XTYY "E0TY%1B%11!(B`9%`7r4p %E0y8)(ER"I%`!B9% m Dt Dt !8# 1! 79#8 & Q9 u 39& 4 1Y# @UU@U@U@UgUU@U 8 I ! 4 7 a !& 79 c 7 5 !## 5& c ! 4#5eY89& ! "T))"I)B1)CE%EID`%BBE0i" t Dt Dt 7 48&!#8 c !Y 4 S 4 S V 8 @UU@U@U@UgUU@U@U@UU@UgU@UU@U T5%fE(0E`7 q4 i%o80`&`41`ET813 i0 "I%`!B)A `lDt Dt u 3 9& 4 9 4 4# 4#8Y V 5Y 7 !8# 1! 7 9#8 & Q9 u 3 9& 4 1Y# 8 ! 4 7 !& 79 c 7 5 !## @UU@U@U@UgUU@U@ DkcB`eEgT%E0XB0e 0n%E0y8)B%%i"BRpBI%k!9%1i0n`1)a IC1)CE%%B5 !8 7 e# Q 7 5&# 3! u 3 9& 4 1 8 u 39& 4 1Y# @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U %0EBB0%)spB0Ek!9!0p"I%`!B9% t B`lDt 4e S5 V5 '5& # 3 8 7 9 45# c S ! 49 V5 @UU@U@U@UgUU@U@U@UU@UgU@U `BT5XB1I1TS 0U1i UET8B%DX`9U1c 1`7iB$( `lB`lDt 48#9 V 4 u 3 x 4 4# 4#8Y V5Y e !8 7 V S V 5& 8 S 9 @UU@U@U@UgUU@U@U@UU@U "`EE)UX0`D01ciDkcB`eEgT%E0XB0UDc %WTY)WI0n`!C1Bi gG BX I a a Bc W PqQQ b c q a 3 4 485 V c 7 5 V S ! 7 9 c & 4 c 7 '8 5# c ! 7 489#Y ! # 3! 8 7 V S V 5& 4 c 7 @UU@U@U@ )k2TQ%X@D)WU`n)B1b7 `D)1EBEgh Dn)T5%(0rB5 "TS )si0X0`Y1XB0)Ik&D)C)i @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@Ug m )m WXE01%1HT5%(Es)BX0$( f ) Bl l V# 5Y '8 3 c 7 489#8! 7 5 VY !8 7 V S V5 ! 3 S 3 2 8 ! V 4 9 ! c 7 489& 4#YQ S c 4 @UU@U@U@UgUU@U@U@UU (0X0`Y1XB0& 1k201I"WT5Du iDHB`5E0CTaEBWk9UDfT76A ) Bl @UU@U@U@UgUU@U@U@UU@UgU@U 8 0 #35 Y S9& S c V 4u q 1C@%o8`4`5UDUXB`5Dy9 m ) Bl @UU@U@U@UgUU@U@U@UU@U 8 0 !# 389 4#Y ! c 7 48& 3#8! 7 5& (1E`QEDc 1HT5%f)%Es)B0X9 t ) Bl ! 4 4# ! 9 VY '&! ! @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U@U@ DkcB`eEX1(%qp " `l) Bl @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@ T1B%sg)k21EX1' 10%f)y5B1p S A 98! 3 4 585 V 5 # 38& 7 ! 7 48& 3 cY# 8 S9 4& 4# 89 S 3!Y# 3 c Y& 7 5 7 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@H)B`5EI)D%iI B)`I)`7E)n%T81E%H1I)1 1m Bl S A 98 ! 9& ! 4 98 7 V SY !Y8 4# 5# 8 7 48& 3#8! 7 @UU@U@U@UgUU@U@U@UU@UgU@w`1B%s! B0C"E`#(0X0`Y0%I`YE)B%1iIHB`5EI)E%uCB5 m 1m Bl ! 3 4# AY S !Y8 4#9Q ! 3 S 3 2 ! 4 4 485 @UU@U@U@UgUU@U@U@UU@UgU@UU@1k2E)0B`e9 "E0TY%n)k20)UDc "DkcB`eE# !21EBft t 1m Bl @UU@U@U@UgUU@U@U@UU@UgUpT1B%s)`7nB0C"E`#(0X0`Y0%I`YE)B%1iIHB`5EI)E%uCB5 `l1m Bl S A 98! 4 ! 9& ! 4 98 7 V SY !Y8 4# 5# 8 7 48& 3#8! 7 @UU@U@U@UgUU@U@U@UU@UgU@UU@@1`cTe%n%I`YEBRqp "Dn1k21B%XB1qp "T%B0E0& " ! 4 4# !Y8 4#9Q ! c ! 3 4 585 V5 ' ! 4#5 eY89 ! @UU@U@U@UgUU@U@U@UU@UgU@UT5%Coa%29B) q9 1EI)By5!0(k&01E`!n`BB`5BrBg Dt Bl 7 489& 48 S9 7 c # 38& 7 1 8 ' 4 Q 7 4 ! 4e S5 5 ! c #9 8 7 V# c 7 e 7 7 48& 3 cY# 7 7 5 A 8 7 V S 48 78 5 F ! c 5 7 @UU@U@U@UgUU@ B`5EfCD%U11Ci0XT`S0`E0%rW"Dq1i 9 p "Di)iWE1)01 m Dt Bl @UU@U@U@UgUU@U@U@UU@UgU@U 0%u(0guD7 y%# DU%I`YE1E)iIH(`&0)sk!in%% t Dt Bl 78!# hp x5 c Y8 4# 5# 8 ' 4 Q 7 4 !Y## 9 ' 4 Q 7 4 c 4e S 5 58 S c ! 3 4!9 A !Y8 4# 5# 8 7 4 4 7 aY @UU@U@U@UgUU@U@U@U (`&1sk!iUDU`BT5CEP`9U1"1`2EBCnE0TY%1E1 0fB`5ET81)I$( `lDt Bl @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@ DkcB`eEgT%E0XB0e 0H(`&0)sk!iD@`BB`5BrB%P9 4 4# 4#8Y V5Y 7 ' 4 Q 7 4 c 4e S5 58 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U@U@UH`1(u! opXsi"TDDft B`lBl ! S A98 4 V! ! S c 5 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U `%0iIn"s%`!`4nXE`!)1Eft B`lBl ! S98Y 8 !! S ! V 4 ' # 5 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U@ D`cTe%@T%E0XB0 %`7B5 f B`lBl 4 4# 4#8Y V5Y e 7 !8 4 @UU@U@U@UgUU@U@U@UU@Ug `!W%By51 "TDDXV `iIp))Ci1g1k2E10UE))P9 B`lBl S V# ! S c 5 ! S 8 c 39 7 99 c 3 4#Y 7Y e # A a @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U B`5E0CTaEg9)B q9 i10%f)y51 m B`lBl 7 489& 48 S9 7 c # 38& 7 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UP"DkcB`eEn"EI`Y%iI1rc1i 9 %1B`ei t B`lBl ! 4 4# !Y8 4#9Q 8 ! 5 7 !# A S @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UgUU@U@U@UU@UgU %o8`4(0)0gh `lB`lBl !Y S9#Y 7Y 4 4# 4#8Y V5Y e !8 7 V S V 5& 8 S 9 @UU@U@U@UgUU@U@U@UU@UgU@UU@U@U@UDkcB`eEgT%E0XB0UDc %WTY)WI0n`!C1Bi gG BX I a a Bc W PqQQ b c q a
5(R( #
B1
k
k
S
2
k
fh
RigSH(-) FSH(-) RigSH(-) FSH(-)
RigSH(-)
B1
B1
AfndCor(k)
RigCor(k)
B1
l
RigDM(k) RigDM(k)
l 1l l
f
l
f Dt
B1
Dt
m Dt
t Dt
`lDt Bl
CE
Bl
m Bl
t Bl
CE
`lBl
a W c I
a
I (
) & # ( ' %$
"!
m i e y l n r i y g i i y l i y r m y y ijll sviy gi hluzuhlf hekfvh ftxksgs tgeiyufxehq{lsihyyg {yl xy hl xyki fhevyfxehugt y sfeiyuses` y y i i m g yikhigyhlg vyy t xl 8m yihi vm f ul fj eih sm feiy yy fses y fhenro Dyu xeh{lu{yy s i{lyxnhnl kr gu gu k{iotnfgusyg nui{g is{y {nskmsgs xg ufhr ige ih s i lj lj l y m r i yl tg {lu{yv y gl l i g y xyfxexrv`y suhduknxn vvk8xitl z k u fly krxn{oug m hxgkvsllg ty l j n # n8 i y l e n t suh tg v l gy i n gy r vyshikhl8srtgkrhivh!sk!iyixw y y yl ikhigyidsgs sl l suh xtDuhi{gfitRisiRl ugi y y gl e e l yy y i wsR hs y ukjsny`8 }tgyz }{ykslUxyomikphikhitgfisifikhiifsvojxnss vixymy sgm xyhfseuiixpi8kry hitgg isy hx8sgh xgtg syse iuy r vt l l g i i r e r p m y g y l l r j y n v n e t`xtq7#8`W87tBWt8z`V x` j hui{gy k umugflsghe keiixyyi khidxyug x hgm vi {puhi {gxnsex s xl omgiikyi kphiivtgy ssr hfhin utgkrfuise i uhisys {g um kil sokl hid iu g}{ghlfxyiksne okjshikihrvn ivt g spvsgy hrfurxey uvm n s gsil kji#kpmlriytswr {nvidky x hivvy fuirtg k visen ir xyoi!ivgyy u i!nm gi yts m m r nj nn oj m i y l p n y r m i i f yi sugiuDfl seuhikvh!oiry r l p m e m y i m y e n g g i l j in ojhi xwir kp{n kyhitgfile sip xyiik{ny{y `iytgfv tgkmvupiyif{l#sukixlkisgisgsu{ysyhhldhkgixlDugghlsrikhitgumhr yy y w y m n i n m e i l j g ivzumi p #sn `isk{wvg sysu hrx {lut kir skrm {yshrxy vnp xyl g xyfxeg vfmit#p sgskhivixpn rgshy hgltgy tdhiy fe gosuiryn ughp sivi#ksl kwi xsyoli }{lu tki iwi u#m s ik hi s`uii vl xpjrg xm zk ivy vlg syp fn xe xyvn xy m {g ughln srr { q vly uig usr zkn ui kyhffsherxgi ihe x#v{syvfviwklpiu iiixyyn syusgxgi wt 8qim s wiy vy txyksrtgkrhiisl ojxn8qsomikpiuivui{yuu!hughlxyugui{wxgituojhiswikp{n g owx hyjo kivp um vyvhzo y momikpi#vinp yk {nii gwi hi g ky v iy km hitgshffseuslzon iy i m i n g n r m i l y y n i l m i y i m n uihxgfssr ixe omvidhipir kp !`un o hrxg urm is{hsg!uio isn kkxi8sn hlr s8{lg xyfurlsefi v{ge tfuue hutgm osgxnkk i {no {tiu8hs{lx ritg hvhg klthuh hxh im xugh uniit` iy khitgsh l j xkr fyi xey m g kyvomikphi } n whhygg yvlg ssy u hr`un i#vin g m y n uz su kijtlohihg{y`syuiv jkxyfseuisgfsuknunig }iiy h ugiyfl }tl xg xyfq{ xe tk{i i!visy o om{iiyki p hikkhni {wtgfvkyegr x!ii`unspri k !{yvx m i m nl m y l i l g ng w i g i {} m nk8gu ihg iy m fizvwihug{m x| x 8 ikunuyi8itgl hrg s{yupn s xhrkyrvl ojqvn hn {gffm v khivisphrgsv!igl u imn ysh xyvuiishg e y i v g {y i l i p e y l y y m y r r p r hoiy# sfkiu n kxl iwsg hi#xuwiDstgx~mh snh }vfi{l h zxfhikr u g s i xyk kjhiw xwsgv8ikhiv tg fise ir koxe xkjiiikghi ifviei ds g xwh ixyg i Dhu {lD Dstzhp iywhvxhg utsr {yqp ihh gg 8 l
% e Y 0 &fE@d !c b &54a@7` 0 G % Y 9 % Y % ( 9 % " 0 % " 9 ) 0 " 9 ) 9 " 0 % !A 5X5) 5!8A W% 4RV#U@1TS @BR" 7!R5Q( P I '&H8EDB@1) 5&87!0 543!1( '&$ #! " % G F CA ) 9 " % $ 6 " 2 2 0 ) " % "
A
|a + b| max(|a|, |b|)
|a.b| |a| |b|
|1| 1
A
(A, |.|)
A = {a A, |a| 1} A A |.|
|0| = 0
|a| = 0 a = 0
(a, b) A2
A
) 3 4 "(
21 0'
(A, |.|) B
|1| = 1 |A - {0}| = {1} = {a A, |a| < 1} A ~ = A /A A A
p: A |b|p = inf ap-1 (b) |a|
//B A
) ) 3 '4 "(
2P
"
|.|p
|A| = {0}
|.| : A
B
G HE
E F E F E 8 C B @ 8 6 %0%0%DA975
/ R0
mvV("g&e("g(H"w!x z y| x ze msRc`yss { bW@ T P b G P Y G bWC S TE C YuA W r q q q G SC G C C YX T S P G YhX $e"VARhi"g@jbRhpudfCiD%&vfCVGv`@wVtpT%$Qs(ef`f`&fC&v!hC !w&aS`WVU%RC&I@iDH C G S E qaubfu&`XfCG g@vaCe0`BF `%e0)`(%aC%`ftem@U %!$`pXD)%&aS`WVU%RC!`%e!%!$DRXP Y bXW WX h@CWW@ Wh YhX P Yon G PC bW PThk Pu YX T S P hh@CWW@ hhC bW bC Tfq C W G SXE !DVXaTbfu&`v0I@G pTaPea@%pC`iDDDaS`WVU&vpC$v&%VC%RCiRb%C Ph@ Y bXW WXFu YX T S P bWC S Gh P P Ch Y $akaPDX j}jlV) qB$fkaTpYvDRXIPgbd`fApTb%cC6phiev6Ex`fApTb%cC gcbC dCh bu S T C C E Y P Y@ C E To P F ewpCvRb%C vfhw!T PW@ PWC P FP G C C T YX S Wh G@u G Y P GhWW &pSfPpD&`X Hq e!IT RhwVC$e@ vP $gVY`f@vRb%4mG Chk T GE PC fq q C T G G TE P !0pCVbfSfT0I@dVCv!RXaP`@ Eh@ C T P CWh Th geq0pCVbRh $k Ch bu S T C T YX S W $fkaTpYvDRXIP&RSIPa`&h WC W b P b G S E u YXW h@CWW@ Wh YC v&HvpCvu %F vVCDXi&aS`9`%`ipgPvDA Ch u S T C T YX S bh bFbC C W q TE PC TW $IkpTaYb&`pXIPi&RSIPa`&l`pXD%v(vxsD`fCVGv`@yRb%spuVbTIE h q G SXE C u YXW Y G@ P %bfC&`vlbg&aS`saPDVX`lcbC h@CWW C YXW C Eh@ C P T P PC `%e @ q aS`i!0g`qw!l%pCVbphy! e!fT sD`"VAphP G@u G bW@ T C bW bXhk l$vfCpTeDsVCG b W u C S S Y G @ T W t u W r T b u P G Y h X q q W P X Y r T X b G S E P $eg@bd&DXE gP`fXeiVbv!sHqadIPVG`a@v!%fCq`(HB``~`f`w`fXpTVbRT$HDpXs!F Q & wVYD`AdvVC&DXicbC h@CWW e%e @ q !gfYpTpbe&dpCfYeH`fXpTea@RbwH`!Rb%C C Y @ T @ W W b P E W X P C YXW PC T P &pSDi!0aCjbph9@ q Chk bW TA $w$vfC mGHq bC bh b YhX !`aXiDH ChkG Y bW $)%pCb vfCpT$vC Wh T C T 0aCcbIP!hVG}F pT%v(vws4q P FbC C W FP G C vfh&s}q Ch Th T C T DakIPpCRbfP!h&aCjb&fSVTG aCRbbl q C bW PX W $ea@D(%C Y P C YX T S P CWh TWtu W r Rh&pSDWVU&vpC0dVbv!s q Chk P G $d%fCVG%pCb P T P PC EWX bW pCVbRh!D!gDDRXP C bWC S Gu P lgPD%&vfuvq%C C E h @ C P T P P C h @ C W W jbDX T f 4}}v s& q !9g`q!q%aCjbRh%iee@ G y| elajx(g | {x { { x { vv(pxH!d&I!|"9"vI!wxH!x z y | x z e6{vV("gw(!v&e` x zx 9"}9"g(H"p!" m Q4g W@ q q W PX Y r sdeee`~`f``fXpTVbpTD(`aXsF 9 wfY`DA$e"fAphd&&vfCfC0s(DRC%!``&fST 4 TX bW@ T P C S S G G @ W rq PWC Ct@ Ch bu S $fkaTpYv&`pXfPT C T YX S W TWh &pSfPpDh 44S bfC&`vs&HpS X WjU&vpCi`%`!F "{zV}ieg&fCVG%HeD!F G S X E u D T S P h @ C W W @ W h C y | G C @ W X Y u YX T S P h@CWW &aS`WjU&%RCQe%e@ Wh uWWX bW b 9`!D`a@p q pT$v&vfC%RC0vspTDpt%``vE !gPD%&vfuvu 4 P bWC S Gh P bC P C YCAWX C bWC S G fCpT%Ruxh $xaCVYD%VC%pCY Y P CW hk T bW TA I@vlyD$gHeqw!pC%pnVbpTvVCVtpT%A GC q Eh@ C Y YE G C Yu Ch u S CE@ P G Ch Y G z y | T P $IkpTaYb&v`p%C $akaPDVX&vV("gVb!icbC u YX T S P h@CWW &aS`WjU&%RCe%e @ qCu YuA Yh h b P `vHjtaTvgP`cX"DpXRb%C aujbfTe$vu 6pT9v{6v&evIRb%C $wH!vdjbws 6qH"mzDp"gxcbC $waPDVX`dVb!`}F G@t G P x z { P Chk WtWC T W r z z P Chk Y G@ T WX pCvaCd!v(DX $akaPDX YW Y bhC W G Ch Y qfWpT$`vcbspWDHd&RSIPa`&h $gVY`f@vcb4mGq !DpXDH h bWXE PC u YXo C T YX S W hk T GE PC bh b YhX Chk G DfCfGvpCb pC$eg@cbDXwaCcbIP!hC bW PW E CWh T GfT v("gsRb%C $lVb!9sDvaS`p%fCpey%aCjbVWVCD`fX`(`aX)%fCs`(gfWpT$`vycblaCD%CDR@VRU`pX9`fXpTevIEVTe@ 6pT9VWaTDDXE P P Chk T W rqPC YXW P G Y@ P Th T P Tt GX b P G YhX Ch bWXE PC bW E Ph P W b@ G G P h bW Rb%C $sjbQs(4!`%e@ 4pSIPa`&h P Chk T W rq hh@CWW C T YX S W bq%&aS`WjU&%RCe%e!v! C Pu YX T S P hh@CWW@ hhC b vC bW T $vfC`X qC YXW CWh P P bWC S Gh P bC P d EW T CWh P YXEWC h C Y@ P PC Tt GX b bC Y D&pSDiRb%C RTi$v&vfC%RCvRTqBC`a@RbfP!icbC aC`vvDX&vaue%RuiRb%dVCD`fX`(`aXdpCRbvl(q f RhP VCD`fX`(`pX)aPDVX`ljb!%!&v&s`q Tt GX b CWh Y G@ Thu WC W r Y@ TW u Y EW T UXh P CWh Th C YX T S P C `gVCt! RhP C`a@RbfP`!vpCgyjb!fWT aS`WVU&vpCiW u q PC SW GX C Yon G G bWC SCh TWXW@E C bWC &`f`(!waC%`fte@ 6i$v&v$Ik`evHjtaTD%!IT %P C GCh P Y h@CWW GFP G C !%!IRTpu`%`@ %fhsq fCaS`XI@gD%&vaCVY`f@v9sr Gh Y G bWC S T GE @ W 75 f((4"! x ` !x9"}}!yz {sH9vV!xvlzy{ z y x { Ph@ C C YX T S P y y| 9"}d(!ygzs!xqzc9aPed!w&pSDWVU&vpCifgVvDa| { x { !x&e(mvw}!szyD m"}zmvD"eaH9vz z z z z e6{e{va" y x x `p$ "R "v(!gz y v{6zVyv9"g{vv`ify9v ) x vgeVy {iaCD%`atvD$DXdD%&vpCbfEaTpYRbzesz bWC YCAW E bWC S P {
A C(A) limn |an+1 - an | = 0 ||an | - |am || |an - am | N |an+m - an | < nN N C(A) C(A) I(A) (A, |.|) (A, |.|) A ^ |A| = |A| C(f ) (A, |.|) /A ^ |.| (B, |.|) M R M =1 (A, |.|) C(f ) : C(A)
nN an
fyD{"" vfyy9"l{v()s%9"}m0vv"g(H"!x { x z" m yR cic4w { Pu YX T S P hh@CWW@ PC Y@E G PW@ PC SW GX C P Yon G@ PC G Y G bWCh Ph PC TE P Yon G@ P w%&aS`WjU&%RCe%eg!pC!DvfCeg&`f`(!%aCvDVt`g%!fC`awfCi$vDcX`pXpP!mjUvqpCv`ftegC q G@ Th@ S G@u G C `9VS!"94`%!IT ! Y%pCea@%vDicbC s4q h b YuWut Wh P ChkG DvpCb bWC S Gu W C z { {x Y G@ GC @ W r $v&vfuvh !0m"vI" H!daPDVX`VCfGv(es}q GC Y Wh YhX %vauiDH C Y G C q TF T P &aS`X&I@!}fF`C 6qH%baCYaEIP!wRb%C C h Y Ph P G Chk bC G T b WXW PC G h Y G { P W b@ G@ (`aX`tcUH`pXVC0$wvVC`f@fApTaY`wcbVCfGvC $akaPDVXmzvvvI"9cbC DVXaT"fhe"A I@$ijb!sy`fCeI@VAaTpY&Rb%C `fXpTefhe"f@gpTxD%&vfCvpCxbgpTfCfG&Rb%9vpCepCpbbllq %pC!`!DVXaT"fhe"A G Chk T W r q G T b P C W b@ G@A G P bWC S Gh P C P GhW PC Yh bh@ C Yh bh@ C W b@ G@ C h@CWW@ Wh P h@CWW G Fu G@A P Y E Wh P !eedcbC e%e@ 64Dfhe"HpDXcbC T D`fApT"%fEfTfpTVb&pSDcb"{"jx""QW q qC b@ G Gh S C YXW CWh PC z y C bh b YhX Cu YuA P !`aXDHivVtpT%cbC puVbVT`Dvu 9$pkgP`fXQB{"gf ~{}| 6{jz!0wpCVb!Rb%C `%ev0phP &aS`WjUu &%RCQW u G@t G Ch Y G z y yx T P h@CWW@ Wh Y C YX T S P C
|.| 0<<1 k k |v| = inf{sup{|b11 | |b21 |, . . . , |b1r | |b2r |}; x = b11 b21 + + b1r b2r }. |.| 1 (A, |.|) A v B 1 A B2 k (k, |.|) A |k | R >0 k |a.b| = |a|.|b| n k || = k T 1 , . . . , Tn A
8 7 2 1 # 9465%430)('"&%$#"!
B = A/ ker |.|
YhX bW PX W `(gD`p@$H%C h@CWW G Y P C YX T S ee@ gRh&aS`WjU&%RCP CWh TWtu W r h@CWW@ Wh YhX P Yon G PC bW P Thk Pu YX T S P hh@CWW@ hhC qVbv!isq `%eiqDHppCv`ftem@U %l$`pX($l%&pSDWjU&vpC9!eeiv! bC bW T f e $vfC`X g qd pT$v&vfC%RCP P bWC S Gh bspTP C C EW Yu G bW Y@ q !9C%pC%!`@ $%VCaTRb`e@ VFDC qT He&vRb%C PW@ P G@u G P bWC S Gh P C P C YX `%!IT spT$v&vfCvpCbyRTx&aS`W CWh P FbWC S G YuWut P G r qC YXW CWh Y G@ P C YX T S P G wRb%C D%&vfCeg@v%`dfhs(`&pSDigP`fXedcbC &aS`WVU%RCQI@HF ECA %DB@
ker |.| = {a A, |a| = 0} p-1 (0) -1 a p (0) (B1 , |.|) (B2 , |.|) B1 A B2 A |.|p 0 |.|p |a + p-1 (0)| R
sup1 ,...,n |a1 ,...,n | |.|
^ f
|f (a)| = |a|
A{T1 , . . . , Tn }
(a, b) A2
|a1 ,...,n | 0 1 + + n A{T1 , . . . , Tn } A A[[T1 , . . . , Tn ]] A{T1 , . . . , Tn } A
(A, |.|)
|f (a)| M.|a|
I(A)
(A, |.|)
f=
A
d
/B
1 ,...,n N
d
(A{T1 , . . . , Tn }) = A {T1 , . . . , Tn } ~ A[T1 , . . . , Tn ]
A A
f
C(A) (an )nN C(A) |(an )nN | = limm |am | |(an )nN | |A| |(an )nN | = > 0 m0 |an+m | = |an | (|an |)n
A
a1 ,...,n T1 1 . . . Tn n
f: A
f: A
^ (A, |.|) A
aA
^ ^ f: A
d(a, b) = |a - b| |.| /B
0 /B
^ (A, |.|)
A
(A, d)
(an )nN
/B
limn |an | = 0 ^ (A, |.|)
^ (A, |.|)
A/ ker |.| ^ B 1 A B2 f
f (A)
(B1 , |.|) ^ A f
B (f (an ))nN
f
(B2 , |.|) B1 A B2 |f | = f Z B
z z x z H"(vm"ggz "`0"DHmea { x { ` }"9"Dw!mzI`d)y{ g"" Db}vvipv%$ eR Dw9I|"9 !"mg!"D{VCVopTpPIP&!`sz y { zx z x| G T S@ x x { D zv"!"i!x svzew z x x | z zx `p"D(ezsH9vp%!egaa!"(Ivapv!v| fyy0eiVp e`vm"g { y x z y}e6{ep0("9"Dl!"m!"({ DfCVopTaPfP&!Dq$v&vaC`v%aC!lgz x z { zx z x| G T S@ bWC S YAhXE Y x z"q "DHmeg { w x z x { D0zvB!e 9Vyy"l(!x z x z { `gzvB!e!x v" { z x y y vg"ipe`(ezyv&vp!"ggpfVv D| a|gz
U Spm(f )(Spm(B)) U Spm(A) k B1 A B2 k{T1 , . . . , Tn } A |.|p A Spm(A) Spm(A) A k A A f1 , f2 , . . . , fn A R = (Ui U )iI U D(fi |f1 , . . . , fi , . . . , fn ) Spm(A) k D Spm(A) sup A f 1 , f2 , . . . , fn A Spm(B) k(x) k / Bi k k Bi k k
i
Vyy" )V"`{ { y z z y &"g H"pe(vDs6v9Vvw` z y z { vDb(IvaD "R (!x v" h b W @ T P W T W t u G T W r q G C W W b Y P T @ S X C h k W C ki T4Ww 9aC$e"VARh`fXpTVbv!&f@iVbe@svIPv`fXpTea@C VW`9`!!mCa $DXyvRGIE $ch c`fXW %puY CWh PC P F YCAhX Wh P &cbVGT v%pbvD`&cbC fC%RC%CRUDRXW u q Go S PW Ph P Chk G $9fP%pCb fP%`fXpTea@VW`9`! GCWW b Y PC T@ SX C TWt Yo SX Wh ThC P { Wh P !qpC`W 9pCRbfP!vyfGT %Q("9"D Rb%C VCvpC%CRU`pXW q Go S PW Ph P C GCWW b Y C T@ SX Wh P !sv`fXpTea@9fWe9D!Rb%C CGCWW b Y C T@ SX Wh q F T PW b C Y Y G WC YuA C P TWt PW E Y b T Y@ P Go !DvDVXaT"g@pfWe9`!%VFDC qTDpuVb$fApTVbpTeg@pY!saubfupT`aXf@4$ub vHjtaTvD b4VC(`fXpTb%pCaP%pC$fWe%fCeg@cbP bW P GCWW b Y PC T@ SX P q q @ S S TX $`pXfP%`fXpTea@qVW`9`!%C qdBs`f&&%C F hy dVYD`A C XXW W@ Yon G c9bdpC%`fte@ U I@&Rb%C G P DX ECA@ Ch TWXW@E W E To WC P `B$fk``v`fXpTbcC vv&Rb%C fC0vpCv C Go S PW mv`fXpT"g@dfWe9`!!0VvDa|qz0fy9vvms GCWW b Y C T@ SX x y| y z zx z x !e!"| zj{"$p}0V&"Hs6` y { y |{ `xVv9vvR"vVy y x e(xV"`{ z w y z z y &"g ep"}$$v6v&vVvD z y z { viDb(Iv9aD "R (!x He aum sRcgcH4 qPC XXW W@ P Yon G PC YXt b@E G PW@ %%c9b%aCvDVt`@U %!dVCaT`Dvau"v&I@e C S S Yt T h T GXE G P &`a@DeI@!!pCVb&VSfT`v&f@iRb%C P G C Thk C XXW W@ Yon G CWh P P G Y P E@W@ T C PC YXW PC TX E h P@ WCu C HVhs!4$!Y9pCv`ftem@U qcbC sq %fClRhxgDe'laS`y!xjhDg!q`Hx%H%!sW h@CWW `%e@ 6G C ! pubfu&`vifCFd G SXE G ECA `B@ PC XXW W@ P Yon G C PC T YX S hhC PuWWX bW b %!Y&%aCvDVt`@U !&RSIPa`&!v!0%`!0$`a@p q Ph@ C C YX %aPed!&aS`W I@w%`Bfv6WcXfC!fW&pSD&I@$%%aC$v`atvD`X$%%pCbfEaTpYRbg%fCpT%Ru%!aCvDVt`@U I@cbC $pkgP`X %%fC!fW%&aS`WjU&%RCP G ECA@ C TE X E TWt T C YXW GFP bWC YCAW E bWC S P P Y P PC Yon G G P Ch Y qP TWt T PC YX T S P G YhX bW E bWXE PC PC XXW W@ P Yon G C VCs`(gD`a@b`g@pY$`vscbg%c9bg%aC%`ftem@U ! C T YX S bhX i Y P { { { G P &RSIPa`&!D tq RhHs%09"}m0vvf@scbC s qDaS`WVU&vpCpCRbvCiDHdaCbfn&`XiRb%C %fhdDpCVb!%augcbC RTi$v&vfC%RCib RTd&pSDdgcb$k phw&aS`WjU&%RCP C YX T S P b E YhX G S E P FP G C q Th Y P P bWC S Gh P C P C YXW CWh PC Th Y P C YX T S CWh P icbC Fv"Vs 9&&%C F Q & pne@ qd !iVCHq`fXRTHvpCvg%!$DRXP VC$ vfERCIPVTfGpTQDX !fCVG%CE q q @ S S P Y C P TWt PW PW bhC PC bW P G ChkT T bh W C G W u T h k W b @ G @ A C h T W G C T W h S b W vaCb$`fXpTefhe"0DIkh 0!d IP0s$`pXP a`vwVC`dbC e9VS!e&!`%!Iw%!VCvpC%C G P YXE P G X C hh@ Th@ S hh@u T PC Go S PW ECA `B@
D(f0 |f1 , . . . , fn ) D(f0 |f1 , . . . , fn ) A{T1 , . . . , Tn }/(f0 T1 - f1 , . . . , f0 Tn - fn ) Spm(A) Di Spm(A) A B1 o k(x) k k A k D = i Di |f | = / B2 A A A u: A f 0 , f1 , . . . , fn A x Spm(A)
xSpm(A)
pCD(DF PX WX YhX r C XXW W@ Yon G CWh T `q !Y9waCvDVt`@U ijbDX f rq((4"!v$ 75 qdvp DV vaC`vGF 9 fY`DAg``qhq&pSD!Vh`gi!DpXg`(wpWDHX o q q S YXC i TX E@W@ T C PC YXW C TX E bh b YhX u Y b W C S C h b @ S b h @ P C P C X X W W @ P Y o n G b W C T P o @ C Y X S W h k C P b W G @ T h k u b W Y P E @ W @ T C P C Y X $v&%$fkaT"9`aX!ecb%c9b4%aC%`ftem@U aCpY$v Vbeg@pYRb`y&RSHIP a`&h $pbpC$vfCe"fA$% $DRXP phg`HeU!%&aS`W T PVCgpC!DpXd$aCpY$`&Q`!%$%%RuIPTpugfhs%`Q%H%!vC W epCVb!fWsfC`DVXDHDpXI@`fXpTea@$vpC%au&I@!Vh`g! G P bh b Chk bWX S WXFbWC S E Y P G rqP@ WCu W Y@ Th T Tt GX b GF W b bW P Y G C TX E h WCu Chk W T TqC YXW b E YhX E@W@ T C Y G@ P Yon G %H%! $i%VCqg`&pSDaCRbvC`(9g``'!laPDVX`qRb%C aC%`ftem@U @ d q q G G `` TX Y C Y X q%VDVDfCfX&eva@SbpT !$hbC vCW pCyaufYQAi%VCC fGu vD!fXhWCWG &%$aubPpT `a@&aS`W I@ VY9!%HDF !Y&iaCvDVt`@U h 4 G C TWh S bhC WX C XXW W@ Yon G CW W ` %Y C W h P W p b ``fXpTea@$vpC%au0(!0VvDa|gz99gI!|e&4vs W b bW P Y x y| { y z aD "R y z { x { C XXW W@ Yon G x y| `& 6{vgvfe!gx&&I|"9!mzIDRdsy{ ic9baCvDVt`@ (!x%Vv`a|iz aD "R } y P C YX m 9W ` U w%!a` lsRP@ pC%co gftceHm@G 4P VCG X W Y n Pvfuv(eppC%VnfTvfEaTRb`&g``qAp%aCvDVt`@U !y`p%`!d$v&vfCe%fXG$DRXi!I`ftaTgVCaTpYvDvD%pDVXaTbDXg!e%e@ C GC @ P Y Gh Y@ E@W@ T C P Yon G PC Y@ PuWWX bWC S G@E bW P C T Y bu SXut WC PW EW C hh@CWW P C TWt W PW bhC CWh P GCh P Y P Y E W P G@ Th@ SG@u T Wh P %C q !fC`fXpTvaC$vgyRb%C v!IpT%aupDX9`pX`Fe9fSe9e!f9scbC T %%&aSvCp$`pXe%fT qPu Y bW P hh@u PRCs`aXibvfCpT%pv$cbdpC%`fteaCRbvl}q(`fXpT"HVh`BsVbe@Qvscb $kpP`d0!&aS`&I@!f fC0S P P h b C C W W Y u b C X W P C Y o n G @ b C d W b @ G @ A C W h T W C P C T h P h @ C C Y X W G C T W h Cp%`ftem@U I@geHw`!iRb%g``qV!q%aC%`ftem@U !qadDgDHsea@$%`DXsfC&v!vW q Yon G G Y@ uWWX PC E@W@ T C P Yon G C PhXW YhX G bWC S@W G SChC jbDRX$Ikv$k T P Ch GCh Chk G C YXW CWh YhX bn S Yon G CWh PC E@W@ T C on G CW hk PW GC DfCVG%pCb &aS`q)DHqpCv&`XE aCvDVt`@U )cbg`HeU!lpCuY %`ftem@U h $DVX%He@ S qCh You G@ u SXu `$IkpTv`ftefCpTaYb&`%`t C PCh P GE PW bXW PC PW YWX P Y E PCu GCAu PW bXW P GF G T b P C W b@ G@A G Ch Y q G T b WX l$fkpTpPDf@vg`fXpT`%l`aX!DHp%aCaY`XvDVX%`%!`fXpTe%fC!`fCeI@$fAaTpYcbC !`fXpTefhe"f@s$pkgP`X `fCeI@$fAaTpY`W C W b@ G@A G P@ Y PX P CW WXF T bWXE h GhC W bWC S h !`fXpTeVh`"&f@Dg@vpCD(ph`$`aCfYeg@pY$`vpCVbTIEVT!vQDVXaT$v&)e@ f q b G S E u G@A P YXE Wh Cht W vVCDXfhe"a`visr
) ( @F !F & ) H C HF & 1 E C ( ) ( @ 8 6 4 3 1 ) ( & $ " ! 0ARQIP20DIG99DB0A97520'%#
k A{T1 , . . . , Tn } A A{S1 , . . . , Sm } k //A A{T1 , . . . , Tn , S1 , . . . Sm } (k, |.|) k //A p : k{T1 , . . . , Tn } k A |.|p A k fA k / A{T1 , . . . , Tn , S1 , . . . Sm } k k(x) A |.|A A |.|p A k
&I!|"9"vI!}z!"({ { { x {x x x y | y { { y | y | "g(H"R"vevD"{zV}iegiV0gBe9`v(ayyg~x(g mzV}ieg0peg(vz
zv`{&I|"9l { x 9"}"g(H"
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
U R (Ui D D)iI 1in k A[T1 , . . . , Tn ]
k (a, f ) k A k{T1 , . . . , Tn } x k{T1 , . . . , Tn }
^ A{T1 , . . . , Tn }A A{S1 , . . . , Sm }
A{T1 , . . . , Tn }
D Spm(A)
A
|f (x)|
/A
Spm(A f0 |f1 , . . . , fn )
k
U Spm(A)
Spm(A) D = Spm(B) A i {1, 2} k
k U
|f0 (x)| |fi (x)|
A f0 |f1 , . . . , fn
Spm(A f0 |f1 , . . . , fn )
/B f: A Spm(f )(Spm(B)) U ^ B 1 A B2 |.| U A A k
|a.f |A = |a||f |A k k
! ('9 h)$H3(
q bC F PC T Y P b Y@ P %fDVtaTq%puvVuaTe"AU VCG fYv!DHg$vfCVGDXpCpCg%aCD%!%v%auDVXaTbahpYRb`vi%fCF TWtu YhX bW G E Y P P bWCuE Y PW E PWXE P G bwpCfYeg@pYVbaoe!I`ftpTpuvVuaTe"AU iRb%C T C T T Y@ C T Y P b Y@ CWh P q P G h X o C T S G b W C Y X o T h k P Y C E C T S G C Sf S VCDHd!fCfGVG&e@0I@gD%!g`($g%fCGgEvvdVCfGVG&`@I@&DXE g@vpCv(vsDDpXDXpCpbbCE Y PWC W r qCWW Yh E bWC S G P bXW WX D%&vfC&VSpTQaCev`F b vC Ch Y $pkgP`X q CWW Yh E G TWtu WX DpXDX&I@Vbv!D!F bh b YhX !`aXl`( C hk GC G hW WX W Y Tb WC C S T h hC $gfP%pb $Q`q%VCa$vd&vfnv! Wh wbC fCVopTgPv`fW`fXpTb`X&vfnv!%`d$eg@p(q G YCA T W EW C S ThhC CWh CuWWX bW b ECA@ bWC S G TA C E XXE GCFP@E E PW@ %DBw$v&vfCfoRTIP!ITWYv0fCVG%%`%0Cw`s q Chk Y PX P bhC WXFC XXW W@ Yon G CWh W To PC d q Chk Y bWX S YhX r $s%RC$HRh!v(DHD!c9baCvDVt`@U wvfCsRb%%f`R$s%pCaY$`&s`Hq Ch bW VWaTDDXE W EW G Y@ PuWWX PWX Y C b Y@ C u S Y@ P GhXo C T S CWh C S S `fXpTb`XwI@9e&D!0H``Bg@!ivC `%puaYb&eg@eH&%fC`(!fCVGfG&e@&&`XE Y PWC W a@%RC%Hsr dq BVDc C XXW W@ Yon G G C G@ Th@ S EC P G Y@ !Y9saCvDVt`@U I@!e9fSe9waCpYb%(RfCgeH C XXW W@ Y@ !Y9baubfupT`BAU CWh TW u W r djbHt!sq b0VCfopTaP%`VWT C G YCA FC XXW W@ b Y@ CW D!Y9bpuvVuaTe"AU h bW T $vfC`X f ((6)$ 2 qPC T Y P Y@ PC v!I`ftpTq%aubfupT`"AU %! CVaT`Dvau"vf@ye9`%`iyRb%C YXt b@E G PW@ h@CWW@ Wh P FY Gh Y@ W %%VCfTvfEaTRb`xs$q P G@o Gt PW EW PC h@CWW G TE P C T Y b Y@ %fCeH`fX`sH`fXpTbH`Xp!xe`@ gVCv!RXaP`@ fDVtaTipuvVuaTe"A U &$s%pCb`XfCdpC$vpC%au%pCY CWh Thk Yh EW G bW P Y Chk TXA C E PC GhW WXW T G@ P W $fY`D!fCfGT`@cbmGH6qDdpCfYeI@e%pEh ECA %DB@ W TW G C S S E TW u P `fXh &DX&VCt!dcbC Ch b G@W@ CW W@ T Y q G T b WXW P C W b@ G@A G Chk PW PX f e $IkpT$V`e9baCjbDpX!@ `fCeI@$fAaTpYDRb%C !`fXpTefhe"9I@$w`pXD(h d q hh@CWW@ WC C XXW W Wh P 4!`%`Q%sc9bm@U icbC vfCVThIEpTpbesHq FY G Y@ W ChkG $ vaCb PC Go S PW %!fC%RCHvC 6G CVvpXaP`@ TE P C XXW W Wh Thk Yh EW G bW P Y !Y9@U QyDvpCb`X&fC&pC$vpC%au%pCY Chk TXA C E PC GCWW b Y C T@ SX $fY`D9!xVCfGTD@dcbsmG46q%DVXaT"g@9VW`9`! W W P GE G PC Th h 4`fXpTVhvfWT 0Rb%l$k C T YX S Wh Th &pSfPpDv0Vb!fWT W P E G `fXRTfhGHVWT 6F Tf
r1 r 2 k k X B1 (o, f 1/p ) X xX |f (x)|1/p U f A r k B1 (o, f 1/p ) X B1 (o, f 1/p ) X G k |k | q k
-1 -1 k{r2 T } k{r1 T }
C XXW W@ Yon G G C G@ Th@ S EC P c9bsaC%`ftem@U I@!`&fS!"9dpCaYbHpVCG aCeQ`F bXW WX RT$v&vfCeg@v%`t PFbWC S G YuWu P G rq b@ C TWh GhXo G P Vhy!`pCei !puVbdfC`(0f@gRb%C sHq pTpP`DX pueg@vpCP Ph@ h bXW Y C XXW W c9bm@U C ` !((6)$ e 2 P G P P G SC C h T TX qC T Y Y d%fC&fSRTiVC%!hdpCaY!b e@ 4vfEDp`!I`ftpTdaubfupT`BUA@ CWh bWC S G Y b@W P xD%&vfCVG%pCah"cbC F Y G q G T S@ YCAh W vgP`fXi`fCVopTaPfP&!DiRb%`DX h b!Y&pC%`ftem@U h $vfC`X f C C XXW W@ Yon G CW bW T y { x v`(IvaD "R `99gI!|e&!x !e| { z y y { z { x { |"9"vIv9Veg%Vz}y{ !Dej{gv{""m `d&I!|"9x!)zVy}esg`|R `9I|"9x `b}vvR !x z y y {x y H9f"g%fggzq!v9Vv(H"xg`|R !z!I`ftpTaubfupT`"A ( `b}vv9pv%$ "R ("mzva9}!!e| x C T Y Y@ y { x x { y Q|"9"vI&Vv(H"aD|R !izv!c9bpubfuaTe"A (!9p"g(vecc9b@ x C XXW W@ Y@ x x C XXW W m m sRcgcfu v4we@ q G C W W CDp%CU sRTHVW`0$vfCpT!Dspq aC`vv9aCeW DX Dk E@ P Wh P T@ bW boX W r YXEWC bX W h Y P P Yon G C h@C T W WC Ch TWh Y TW@ phd%aCvDVt`@U ! e%vpEIPe@h vx$Ik9aCvfne9S CW YW b G bhC W r bXW Y PThkG Y E P h@ T G P h ipC!HvpCvu y!%H&s$q pu`wa@%RCD`a@phbHphaYcb&eCaEfP`@VCsRb%C sq aCvDVt`@ qVCvpXaP`@ Yon G G TE P Y P h@C T Wh T C hk q q S YXC i TX bWX S W phe%vpEIPe@viaCcbIP!hqVGT $dBs!s vaC`%9F hy QfY``AaCpY$`&sr Gv`fXpT"g@xfWe9D!qDk CWW b Y C T@ SX Wh Th q P P C EW G@ Thku CWh bW bo WX FP Y@W P bWC S YAh E Y P G Y@ C YWCtWC Tt GX b G C %aCjbpTw!C%VC`"VA$Q$vfCpT!DX `y%geHea@RbxgPD%&vaC$`XpCi%fCi`vau!%`%fC``fX`(`aXI@! VbW T Uh S WX !v`qT q !IPv`fXpTea@%fWe9`0!xVCvpC%CRU`pXifCG C GCWW b Y PC T@ SX PC Go S PW Ph P PW bX qP Tt GX P H`pX`i%%fC`DVXDHDpXbU CE YhX Ch bW E PC PC XXW W@ P Yon G C C T YX S Wh C TE P@ `(VWaTf DDXcb}F %!Y&pC%`ftem@U !&pSfPpD&0%VuvpXaP`4F W b@ G@ C qP G T S@ YCAh DVXaT"%fEfT`&W vVCfopTpPIP&!`waPpbvD`X aCpY$v%fCfoRTaPfP&!DxaP$%%pCDXpC%fCi`%pu!Hv`vfCfGvCcbC bWC P G T S@ bWC S YAh E Y P G Y@ C YWCtWC G E P Y P Tt GX phfC`DVXDHDpXb @ q u C P G T !Q%fCfoRTaPfP&S U@ YCAhX PC Go S PW !`aPpbvD`%!ifC0vpCvC 6G a@!HvaCs)`aCD`"VAph9VCD`fX`(`aX6bU I@x!qQg@vpCP YW Y W rq bW@ T P Tt GX G C TWh S Y fC0vpCv C Go S PW z { `cvgp" {"$aeaw(!&"gw% z x x y y y y { z { y z e"{~xv({ p"V|}"Dp"Vv&vvv R!xg"}epDeaV"a|m(!x {{ D6zvB!eVVvDa|g"v x y y| z z z zx DH9fy`{vq!ez gzxD{%xVyy0ewV`ve6{vggvm{ Rvx`6{zefqzx`&Vyy"df` "!vpVv"{vBH"gVgBem { y z z y z { | y { { y {x y x z y yle{!vvqv&"{zgvgvH"sVv`mzy `6{zeDxVv&vvv R&D { x z z { | y | y y fy9vR"v!x { x (e`Vlz !gx fC`DVXDHDpXIb }wVv&vvvw!x z" y Tt GX y { d` qs H&&%C F hy VYD`A q @ S S TX pT$v&vfC%RCv4RTfCVopTaPfP&!D$v&%pCDX%aC mcbmq %ey9 iRUXc c`4pXhP w! P b W C S G h P b C P G T S @ b W C S Y A h E Y W g C P fWT @ R ` C h P C S P ysfCVGfG&e@W u `fWe9`h { T S C q C T @ S X W b!Y&pCv`ftem@U h $vfC`X f 6H6H C C XXW W@ Yon G CW bW T 2 8 7 2 1 # 9465%430)('"&%$#"!
Ouv(Spm(A)) G Rat(Spm(A)) f (Spm(A), O) A Spm(A) Spm(A)
Spm(A{f -1/p T } T q |g )
k
nN B1 (o, ||-n ) k
k
(Spm(A), O) k
D(f0 |f1 , . . . , fn )
k
/ Rat(Spm(A))
Ouv(X) P(X)
G Ouv(Spm(A)) Spm(f ) Rat(Spm(A)) Ouv(Spm(A))
A{f -1/p T } =
|f | r
k
A
X = Spm(A)
k
A f0 |f1 , . . . , fn k Ouv(Spm(A))
k
X
Cr1 (o, g 1/q , f 1/p ) X Ouv(X) k{r-1 .T } = k k f= k
A{T } f |T |g(x)|1/q |f (x)|1/p f = g p = q
Spm(k{T }) 1 Bk (o, r)
U Spm(A)
B1 (o, r)
f, g (X, O ) B1 (o, f 1/p ) X
nN
f=
A
an T n A[[T ]]; lim |f n ap | = 0 . n
p
nN
B1 (o, 1)
(U, O)
G
an T n ; lim rn |an | = 0 .
k
D Spm(A) D = i Di
k
k
k
B1 k
xX B1 (o, f 1/p ) X 1/p 1 BX (o, f )
n
k
X
B1 (o, r1 )
n
f A
k
k
X
X
B1 (o, 1) k
Rat(Spm(A))
(A1 )an k (A1 )an k
X
k
Ouv(Spm(A))
X
Ouv(X)
G
/ B1 (o, r2 )
O
(A1 )an k
k
(U, O|U )
G
p N - {0}
O
U = Spm(A)
f
f A gA Cr1 (o, g 1/q , f 1/p ) = X
A
(Di D)iI
G
X
V9 2 2B c 1501p 59CSCB i 0 9 P99 0 PB ( ( 'G P '% Q 0 9G f T 9 Q P a ! ( c P 2 ( P G P99B Q W V ( 2 2 ( TE 0 Q G P ' @ 2B99%E ( 9B @ !9 7 ( ( 2 2 0 ( '% !! 3D34$"3gCS33)#3)"RB3e#3A"53bed"&"3b5a13H#34$C`55QYX8)5'$! 3U3SRBD3)IH#5GCF$""3DCA$" 86)5'4! 31)&$#"
Spm(f )
X
k
V PB% 2 R ( G'% ( !9 W S PB% 2 REG 9 2G'QE!B% 9 (B 8C$59CB 3P $)"3"$" XAC"459CB $1@ "30 5$))V$`RBQ 1C0 2G T S PB% 2 R 0 ! (BG Q 2 Q '% ( P G ' 0 !%B @ c ' 29B P !9 $59$1@ UAC"459CB &1`$hC$4! 3T 4! &5c52)"e33 7P ba 4! )49 3e$"C3&)% I 4Cha #" G P9 2 Q Q ' @ '! ! 2 ( T @ D 9 Q P 0 ' I9 P ' 99 34e5)C5')I)$`RBQ U3DIb9! 30 "533h&c P3)49 $e30 ! ( BG Q 2 Q '% !9 8! G ( W V P EG @ P !9 ( 2 2 0 ( '% !! 2 ( @ D !9 $dC4$! T 4! &5c52)"eRBQ $" HU Y)3F"I349 #")5'$! 33)"A$$U 3V 3T Ib9! 30 #" C' C% 5Q h"&"3h5a33D#A5043)"S)h97531) (3CB &$ B 9B ! ( c P 2 ( P G@ 29 ( '% ( ' 86 4 2 0 ' 9 ( %
M Rat(Spm(A))
hh@C T Y C Yot Th Y G P !`%vpEIPe@vpud!auVb!!pXVCgRb%C h@ T Y eCaEfP`@auC q G@u Cu Y W YC S T CW `%!IT %aSvC`fXRTgPv&&VS0h jbDX f 4}}v s& T z VyDHvv(m|plz x { | "ggvI"6v(q u mm QQ4g bW@ T P C S S G G @ W r $`BfAph&%VCdfCsHq W b TE P G Y@ `fXpTef@v!RXaP`@ 60e Y phP h@C T Y Wh TWtu WX TWt C b C CW W@ @ Su `%vpEIPe@vpuvVbv!`)Fx(cx!9b9vgpEmPU h Vb`X W T q G T b WXW P C W b@ G@A G Chk T P G PW@ EWX Y PX P W r TW DVC`f@fApTaYiDicbC QDVXaT"fhe"f@DaCjbRh&I@edD!a@%RC$HRhs(f qt C b C bWC S G@E G P@ Su PC G E PW@ TWt C b C P@ Su PC YXt b@ Ph P G C W P GE G bWC S G P P (cq$v&vfCe%!VXl`9vgpEmPU %!dfCVG%Cqe(c!q`9%gpEmPU %fCaT``%puevmEcUDRXI@s!Q`fXRTHVhvVWT $%%VCfSpTRb%C Yh EW EF G T b P C W b@ G@A G Ch Y %pCb`XwCDVC`f@fApTaYcbC !`fXpTeVh`"I@sDakgP`X C T Y Y@ CWh E P TWt C b C @ Su Wh Th q !I`ftaTsaubfupT`BAU sfCT!pXpPD@ Hs(cw!09%gpEmPU 0 $k dq Ds Dc W b@E b G@W `fXpT"%HVtpT$V`e@ 4 YvpCb`XwaCfYphaYcbH`XdDVXfGewDDeg@``a@`Cwew4qgdVbeg@pYRb`ee0&DX0auvfpT`vdRb%C DX`9vgpEmPU h EW Wh T PW E PW G@ PhXWFC Yt Y@ E PW@ T Po@ h@CWW@ Wh C S S E Yu PWXE P P@ Su P!gVCaT`DvpuevI@G C YXt b@E aCe&De`&v#isDq bXW WXFC S" S C C bXW Y P PC T Y P b Y@ PC YXt b@E { z CW vpu`wa@%RCp!I`ftpTppuvVuaTe"AU %qfCpTD`%pu"%@ qDej{wmv{""aU h C GC @ Y vfuv(eQa@%RCP gPvD0&pSfPpDh 49VCS !I`ftpTpuvVuaTe"AU 4`&v#&s qDb}vvp vD eU &vfuv(e@ YCA C T YX S W TWh C T Y b Y@ CWh FC S" S C y CWh C GC Rb%C P P Yon G C C T YX S W TWh C XXW W@ Yon G C qC XXW W@ Yon G CW T pC%`ftem@U i&RSIPa`&h 4ifC0S !Y&9pC%`ftem@U W u D!Y9bxaC%`ftem@U h Vb`X f
4 6F C 1 F 1 4 1 UIt!C 79a@ AC U( & ( 4 6
A A A A han (X) k A i: X X / An Spec(A) = Spec(A[T1 , . . . , Tn ]) A han (X) k Rig : Schtf /A A k k k B X an / VarRig/A , X VarRig/A han (X) VarRig/A k k k A Spm(A) Sch/A I A[T1 , . . . , Tn ] /B
qC T YX S `&pSfPpD&`pXfPT Wh bWC S Y PC b TX C P Go S PWC P G Y P C E Y P PC Thk YCAhX W YC S T CW hk TXA C b Tu S T PC wD%&vugE`XscbwaP$VWDH0!w%fC%RC%VCwph0DVAaTb%cC6phwRb%$aCRb%`D`fXRTgPv&&fS0h $wVYD`def@!&fSdcbmG y { Vv{g"lgz { 9gI!|e&wVy { zx z x| x y y z z (e&e`&!"m!"&vsvz ` $v9{ RH"}mea Vv{gI0e`9z"!" !mgzIDddy{ y { x x { pCpbvD`w`fXRTgPv&&VS9(!9fy`g|szxDej{mv{""m D YCAhX W YC S T x | { z { 9I|e&) `q %vmYaS `WmC fXpTaP%scfSgciRb%C Cu Y C R & CW 4h wP S T fCeD`eI@!xI@ pTv"|v0cbC G@WXt T G P P PC T Y P b Y@ C C T YX S W hk T WXFCh You G %!I`ftaT%puvVuaTe"AU pSIPa`&h $wVb!``$IkpT%`fte@ VCaTpYv&`vDv&Dl}q !v&aSvCdaubfupT`BARU`pXwcbC $s$`p@IP!%&a@%RCe`H%$vD`X `%&pS%C`fXRTgPv&&VST ! bu SXut WC C S SX C Cu Y Y@ Ph P CWh P Chk bW T WC Y Pho@ WXFbWCAh qCu Y W YC S G C P GF q q W TWtu G PW TWXE P G Ph G@u G GC `%!IT Vu%He@ bW Yu EG@u T Wh TWtu YhX bW G E Y $vaCvDXe!I0pfYv!sDH$vfCVGDX%aCqRCP %fC"%f`f`Q`fXpTVbv!df@!fC pDVXaTjb`vp%fCH`X f y { x x "`!!e| 9|"9"vI{ Vv{gI0e`wH9vp%!egaa!xmzIDRd y { z x x { zev { { zx z x| x 9I|e&Vy !e9!evlymz ` y{ %&pS%C`fXpTaP%&fS(!VvDa|wzQ``j{&mz{eBm ` Cu Y W YC S T x y| { { 9I|"9 h q YCAhX W PChk CWh PC bW Yu P@hk P GhX S P G YhX bW@WX P YXE W DpCpbv``Q`fXpTRb%$gRb%laP$vaCv`XmEVURTD$i%fC!!&%fCiDHspC$eH`(RpCaY`v`fXpTpbvY pCaP``P @ m vmq $bgP vaCvyRuY D X cicP VCH4w&S W E g G X U PC P G E bC TWt C b C P GhX PC YXt b@E G bWC P YXt b@E C CEW G@ Thku CW P !g%fCVG%Cvw($cdg%fC!!&SU %!wVCaT`Dvau"v&I@dpCaYD%yVCaT`Dvau"vdvvfCe"fA$%h Rb%C fCT!pXpPD@ E P fC!!&SU Qs$vaCbDXVC9$sd"s H vaC`%0F hy 9fY``ApCaYDD&wsD!c9bpC%`ftem@U Rb%C Vb`X GhX Wh Thk Yh EW G Chk q q S YXC i TX bWX S W r qC XXW W@ Yon G CWh P T { z y x ` yya|`y4|aiV!e%9m !x g"" 9|"9f "vI{ zev { y z xsv9`(IvR D&e| D Vv{gI0e`v9p!"ggp !&mzIDRdyy{ QjHpvgq|p y { z x x { z zHa%gR{Bvz xz"iw %f!"&m !x zw``j{gmz{eBm (!x x { yx { z { He m m q sRcgcH4w `fXRTHvpCvDRX$pTfWe@ W PW bhC W P Chk P T aCeW DX G $k bX W Ch Y h@ T Wh P phP `%CaEIPe@ Rb%C VCv!RXaP`@ TE P GCWW b Y C T@ SX Wh Thk h@C T Y G GhX Wh b C XXW W@ Yon G CWh PuWWX bW b v`fXpT"g@pfWe9``$de%vpEIPe@vpupVC%F fC!!&SU vC c9bppCv`ftem@U 4%`!$`a@p q !`aXgDH bh b YhX pCb`&`vEVUpT`$pjbDRXP E@ SX P@hk T Chk G b PC XXW W@ PC Y@ DIPvaC!c9b)%!p` VCVopTpPIP&!`)D%&vaC$`XpCb&!DqfCVG%RTgv"m|{"xpCVb! G T S@ bWC S YAh E Y Wh C S@ GC P T Rb%VCfGsBv!Y9bm@U HqaCD9s`qpCpbv``XpCsRb%gfCfGvC $pkgP`fXimv`ieaR{"x0aCjblcbC !I`ftaTqaubfupT`"AU W PC G qPC XXW W CTWt Yo SXW Wh Y@ YCAh E Y PC G h Y G z| T P C T Y Y@ C qPW EW C PC XXW W@ P Yon G PC G S E G Y PW b Th Y h@ WX P Y E PC XXW W C Yot Th Y %`fXpTb`Xd!!c9b%aC%`ftem@U %!aubfu&`X4%VCaT`pXvaCgjb!!pXQe`(p%aCpYDX%!Y&@U !puHgjbuaX C q &vfnRTIP`aXpY&h 4HRhaP%! `@ vC !I`ftpTw%aubfupT`"AU !v!9!xauVb!!aX&VCG C S T b CW P PC h b PC T Y P Y@ hhC C Yot Th Y a@%pCesvVCHd%aCjb&fSVTG Y bXW W r qP TWt P T P%fCwv&!`@ G bC S VCaT`Dvau"vQ@ v!Y9bs%aCvDVt`@U !fCpTD`vau"%I@00vpuD(DX ipC$vfCe"fA$%dcb&VCfGs(v!c9b@ YXt b@E qPC XXW W@ P Yon G PC YXt b@E G C C PX G bW G@ Thku PC G qPC XXW W P Y@ PC C T G YXt b@ h P %aubfupT`BAU %&VW%VCfCaT``%puevmEcUP DRXf@G bs%fDVtaTpuvVuaTe"AU %!fCaT``%puevQf@G C PC T Y P b Y@ PC YXt b@E pC`sr bXW W
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
A
M
k M A A f0 |f1 , . . . , fn Ouv(Spm(A))
X A Ob(VarRig/A) homSch/A (Spec((U, O)), X)
W X D
k
A
(i, j)
a(M )
VarRig/k
a(Mi )
Spm(Ai /Ii )
/D
k
k
X
Mi
^ X S Y
/ Spm(Ai )
Ai
U X
(X, OX )
Y (Ui = Spm(Ai ))i
k
A
k
j: W
A
O(-)n k a(Ii )
/X X
M Rat(Spm(A))
Q
M OX (Ui = Spm(Ai ))i X
Ii A i
i
Y f : Y
/X
Pi
Afnd/k VarRig/k k k X k
(Ui )i
/ k
i O(-) k j
0
/X
A
i
Y
a(M )
k
Ui Uj
X
Y X Ui
D U
I OX
M
M
k
VarRig/k
Y
D(f0 |f1 , . . . , fn )
A OSpm(A)
k
M|Ui
/ Y X Y ^
/ Ui
S
U
dq Dc VpTea@!0DXE b bh S S b @W Y P GCWW@ P E@ P C C T YX S h TW G C CEW T G Y bWX S C EWX Tt P PW@ TXAW $WeHvaCg%fuveiC`HRCU &RSIPa`&!puVbTIEh 6sbvvaCcbIP!hC 6g%pCaYDD&!D!Vb`e@ %mGq ` VCD`vC WCE Ph P PCE@ P C C T YX S G Chk G C CW W@ bWC S YAhXE Y W b$vv`R@fcUH`pX%v`p%CU !&pSfPpD&fC$sIPvaCb !09b$v&vaC`v%aCh bC C Y@W P bWC S YAh E Y W !Qa``a@Rb$v&%pCDX%aCh EWX bW T D!$vfC`X f q bC vfCev!fXicb`fXpTRb%$&@ WC G@E G PC W PChk
(Di , R|Di )
an
aupY`vfC)VpTea@!0DXE Y@E G b bh S S bW@W $eHvaCY P GCWW@ PCE@ P C C T YX S Ch TWh Wh ThC hk Y bWX S PW G@ PhX Vuvew%v`p%CU 0&RSIPa`&$fkpCRbIPvVGT $w%pCaY$`&H`fXVG`sDi q gP`fXe&s(q dfuT!pXpPD@ Y G@ @ W r E P Y P bW Yu P@hk Yon G phdpC$vaCvDXmEjUpT`HDwpCv`ftem@U I@G aCesr bXW W q pC%`ftem@U I@pDpX%!fpTDXgbC Yon G G PW Yu PW E jbDX q e!fiy`}t !waSvCc9bm@cUDRXlRb%C T G@u T Wh Y@TW u C u Y C XXW W Ph P Wh P aPDVX`HF Y G@ T q Cf T Y X S h W E P C W &pSfPpD&!`fXpTb%pC0h Vb`X T q`aueHvpuRb%C $akfPs$vaCvDXmEjUpT`f@ $qaPDVX`Rb%C eCaEfP`@C q !d9bRb%`Dih 4Q`fXRTfhGHVWT 6G Y@ P P Ch Th bW Yu P hk Y G@ P h@ T C CW W@ YCAhX W W P E Vb`X f T q T E Y P G Y P bW EW E P W rqb G bWC S Gnt P DpuVbVAaTb%cC6ph&f@gphpCaY$vCH`XpCsH%"I@i$v&%VC%!HgRb%C PCE@ P C C T YX %v`p%CU !&RSIPaf `&S fC9$jb`@!$v&vfCfGT`@fC`v%!`fApTbcC"fWcbwDp `R$jb`@&C q G Chk T h bWC S E GhXEu C E T PC d q Chk T P@E G Y T b C b W P `%xVC%pCVbeg@pY&!&phlmG 4}}v s& z "6{vgve { zDp `R "{"g~{}`a| p dq z y| y zx z x `(IvR !ew!" v"m|09vg !x ze m sRcel)ys { q W b Y@ P C Pn bX G PW P T Y `fXpTea@`vpu!RCp`H 6`p@`a@A C G Y EW Y@ bW bo P Cu YC E Y W b@ G G FP@E CE PW@ &aujbfTeI@aT`pXb`Xd`$vfCpT!DX %%ggEvgpC`fXpTevIEVTe@ 64%`%&v`s Rb%wDRXaP%!mVUT"g@jbph%au&VCyRTdDpXD`ftd`q PC Ph PC E b G P Y G P PW YXW T PhX q P@E G Y T b C EWX b W P `%VClvaCjb`a@aYi!iH`!Rh}mGDq bC Ch GCE YhX PWXh Y@ S Dk I@vviDH&D$ak`9vC S dq $p Dc W b@ G@ CWh T PWXE PW G@ PhX qC XXW W W DVXaT"%fEfT`wpCfYahpYRb`vg`fXVG`g` `!Y&@U h bC @ Su v09%gpEmPU h $vfC`X W bW T qPC T Y P Y@ C P YCAhXq P PCu Y PW YC S T PC Y P P@ Su C P YCAhXq P PCu Y PW Y S v!fDVtaT%aubfupT`"AU ygd%aCRb%`D!RpCY%%aSvCqH`fXRTgPv&&VS!iphg`9vgpEmPU !qgd%pCpbvD`! RpCY%v&aSvCqDVXpTgPv&fC &VST P G TXAW Chk bWC S G@tu PW bX %fCgVCD`vC $y$v&%VC`Dvs`aXei8q W vC G Y b@W C T YX S T Wh @ WX q vaCah"&RSIPa`&DRXIPxdxD`fF`C 6qT F%%!Y9@U pCaY$vC PC XXW W bW PC T YX S P G bW@ T P P T G@ P PC W PW b G bh S S pSIPa`&q%fCq$e"fAphs%aCVY`f@`vaE%!iDVXpT%pChC 6waC!&DXE vaCbDXfCw$$v&%VCG Yh EW G Chk bWC S VUTD@yVtaTv`s$qes DcivpCb`XfC0DX 0eq E C YuA W r d q Yh EW G C P T@ PC WC h Th G Y EW P !RTHVW`pRb%&%!VGT $pkIPqfCVG%VCaT`aXbDXpcbC W b TE P `fXpTef@v!RXaP` @ j}jlV) q h@C T Y G bW P Y Y ChW boX P T@ C T Y b Y@ e%vpEIPe@vpusfCipC$%RCpu%pCsvaC!`(RTHVW`s!I`ftpTspuvVuaTe"AU @ q bW@ T $`BfAphP VClD`f@fG`v%aCi%C fYv!saC`vvC P G bW GXE Y W TWtu YXEW EWX bhC W rqPCW W P@hk Ch Th P Y@ P hhC bW D!l!%H&sD%%9bm@VURTD$i$pkIPpu`vpu!%s$`pXP H`fXpTbRCgPvaC$VWpVC"Dpu`vpuipuD(phsfhyRb%C W $akaPDX PW E Y b T P GF Y@ P PX P P G P Ch Y q h@C T Y G bW P Y e%vpEIPe@vpuwfCdpC$vpC%au%pCY Chk TXA C b Tu S T PC DgfY`D!i"I@!%&&VSiRb%4mGq bW@ T $`BfAphP VC$eI@VGDX%aCQvC !I`ftaTsaubfupT`BAU ijbHt!sHv9b@ P G bW G E Y W C T Y Y@ CWh TW u W rqPCW W aPpbvD`%!we ! IPaTe}$v&vaC`v%aCh YCAhX PC Y@ C T Y@ bWC S YAhXE Y W jbDX q Y @ P P C h k Y X o P W P X T f Dpu`vpusRb%C $g`(e@ 4H`pXD(h f 4}}v s& ``j{0v{""m { z x | z }!9"dfyDHvv(m|pz x { | y { z zx z x eagvI"6v(Vg yia|i`w&vgv !"m!e tm m sRcel)ys q YhC G bh b TW u h@ T Y G W To CAh Y WXFCW W@ P Ch Y DpCv 6!DpX4t!QeCaEfP`@auVC%VCw``aXpYbaCQ`!`&cbC $pkgP`X q h@ T Y W `%CaEIPe@auh TpfWes$vfCpT`sH%%aCvDVt`@U !&pSfPpDh P T@ bW boX W rqP Yon G C C T YX S W bC C XXW W@ Yon G CW F TWt C b C Yon G CW v!c9baCvDVt`@U h `fC(c!pCv`ftem@U h YhX DH C Y G C bWC S G@E G bW P Thk P Yon G WC P GCWW@ PCE@ P C &pSDXxI@9!$v&vfCe%fXdDDRXl$spC%`ftem@U vw%fuved%v`p%CU ! P&RSIPa`&p!fC%RC%C G C T YX S PC Go S PW pCe9`X bXW W bC ph(HgHcgiRu$HRhl$%%VCvpClcbC T f Y PTWt C b C PX P bWC S Gh P P j}jlV) q q q G SC p VDVD9VC%!hC 6G C C T Y Ch b G@W@ T Y !xfDVtaT9$fkaT$V`eiaCjbDpXf@G ECA `B@ aC!%pCaPDVX`!%Hssq YW Y Y G@ bhC W r C P@E !0`vVCG Y T b C EWX b W P TFP Yot Th Y P G C PX vaCjb`a@aYlqH`!y&phfG"vpuH)aPVb!!pX%fCq%pnpP$H C S SX G@u G C TWt b YuWut T S CW &Dl$q e%fT 6!ifClCIEaTpY"g@vv`fCVGfG&e@h `B@ ECA 8 7 2 1 # 9465%430)('"&%$#"!
G A G j : X0 X s : U = Spm(A) (Pi )i X R X = A1 Spec(A) X X X an OU han (X) A X an X X an Rig k
an ^
homSch/A (Spec(B), X) homSch/B (Spec(B), X A B) homVarRig/B (Spm(B), X Spm(A) Spm(B)) homVarRig/B (Spm(B), (X A B)an ) (-)an
X U V1 = Spec(E1 ), . . . , Vr = Spec(Er ) f Di Vi
han (X)
A
han (X)
A
X an
/ X an U
C
(X A A ) X A A A U = Spm(B) A
homSch/A (Spec(B), X)
X
Xian
X
A
/X
X an
A
A
X
A=B
(Di , O) A I (U, R) (U, O)
an ^
I A Spm(A)
Xi X j (Xi Xj )an k
VarRig/A ^ X an = (A1 )an k Spm(A) k
D1 = Spm(F1 ), . . . , Dr = Spm(Fr ) X
U Ob(VarRig/A)
X
fi
f
fi f
B
/ homVarRig/A (Spm(B), X an ) .
(Xi )iI Xian
/ (Vi , (j O)|V ) i / (X, j O) / (X, O). X X /C A
j O X / Spm(A) Y = X - X0 Z = s-1 (Y ) X an hI A A(h) = A[T ]/hT - 1 A(h) R(U ) = A(h) / (X, j O) G f : (U, R)
/ (Vi , O). X Ob(Sch /A) Spm(C) han (X) (Xi Xj )an
tf
han (X)(U ) / Spec(C)
(A1 )an k
U = Spm(B)
han (X)
fi
A
homVarRig/A (Spm(B), X an ) A=B
A
G
G
/A
A
X
/ Spec(B)
k Rig
X an
B
U
dq $ Dc d dfY`DA"}"X#`v{}$saqwI@wDea@pPDd!%pCb`XwfCgVY%pC!DiDH TX x " x G Ct P@ C Yh EW G TW boX YhX W E P W E C E Y G E Y E W X b h C ( `fXpTbphaYcbH`XdpCpbbvlvfCfG`XpCsD!i!v(iq s`r3W q !(%DVXaT"g@sfWe9D!Rb%C F !H fPaTe} Rb%`DqRb%C pTw$VYD`A CGCWW b Y C T@ SX Wh P CT Y@ YCAhX Wh P P Chk TX C E PC Y bXW W G Ch !ifCfGT`@ycbmG`q g@vaCe&DX i$k C XXW W G Y TE P@ T G bhC WX bC C XXW W@ Yon G CWh P !c9bm@U fC)%VCv!RXaP`(Vhq!v(&`yvg!Y9baCvDVt`@U yRb%C aCvDVt`@U f@$F i(ci!gbi&(%&pSDXd9vgaEPU D!l$eg@b$%$v&%$aXfTu q Yon G G TWt C b C C CW W@ GC Y @ Su Wh uWWX bW uFbWC SChk Y E CW W 9b@ G%aS`X09vgpEmPU VCvfCT!pXpPDfhi!%Hs}DVCxDVXaT"a@$%RCpug$%%$fkD`fX`(`aXijb`@Qvw`sbdfCVG%VCaTD%SW RCaP%iRb%C $k C Y @ Su G Y E P@ T G bhC W r q TWt W b bW P Y C bWC SCh Tt GX b T WC EWX C G b PC P Ch fYeI@GcbmG4`fC0Hc!w$%%$fkD`fX`(`aXpCv`ftem@U 0`!Rb%C sq T E PC q TWt C b C bWC SCh Tt GX b Yon G CWh EWX P C bW bXhk Wh P !d$%VCaTe$wdRb%C aCvDVt`@U @ q Yon G W b bW P Y CW TWh S C XXW W@ Yon G CW T `fXpT"g@$vpC%auh QfC0x!Y9bpC%`ftem@U h Vb`X 6q%VCaT$vpCpPRb%C D fCfGvpCb G bW PC P Chk G VC`9VS!"9dv&aSvC`fXRTgPv&&fSh 4RC$HpfPsHq ph"I@iRb%4fGT %fF`C 6q!$`fW pXVCG G@ Th@ S Cu Y W YC S T CW PX T W r Y P b G PC P q TFb P P Yon G P G P G bW PC P Chk T W r TWt W b bW P Y Cq P TWt C b C bWC SCh Tt GX b Yon G %aCvDVt`@U VC}pT}%VCaT$vpCpPqRb%C $0g@fY!sHq dx`fXpT"g@$vpC%aud!!RpCYssHcw!D%&v$Ik``fX`(`aXiaCvDVt`@U CW h 4 GC Y EC P G Y@ uWWX bWC S G@E G PC Thk bC E@ SX P@h GC Y @ Su P Wh PC d { z z v&pSDXiaCpYb%(RdVCqeHD!q$%%VC`v!fXqRb%($vybD&`vEVURTD$k v&pSDXd9vgaEqRb%yBse6{eHv a(wDyp%aCYyia|x`%aS`Xi9vgaEmPU W %fP%&pSDXD9vgaE%!dDveD%&vfCeI@$fAaTpYd$v!%pCvu %`fXpT`w%l | q P { z GC Y @ Su q GC Y P@ Su P PC P@E h@ bWC S G T b bWCW b P PW bXW PC qfCi`fXpTea@$vpC%au!g$v&v$Ik`DVXDHDpXs`bC9fCVG%VCaT$vpCpPsaCvDVt`@U icbC sHq d T W t W b b W P Y C b W C S C h T t G X bu E W X G b W P C Y o n G C W h P C Ct@ !``&fST 6G g@vaCe`F &Hc&!$v&%$fkD`fX`(`pX0aCvDVt`@U 9%D!$ea@s`fCw`fXpTea@$vpC%au9!$v&vDIk`DVXDHDpXb Y bXW WX TWt C b C bWC SCh Tt GX b Yon G CWh CuWWX bW b q TWt W b bW P Y C bWC SCh Tt GX Rb%wVCfGvfCpT$%RCaP%gb wHcw!g$v&v$Ik`DVXDHDpXiaC%`ftem@U aC!`aXsDlRd fCVopTaPfP&!DR@!eHsv`I@&vaCRb%w fCVG%VCaTD%RCaP%RCDf` PC G bW PC C TWt C b C bWC SCh Tt GX b Yon G bh b Chk G T S Y@ uE G S Y PC G bW P X q W PX Y r TX G T b WXW T Wh P E T P `fXpTVbRT$HDpXy4F &VYD`Ag`f@fApTaYwDjb`@cbC spcqjaTb%cCeVWRb%C C T YX S G q T PW@ Yu C &RSIPa`&&VC VFDC q4F `pX%v! Y h T T W h P @ P C h Y G z T Y P G { z z | z { y z T Y T W t C b C vpCfPfA!iy`HqcbC W $akaPDVXwfyy{vvaaCjba@%RCwVCfGs$qHs"{"mvvv}y`lH9vx`pf"a|mpCVb!g@vpCP 6Fi(ci! G`%!I`D"0! VYpTfPDg!v(`}%Vh!T f q @u T Wh h@XW C TX E bhC WX FP G C W b bW P Y CWh bC S@ `fXpTea@$vpC%ausv`bC fF`C qp`$Ik!`m@U fC`DVXDHDpXI@9`(pCvVnDXiRb%fCVG%iRT0s0a|`9v&vx`aV"a|mz pCVb!iRb%QpC%`ftem@U W q T FCh T Tt GX b G YhX b G S E PC GC P { z z { y T PC Yon G C q Yu C Yh T T Wh P@ P F h@CWW@ Wh PW@ Chk YW TAW E W r aXv%!i%RCIPVA!DiRb%C W eeHe$a@!HvfC$DXs(q u a@%RC$HRhD`fCeI@VAaTpYcbDVXaT"fhe"I@yT f qW `W Y PX P WX F G T b PC W b@ G@A G Gh WX jbpTfPDgE `DVC`f@fApTaY`0cbC ! `fXpTeVh`"f@&$akaPDX TX WXF G T b WXW P C W b@ G@A G Ch Y
) R0(
& #6 00A6 ) ( " @
) ( C 0BD(
@ 1 4 ( 73711 7@ 7 G%70#! 7x ( F & H 4 H ( & $
1
j}jlV) qC bWX Su PC W PX Y `vaupY$D%!icbDVXaTjbpTD(`aX @ PC u G@W bWX P Su `@9vgpEmPU !9&pSfPpDdH`%pC!Dd`}%%&pSfPpD%C$eI@VGDXpCY C C T YX S Wh PWXW boX PhXW FPC T YX S P E bW G E q Rb%&VtpTb$fe`@ G$`! W s4q P@ u `9SvgaEmPU !&pSIPa`&S&$fkh 4&VtpT$fe`@ jb`a@bu C C T YX Ch TWh W u b G@W G T C T YX S G Ch &RSIPa`&&VC&$k hA @ W &srq PW EW PC Y X `fXpTb`Xi%!wh`( b vC ECA C `B@ ! fCfoRTaPfP&!DiD%&vaC$`XpCpCRbfP!vmG G T S@ bWC S YAh E Y Wh ThC q bC !9bD%&SvaC$`XpCYh bV`pXe%vC(v0sDVXaTjbpTD(`aXsf@qpCfY!!g!wfq!pVYaTRb`slpufP`pRb%mG C CW W@ bWC YAh E W T PFb C W q W PX Y G Thu C G C Y@ T@ PC q %fuve%v`p%CU !&RSIPa`&dfCgeVb!HVWT P GCWW@ PCE@ P C C T YX S G Y@ Th P Yon G C C Y S @ b @W P Y %aC%`ftem@U !pSIPTa`X&vh`0$WeDXHp%aCaY`XE P@ Su `9vgpEmPU !&RSIP C C T pUY`&s$kIhWh 4DVXaT"%EHjtaT$V`e@ 6bR%C X S dCh TW W b@ b G@W G P P T P Y@ C C T YX S G Chk TXA WX F bWC C!fDVtaTYw%aubfupT`BAU !&&RSIPa`&VC&$jbD`w`4`aC$v! Uv%ua`fXRTaPPhaEIP!x@I!CHvh$%puDX&S&`lq uE Y W T G C EWC k PW E C SX G@ G @ T tu C Cu Y C XXW W Ph P eu!fT YeH Wv! !0v&pS%Cw!c9bm@cUH`pX0fCG H`pX`i PW bX q T E Y G@ PC d q Y@E h W b bh S S E `aCfYeI@GaPDVX`cbqe `c&aupY`v!`fXpT"g@!&DX&@ C PX S E YhX Y G@ YW Y W r vpuD(DX9I@G `(qaPDVX`0g@!%pCs(q T Y@ YCAh G Y P GhWW C S SX fPaTe4 pbvD`X 6`F phgVC$e@ vP &DlDq
/ (Spec(Fi ) - Zi , O) i Spec(Fi ) - Zi Spec(Fi ) / Vi - Y i / Vi . h Zi
PW@ hW bW E P evaCDDXRb%C C T YX S Wh &RSIPa`&&`X
P@ Su C W P Y@E Y@E Wh EWX `9%gpEmPU Q%VCpT%auRb`vsauaYevdH`! b$vfCpT`sq W boX W r bC pCD(s4q epCgPv`fW``9VSDRXP vC ` !aCVY`a@$v&vfu&`v0fCG PX W r Y@ YCA T Ct@ T W b PW@ C T bWC S G SXE pu`iVbe"B`X 6$gDVX%He@ S DC%pCRbfP!vC `aXpY$Dq !&iRb%`DQcbC bXW T@A@ W G Chk PW GC q EW Th G PW bWX C CW W@ YCAhX Wh P Chk T h bW TA Y TW DVbe@Q!i$vfC"pXpuVbTIE h
X0 A Vi X 0 X Z (Fi )(h) o O Fi o
i i
dq D Dc puaYe%fC}VpTea@!0&`XqD`%pCY pCv`ftem@U !&RSIPa`&9!wpuVbTIEh 6lbivvaCcbIP!hC 6s%pCaY$`&i!&%" 9`y$vfC$%pC($l Y@E G b bh S S E bW@W P Yon G C C T YX S h TW G C CEW T G Y bWX S C C S S h@ bW TA Y Thk C
q
(Vi X0 , O) O A i
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
k
=0
Y
k Limn A/A. n
A k A, |a|p 1} A Spf(A ) k U
A
V(h)
A
A A
(Vi - Yi , O) Spec(Fi ) - Zi
Ui
/ A[1/]
Spec(A(h) )
k
G
/ X an i
k
(Vi X0 , O)
k {T1 , . . . , Tn }
A
A
(Ui )
X
X U
k
/ X0
p : k {T1 , . . . , Tn }
(-) : SchFtf /k 0 U / A[1/] U f k
/ (Spec(Fi ) - Zi , O)
k
Ui Ui
(h) A / X an U - V(h) 0
k
k
A
k
Vi
Xes p : k{T1 , . . . , Tn } k
Spm((X, OX ) k k) X
Ei .
f
/ VarRig/k , //A A k
(Xi ) Ui = U - V(hi ) A
Spec(A)
Yi = Y Vi
/ (Fi )(h) .
A
X X //A
/X
X
Ui = f -1 (Xian ) hi A / Xi Spec(A(hi ) )
(X, j O)
Zi = Z D i
k
k
X
Xes
p
X A k
(X, OX ) k k Spf(A) k Spec((Fi )(h) ) A = {a / (U, R) f A Aes A
P `X VCG Y@ Y@W P bWC S YAhXE Y G EWX C S@ `9ge`a@Rbl$v&vaC`v%aCsfCs`!gb&!D4mG$q CGC Y G S E G P b E G Chk PW GC !(%&pSDXaubfu&`XdfCqRb%C pu"I@Gu w$lDVX%He@ q ChkG $ vaCb Y bWC Wh ThC TF YCAhX PX %VCaT$vdpCRbfP!vVGRb%`DRu$HSRhP Rb%le%fbC&Dlpq PC G@u T E C S SX G@u G C b E G P Chk bC CW W@ P e%fT 6i!au"I@G%u 0Rb%C $vx9b&cbC Chk Y PX $0%RC$HRhP bWC TAu C T YX S G Chk Y bWX S C Tt P $vfv&pSfPpD&VC$9vaCpY$D9Vb``@ %smG I@vC`(x!v(sl`fCfoRTgPvD$fWxRb%C G E YhX bhC W r q G YCA T P Vy{vvgyz~x}l`s!v `fyD%9!pv&vVyvp!x"(ygzvVCfopTpPIP&!`g$v&vaC"I@G%gzecyysa|`yvz4vB}} `s &x z y | { x x z y z G T S @ b W C S b E u { { z z l !ev " { ! xy v z z zx H9veVyi!" { z y z { ``j{v{""m (!x pv6{HvgiVyD%9!x z" m sRcel)ys wd fY`DA9pC$`BfAphQDVXaTjbpTD(`aX T X b W @ T P W PX Y I@xs`fCeI@VAaTpYDsRb%C !`fXpTeVh`"xI@$DpX%RC$HRhs`!``a@Deg@e0C! xf@9 $ak`qBpSIPa`&wbC G @ W r q G T b W X W P C W b @ G @ A G C h k P W Y P X P P h X W F C Y t Y @ E C W t G h P h d C T Y X S W h b oX Wh G@Wt YXt b@E G bW G@ Thku P vcC!`fCeH&VCaT`Dvau"vf@pC$vfCe"fA$%Rb%C F G T b P C W b@ G@A G Ch Y duWWX YX Go S PW DVC`f@fApTaYRb%C !DVXaT"fhe"I@9$akaPDX q`!g`xfC0vpCvC W P GE G Chk T GE PC `fXRTHVhvVWT 0$dVY`f@vdRb%mG Wh bW G@ Thku q E T P 0aC$vfCe"VAD4VFDC qTpCb%aCVY!Rb%C FP G C q G@Wt E P vHVh&s(DVC``XcbC "D { x { y y x Wt P G 9I|"9!9zIDRdqy{ ! Vv`99V&v wfhwgz Vv`9 !x sx9"a"&e y { x ~y{vva0VyD%9D&%9"l(vV(i{eDaegRge9v z { y | z x z z v `VyD%9`&{"$p m"g fy m"}q' z w Dej{&v{""m D9I|e&evI!x ! yga|9Dyy9"&vgv ! { z { { {x { z x GnX S y| x z| { z { `VC%!!&Vyv`a| w ecv"|v{Bvzv`iegRg{"xc``j{Qv{""m (!x z" q C E Y P P C b @ G v b g Pw `DVAaTb%cC6phicbDW VXaT"vIE fTe aC@ mRbCvsfCGF pTR$vcW pCgpP cP 4cb C bE C Ch Y G Chk bWX S W r $pkgP`fX$aCpY$`&s(q IPv&pSDX0&%gaE!&RSIPa`&!``9VST 6dfCT!pXpPD@ GC Y @ Su P C C T YX S h Ct@ G E P b TX Wh Th $fW`($k W b@ G@ CW Y G@ PX T W rqbW E Ph P Ch Tt GX b E@ P DVXaT"%fEfT`h 4aPDVX`RC$HpIP!QsH$vCDR@VRUDRX$Ik``fX`(`aXwC`HRC G g@vaCe`6F%aS`X09vgpEP Y bXW WX GC Y @ Su U Rb%C T f q vaueHvpuPVURTD$)vspCb`&`vEVUpT`$y!I`ftaTypuvVuaTe"AU yfCT!RXaP`@ l($cy!v&pSDX9vgaEmPU i!$k Wh P C Y@ P@hk bC E@ SX P@hk C T Y b Y@ CWh E P TWt C b CGC Y @ Su Wh Th 8 7 2 1 # 9465%430)('"&%$#"!
Ui k k X = Spf(A ) x X X |X| X Mdles (X) Mdl(X) k k X Mdles (X) X Mdl(X) X N N - {0} X X X k I = (f1 , . . . , fn ) A (f1 , . . . , fn ) Proj(nN (I )n )
N
Mdl(X)
X
X
^ Bi = A [T1 , . . . , Ti , . . . , Tn ]/(fi .T1 - f1 , . . . , fi Ti - fi , . . . , fi .Tn - fn )
j=i Ji Bi Bi / Bi [1/] Ji m .Ji = 0 (Ui )
Ui = Spf
Gr0
nN
(I )n
Mdl(X)
k
1 fi
X
k
X
Spf(k (x))
/ /() .
Mdles (X)
Bi .f1 + +Bi .fn = Bi .fi N (f1 , . . . , fn ) / Ci [1/] A [1/(.fi )]
q
C YX S T bWC SCh TWXW@E P wpD&`pXfPg$v&vDIkDe%gRb%C Chk bWX S TEC DaCpY$`&v%l(q Chk bW TA $&D%VCqmGq Y@ uWWX P eD!Rb%C C h@XW eD"iC q Chk G Y bWC Wh Y G $vpCb vfCpT$vgP`fXe@ CpRbIPvmGDVXaTjbpTD(`aXxI@`%pCxI@iD`vg`wv(c!e!IwjbDRXP $w$eg@cbHVWDRX$Hh ThC q W PX Y G C CAh Y G PWXAC E@ bC TWt C b C G@u T Wh T Chk bW P T Wh PW PX f q $v!Ivu bWC TA Chk bW TA $l$%VC(mGDq C C S S C PC WC q G YCA T EWX P !s&%" s!qRb%iv(mG``fCfoRTgPvDVWgH`!cbC C T YX S &RSIPa`&QC qd Ch Th G YCA T bW TAC W EW DakIPVCVopTaP%`fW$vfCv! `fXpTbH`Xf@G ` RCgPvD$fWvDX RT$9D%VCgmG PW@ YCA T W G P Chk bW TA bWC S T E @ W rqC T YX S CE C h@XW $v&vaCfYeI@Gs"D&RSIPa`&yvqeD"yfCG PW bX `aXei3q DH fCT!RXaP`!$k YhX E P@Th bWC TAu E Y P C T YX S W PX T W $v!IvjaTb%cC6phpSIPa`&h dpCD(pfP!sr
fj /fi Tj fi Bi / /() / Ci / /() / / Ci = Gr0 [(nN (I )n )[1/fi ]] , Ji / /() Ji
A [1/(.fi )] Ji [1/] = 0 m es (Bi / /()) Ci / /()
(Ui Uj )
^ ^ Spm(A{T1 , . . . , Ti , . . . , Tn }/(fi .T1 - f1 , . . . , fi Ti - fi , . . . , fi .Tn - fn )) = D(fi |f1 , . . . , fi , . . . , fn ).
^ D(fi |f1 , . . . , fi , . . . , fn )
/ X
X
X
/X
X
k
X
^ D(fj |f1 , . . . , fj , . . . , fn )
q
v vwsp~x}zy!x Vv`& z x y | { y x { z | x { z y dx !mzI`dsy v"|vvxmzD"gR{BviDej{&mv{""m (!x 9Vv`&Yv` z v" z { 4 m qd Q43 g fY`DAyq x9bifCv!!&S h 4xfCVopTaPfP&!Dxpu"I@Gu h 4x9bge`a@RbD%&vaC$`XpC`HeB@ S !0vaCbDX9q VC e@ T X C C W W @ G n X W G T S @ b E W C W W @ Y @ W P b W C S Y A h E Y h @ h @ W C Y h E W ( G `Y WX P Y C XXW W W Y@W P bWC S YAhXE Y bh b Chk bWX S P PC E bWC Sh Y `(p%aCpYDXE c9bm@U h 4xge`a@Rby$v&vaC`v%aCq!DpX$saCpY$`&sHRhaP%!mVUTq$%`ate @ 75 j}jlV) 6qHdHcd!iRb%C e% d(cscb4$k HI (()4"!`D@h "X W TWt C b C P Y@E TWt C b C PC Th C ! W b bW P Y G bW boX WXFPuh Yt hh@ bW P@ W b PW b Gt P G YhX DVXaT"g@D%RCpuI@gD%VCaT!``%%!Da@Dx!e$ea@pPDs!q vC H`fXpT"g@pYVbl%fCgDH bfEaTpYRbs&pSfPpD&iy$%%pCfYeI@GgRb%C &RSIPa`&C q P C T YX S Wh bWC S T E P C T YX S pTP ECA %`B@ P bWC YCAW E bWC S P P Y P PC Go S PW G P Chk G %CaD%`atvD$DX$v&%pCbIEaTpYRb%fCpT%Ru%!xfC%RCHvC 6Rb%C $iVCfGvaCb Y W b Gt phP `fXpT"g@pYVbQI@G H`pX`i 6qH9Hcb PW bX TWt C C P h@XW W !9Rb%C eD"v`X q phP fC`Dv$k Y TXAWC Th W b bW P Y G PW Yu PWX `fXpTea@$vpC%auI@0`aXv!IpT`l 6qt9($c&$%%$fh fkD`fX`(`aXb&fCVG%VCaTD%RCaP%9DakIPxfCw`fXpTea@$vpC%au9!$v&%$fkD`fX`(`pXRb%C TW C b C bWC SC Tt GX b C G bW PC Ch Th TWt W b bW P Y C bWC SCh Tt GX b P Ch $k pTpPHewP`$gkYe9vC DVXaT"g@pYVbpCpbbC&!v!Da@D0fCdD`vaCwvC Ph@ WXh @ S q W b Gt E C Cuh Yt G bW@W Y W pC%`fte@ 6`DpXaYbaCs q Yon G G CAh Y W r YhX DH ECA P Y P PC Go S PW G P Chk G `B@VCaTvupS %!ifC0vpCvC 6cbC $VCfGvaCb Y phP VCfGvaCah"DVXaT"g@pYVbxh iD!pCD(RIP! G Y b@W W b Gt CW EWX PX T Wr ssq b vC F ECA `B@ P Y P PC Cu Y VCaTvpu%!i%&pSDX pC%`ftem@cUH`pXf@VtpT$v!IT vP Yon G Ph P G C bWC pC%n`ftem@U @ `fC`fXpTea@$vpC%au!scbC Yo G q TWt W b bW P Y C P Cuh Yt Yon G G Chk Y bWX S CG YuWut P G %!Da@DlaCvDVt`@U I@$%pCaY$`&!!eg@v%`pfh a@%RC$mG%DVCWDVXaTb"g@D%CRups!Rb%C aCvDVt`@U f@sD%pCaY$`&g $vfCvaC`TlBqp(cp!Rb%C $$vpC%auH`VtpT% Y P q T t W bW P Y C P Yon G G Chk Y bWX S bW TA Y EC TWt C b C P Chk bW P Y PWX Yu q T E bW@W b T@ S PC b G P Y G bW YAhXE %fYeI@Gi$eHvpC$fWe9iRb%g"g@jbph%aufC!F $vaC`v%aCY P G Chk uWWX bW b %fCwD`!$`a@p b vC fP%`fXpTea@}fWe9`! %!`fXpTb%pCaP%pC$fWT DHp%aCaY`XE GCWW b Y PC T@ SX PC W E Y b G WX P Y FP G C vHVhy
G a1 ,...,n T1 1 . . . Tn n 1 ,...,n 0 n r-1 (nN (I )n )/ /() r (K Gr )/(K Gr+1 ) Z F
^ D(fi |f1 , . . . , fi , . . . , fn ) Ji A n nN (I ) F
G
/ / (nN (I )n )/ p : A {T1 , . . . , Tn } /() k {T1 , . . . , Tn } Gr n a1 ,...,n = 0 i=1 i r - 1
X
X
Y
Y
nN an .vn (nN (I )n )/ /() (nN (I )n )
nN (I )
X
K
n
k
A
an k Fr
p
A [T1 , . . . , Tn ]
Ci
k vn (I )n
Ti
k
fi
X X red : X
(nN (I )n )/ /() Limn |an | = 0 an = 0
X
/ / nN (I )n
X
X
X
Mdl(X)
K
/X
X
/X
k / |X|
q bW@WX P YX l$e`(RpCaY`vE geHea@RbgRb%`DX ss$IkpT%vDwaCI@qRb%C %DVXaT"g@sfWe9D!wC q Y@W P YCAh G C Ch YuWut Yot G P GCWW b Y C T@ SX G@u `%!IT 6G W vC C b GE PW bX G@u T C S SX bW YWCtW puef@v%u 6G `aXeY6q`%!I&&`vE $eg@!%`%C ECA `B@ Chk PX P W r $RC$HRhsq Y P GCh P Y C YXW CWh Cht W r GCWW b Y C T@ SX Wh P phiVCfGvfpT%aug&pSDig!&sDqv`fXpTea@gfWe9D!xyRb%C X P@E G Y T b C b W P `pDvifCsvaCjb`a@aYg!l&phHmG 4}}v s& vwss~x}| v x y z y xvmz { x x z y x { e I{")z"!"!yv D Vv`9!mzIDRd yy ` Vv`9y!"m!" D&("9"D!x z9Db(Iv!x { y zx z x { z { He % m lclss u cqVpTbcCRhvsaC`aXiRb%IPvfCpT$vpCaP%%fCv&!&pSfPpDg!DpXd$$v&%pC"I@!&fSijb!%!%sr E Y P bC Y Y PC G bW PC P GnX S C C T YX S bh b Chk bWC S b Tu S T Thu WC W j}jlV) qC S S D&%VCG h CAh Y G CAn E@ TEC C YX S T PC b Eu W P C G T S@ Y G@ P !d`%pCI@DvgD}v%lHq spDDRXIPgcbsau"I@Gi`pXibwVCVopTpPIP&!`qgP`fXegcbC G@u e!IT q bWC S Gu W Ct@ G P $v&%Vu%wh 4&``&fST 6Rb%C Chk G Ds%pCb pCRbfP!hpmG4q pCv`ftem@U f@9sgPvaC"g@v%`w% T C Yon G G C Yh b YuWut PC bW T $vfC`X q TWt Yon G CW ECA DVCaC%`ftem@U h %`B@ bC Chk Y PX P EWX bhC W r q G@E G bWC S G@tu PC uEWXW $%RC$HRhdD!s!%Hs(DVC`v!VXg$v&%VC`DviRb%v`vu 6fG C Y@ CW E P G Chk bWX S TEC Chk Wt P G GnX S Wh TW boX YhX hh@u T P G bWC SC P EE P Y b E !0VCaTRb`!DXRCxI@D0pCaYDD&v%lq $fhfCv!!&sVYvpC``( e%f%fC$v&vDVApTpPC%RhwvaC"I@G%u 4 gP`fXeRhmG}q Y G@ b W P Chk G b YCAhX bC TWt C b C $wfP%pCggPRb%`Dwv(c! aP$vaCv`XmEVURTD$e%fs%!pCfYahpYRb`vdv(`X bW Yu P@hk hh@u T PC T PWXE bhC W F~ !F 8 sep6q(c q q Y@ r TWt C b C pbvD`}`%!IxspCaYD%Cd! !wVCfopTpPIP&!`D%&vpCef@v%h DQyVWD&qRb%C YCAhX G@u T Wh bW E C C G T S@ bWC S b GEu W hk Wt P TX S P C k PW PX P C T Y@ bWC S YAhXE Y W D h gH`pXD(phbC !4 fPaTe} $v&vaC`v%aCh jbDX `v`vu 6d!fCpTpbed!DXRC&I@0DRXaP`pr T quEWXW G C Y@ CW E P G PW P@ q Chk T G ECA@ W T bWXE Wh PC EC $jb`@dVCs%DB`fXpTbfE`g@pY$`vfC gcbTl(q C Ct@ G C YXC W !``&fST 6d!ggP`%!vC bD%&vugE`XgaPDVX`scbC $fW`(C q $vfCpT$`vE $)%pCb WC S Y Y G@ P b TX bW bWX Chk G C u Y b TX Wh YCAh b bhC WX PW@ PWC P ! &pS%C$VWDHivD`aXpYs!v(`HF HepC%!wRb%C C S SX @ Su P h u Y b TX Wh bWC S G Y b@W P b TX &&`lq 9vgpE&pS%C$VWDHs$v&%VCfGvaCpheRb%C $fW`(C q !De9VST i`sDVb`pX0)$k C Ct@ G PW@ P@ T P CW Th bW E Ph P Ch Tt GX b CE@ P G C u Y b TX Wh YCAh b bhC WXF P YCAhX W YC S T CWh T b dD%CDR@VRU`pX$Ik`DVXDHDpXv`p%C 0! aSvCw$VWDHwwv``aXpYw!v(DHDpCb IEaTpYRb0pCpbvD``fXRTgPv&&VSdVbea@vu T f q$v&vvaEDXH`!D9bbVpTbcCeaX4v&aS`X TWt bWC Su Y EWXFCW W@ C E Y GC Y @ Su 9vgaEmPU Wh Y 9g@vpCP FP@E G P E P FbC C W qC T YX S T Wh P Chk Y bWX S C Y W P %`vVCdRb%C eRT4%v(vs(`&pSfPpD&`pXfPwRb%C $w%pCaY$`&!a@9phmG(q C a``a@Rbqpbv``X 6%`B@ Y@W P YCAh G ECA C W E Y b G TA C bWC !iDVXaTbRCgPvaCDfWT 60 f@dVtpT$vfT vP FbWC S TE Y P G $v&vpufPv%aufhgbC
C W b Y E CW PX T W r C P PC h@ TWtu !Q`fXpTep@IPpTDpXb`@dh 4pCD(RIP!s(q !gHRhaP%!efCv! W E P G C Ch b@ Su `fXpTbRC&I@!$fkaT"9vgpEP CHvaCv!D@ s$`vaC%vaC!`X EW Yu G bW@W Y WC hW bo C u Y @ Su Ph P G PW Yu PWX ! aSvC9&%gaEPRU`pX&VCs`aXvfpT`lq ECA %DB@ C Y G C P &pSDXI@!yRb%C G@u G e%fT 6"F C S SX TWt C b C P &Dl$6qqHcq!yRb%C G@u `%!IT 6G F TWt W b bW P Y C bWC SCh Tt GX b bW DVC9`fXpTea@$vpC%auw!q$%%$fkD`fX`(`aXl$`pXP bC &&`lHq C S SX W b bW P Y CWh T `fXpTea@$vpC%audVb`X q eiVb!fWq%aCvDVt`fG @U Y@ Th T P Yon C C T YX S !x&pSfPpD&9VCG aCeW s%%fCfGvfCpT$vpCpPd%$Ik`DVXDHDpXw%aC%`ftem@U !v bC %`B@ bX W r qP G bW PC PCh Tt GX b P Yon G hhC ECA b vC Chk Y PX P bhC WXF G@E G bW u W PChk $%RC$HRhsv(QD`fCe%!VXi$eg@b`fXpTRb%D9@ T W t P C h k Y X o b W X S W qgRb%C $aDH`@ daCpY$Dsr 4}}v s& vwsp~x}zy!x x y | { x z y z { z z x `iVyD%9!ggzim"fy Rge`sia|&`vzvB!e y { x { { { "`!mgzIDdsygzsz gg"VsVy{gvBHgz y { x || w z y { vvB`xUv y{vvg v&e"&vgv `0(Iv&I|"9!x z" { j}jlV) q G T @ W C v b u W d aCY X W `fCfoRTaPIPS `i$b v&aC"I@mG%E Q`4vgvC S h E Rb%gVCVopTpPIP&!`saP$v&%pC"I@GQv!vpuD(DXxI@$sVbe@VCwbDRXaP%!mVUT0&&vfCfCieHgaPDVX`s!fhGDXs4q ph9b@ PC P G T S@ bWC S b Eu hhC C C PX S E G Chk T G C Ph PC E C S S G G Y@ Y G@ b EW E W r Y P CW W Rb%C v&pSDX0&%gaEPU fC$$v&vfCVGv`@Vb``Qs(q P GC Y @ Su G Chk bWC S TE TXA W r PW Yu PW E DpX%!fpTDXbC VCVopTpPIP&!`e%fT 6%C G T S@ G@u G W C b GE PW bX !&puef@v%u 6G H`pX`i q TE P@ Y@W P GCWW b Y bWC S YAh E Y Wh Y@ YCW W Y P bhC &VuvpXaP`geHea@Rbpv`fXpT"g@s$%%pCDXpCdewv&a@pC!v( bWC $v&S UaC`v%aCwC q P C W W @ G C Y P @ S u YAhXE Y PC Y@ C T Y@ bWC S YAhXE Y W Vb`X T f 4}}v s& v9bIPv&pSDXl`9vgaEmPU %!geH } fPaTe4 $v&vaC`v%aCh x y | y x sp~x}Vv`9!sz z y z { z x y { x y { y e`eVy Rl"D `sHssa|`zz"!")"`!g"VemzI`d y `Vv`&!x z" y x x | | w { z" { !xVyv"{v9"((!d y"Dx%Db(IvR !x Dm yR c`yss ` { %D%"| z 4Vyv svgeVwm"a6~zycglz yx y z y {"$pm"g9%v"m|``j{mz{eBm }!x z" z { { m qcysy q PW@ bWC S G@WXt T u YC S C e$v&%VC``DeI@!`atv&&VST ! PW@ He Ch b@ Su P EW Yu G bW u C S S $IkpTe&%gaEC%pC%!D@ l$eg@bs&DXE Y PWC bhC W r GC Y @ Su q%RC%H!%Hxs!q %&pSDXd9vgaEPU VCG a@%pC`xs%aP$%%pCDXpCY Y bXW W rq bWC S YAh E PC TX E h bW@WC u T PC bWCuE Y C S S G h CAh Y G PW@ C bh W E PW E G Chk bWC S TE bWX S W !jhDg&!gpC$eHv(v!fWcb$%!%v%aul&&vfC!qDvaCsf@)`gvpufPfTfGpT!0`fXpTbHphaYcbDXwf@q$s$v&vfCVGv`@lpCaY$`&sr j}Vyf q ChkG $vaCb u Y GC Y @ Su Ph P Wh bW@WX P Y &aSvC(v&aS`Xd9vgpEmPcUH`pXD`DHp%aCpYDXE GC Y @ Su P G Y P G@u bW Yu E h@C T Wh bW TWtu %aS`X9vgaEVCph4e!fT 4g$vaCvDX`%vpEIPe@g$vpCpPIPv! PVC$gfY``"I@!%&&VSiRb%4mGq G Chk TXA C b Tu S T PC PW@ C YCA T Ct@ e !dpCaP%`VWdDe9VST G aPDVX`RC$HQsHq Y G@ PX W r C G@WXt T !wfCeH```f@I@G ECA %DB@ u Y C XXW W Ph P h@ bW@WX P Y &pS%Ci!Y&@RUDRXel$``(p%aCpYDXE C G@u G PW Yu PW E ! `%!IT 6qDpX%!fpTDXF bC !DpXq`r bh b YhX q C ! bWC S YAh E Y W $%%pCDXpCh Vb!!xv&sq vC ! Thu WC W r b C bC PCW W@ bWC S YAh E Y PC CWWX P W %&paP$v&%pCDX%aCp!iD!iRCxsr 4}}v s&
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
an )
f1 A
b1 , . . . , bn B ai A (a1 , . . . , an , N ) A
w Z/ /()
Pn / A /()
Y = Spf(B ) k
(U ) Spf(A ) Y
(X) Iij Z
X
^ X k Y
I = p-1 (0) ai A
B = A f1 |f2 , . . . , fm (f1 , . . . , fn ) A Z
X
i
Z
X
X
X
^ (Ui k Vj = j ^ X k X
A
Y
Spf(B )
f
f
X
U = Spf(B )
Spm(B)
/ / B p : A {T1 , . . . , Tn } A = A [1/] Z [1 : a1 : . . . : an ]
B
XY
f: Y
(g1 , . . . , gn )
^ Spf(A k Bj ))i,j i
X
k
k
X = Spm(A)
w
Ji
X = Spm(A) UX w Iij /X Spf(B ) k w f A s x
w
Ui Ji O U
(Ui = Spf(A ))i i ^ Xk Y ^ Ai k Bj Iij Iij ^ Xk Y
U = Spm(B) fi A U (Ui )iI Z ^ X k X k
I OX
/ Z/ /()
Z[1/]
Z=XY
Ii = (Ji )|Ui Y X = Spf(A ) Y = Spf(B ) B A N B N N .bi B
B = B [1/] I[1/] Pn = Proj(A [T0 , . . . , Tn ]) A A
^ (Xk Y)
U
Z/ /()
A
A
U U
X
A
k
/ Spf(A )
B
{x}
U
XY
Ii O U i
^ A k Bj i
Y
(Vj = Spf(Bj ))j
k
s
Mdl(X)
g1 , . . . , gn A k
s (Z/ /()) X
e: A
X
Spf(A {T1 , . . . , Tn }) Spf(B ) Spf(A )
U0
Spf(A )
X1 U1 X1 X / B x X
^ (Ui k V) (X)
Z
(T1 -a1 , . . . , Tn -
Spf(e)
Y
X = Spf(A )
^ V i = Y X Ui
Y
U0
Spf(A ) A Z
Z[1/]
U
f X X
j}jlV) q G YCA T PC d q DVCVopTaP%`fWRb%q` Dc Chk Pn bX G T bW E EC qC TA PC d q Y@ Yot G bWX $pC%ae($ ijb!pCaY$`X)Tl}`fiRb%` Dc`pC&I@iDD! C u Y b TX Wh Y G@ T Yh b TX !&pS%Ci$VWDHaPDVX`sjbpWDX $fW`(C q PW@ ChW bW E P@ P W GC FY G Y@ W ECA@ C E XX `&vpC$DXd`Rb%C &VCfGv}vfCVThIEpTpbes4q %`B&!ITWYvE fCeI@Tp&aCiI@G G Eu P Yot F `vaCDDXRb%C PW@ ChW bW E P Ch YuWut Yot $fkaTvv`daC0I@G ChkG C TWt W PW b C CW ECA $HvaCb VC`fXRT%pChgh %`B@ C T YX S EWX ThC C b G bW PX SXE CW &RSIPa`&D!ipCRbfP!vmGeq ipC"I@pC$`p@$H&`vh 4 CHvaCv!D@ g`wRb%$k EW Yu G PW@ PC Th PW@ u Y b TX Wh T q T bWXE G PW PX ` &pS%Ci$VWDHsjbDX `aCfYeg@pY$`vVCi`pXD(h q !s%aC"I@g%aC$ea@D(DXE C P b G P bW PX S P C E W Y u G P W @ P C u Y b T X W E h h k Y b W X S C bf W P !iCHvaCv!D@ sHelRb%C W !&pS%Cs$VWDH9h`@ $s%pCaYDD&i!l&phHmGeq PW X `aXefb q Y P Y Y P phiaC`aXyRb%C Th PW@ PWC ECA Dk eRCHv! %DB@ W b@E E@ S E CWh YCAh b bhC WX DVXaT"%HjtaTbD&`Xg%`DpXaYs!%HDF epHq Y@ r Y P P Y Y phi%aC`aX b W C P b G P b W P X S X E P G C h k Y C A h Y @ A W r q G Y C A T T P d q C h}k W P X P F b W C S C h k Y T E $`pXP !9%aC"I@U pC$ea@D(&`vi%fC$xvD`aX"sq`fCfoRTgPvDVWxVb`pX9"~`Rv$9P' `pXD(phyD%&v$!pXVv%u S qb G P %"g@jbRhpuY fCG `X qe Dc` Rhx"D$%Rb%C $0$%%VCfGTD@Qa@%VtpT%$0vaCbVCxC q d q Y@ Y P uXAWC P Chk bWC S E YC YuA Yh E G b TX Wh CWWX $fW`(x`! PhXW Ch YuWut Yot G Ct P@ Y@ Th DDIkaTv%`pCHxI@i0`ea@aP`we$k fP%aS`X`P 9%gpE&!&&RSIPa`&dVb!!w%s q GC Y @ Su P C C T YX S Wh Thu WC W r WC W b b TX G bW@W bWX C G Eh T bW PX SXE G uh G@ Y Y C Y YE G Y@ %HvpCvu %P $VWDHfC"F $e%pC$`vE !gVCVoaTb!puaYfYypC$`p@$H&`vdf@w$fkfT`qaubaC`aX!pC%pnVbpTvfCpepr q C T YX S Wh WC Ch TWh Y TW@ S CW W pSIPa`&&0%$fklaCvfne9gh 4%pCbu %P C PX S vpuD(DXE @ q P@ Su C C T YX S W bW TA Y b TX C qu Y b TX W D&%gaEPU !w&RSIPa`&Qh q$vfC"pX$VWDHwlH`&aSvCq$VWDHh b C TWt W PW b C CWh P bW T qPu Y b TX P G Y P C E Y P PC G hk Y bWX S C Y W P T F YCAh vsC F !fCDVXpTvaChxRb%C $%VCDX f %%&pS%C&aP$fW`(9%fCxph`fApTb%cC6RhiRb%VCfGvC $xvaCpY$`&!g@9RhwVGpDpCpbvD`X W YC S T CWh PC d q C S SX `fXRTgPv&&fSwcbie Dc&&`l q Y P P Y Y bW C P b G P bW PX S E P G Chk YXo PW PX Rhs%aC`aX$DRXP !w%pCef@g%aCD`p@$H&DXw%fC$g`(e@ 4H`pXD(h f q"~`Rx!VCpD&`pXfPT 4dpC%pnVbpTvVCvaCpY$DpCRb%aCmG!qeI@VAaTpYgRb%C d q C T YX S Y YE G Y bWX S P Y G T b P TDk h C P@E G Y T b C Y G@ b W P qbWC S G@E G C YuA P YCAhX W YC S T CWh PC d q Chk T !w`%fCs%pCVbeg@pY0!gP`fXe&phmG(%$%%VC`v!fXVtpT%pC0pCpbvD``fXRTgPv&&VSdRb%" Dc$dVbe@C q b vC bW@ T P bW G $e"fARhp$vfCVGDXE PW b@E bWC T P G GhX DVXaT"v(jtaTD%!IVCdfX!!&S UY vpCpCP PC En P qPCW W@ YCAhX PC Y@ C g%H%C v9baPpbvD`p!seH ! $%%pCDXpC9`!ipC9`"Feg@vW bWC S YAh E Y Wh CWWX P WX G Yu Uut P@E G YhX r d q q C S S G G TX v`4DvyfC DvqeVDVD&&vfCsfCfY``A Yh EW hbW bW P Y Y C S SX vpCb`Xd D`a@$vpC%au%pCs&&`vE C W TW u G T b `fXpTVbtif@DTIEypCfPfTfGpT!h Wr sHq bWC TAu C T YX S h@ WX P Y E C En b EFCW W@ P Ch Y d q $%!f%&RSIPa`&exDHp%aCaY`XdgpCRbvC!`&Rb%C $akaPDX q` Dc pCpbvD`Q`fXRTgPv&&fST xg`(e@ 4gjbahpYRbDXsH`v(ph$%%VCeDpXgRb%wpuVbVTfG`a@`vpu!wpC%pe( YCAhX W YC S G YX o T PW E W rqCh YC P bWC S Go@o Y PC To Y@ P C Pn bX 4}} v s& | !gxp(g|p(qz !xQm"V}yzw!v DVv6{v"vme""gm!"m{dv&vV!vvsvg{&I!|"9"vI!zd!x z y | {x y zx { z | zx z z yx z { { x { dq ` Dc Dej{v{""m { z { x { x { { `xH"}"gmz"!" "{gv&}!mzIDRdy %Hs z z | z z e6{eHvv(y`svv"|v&vgv !x vy9"l9av !x z { He b W @ T P W P X Y G C ` o P s h @ yP &pC$e"fARhQ`fXpTVbRT$HDpX&f@d!fW pXm%(CP lH`RW aXcY e`gDsisXh T j}Vyf q VYaT`vi`(wpCfYea@pPC%l!Y9bm@U !daCDVCgRhfCVGv`@sC%pCaYhpugeqgP`fXe@ YAhXE YhX T P EuW PC XXW W C Yo SXW G Y P TE EW Y E Y CWh Y@ Y G bW bo P C S YXu b qC T YChC WC P T G bW P T u P qCX u S C S" S G Y@ W Y P@ CWXE P G bWX S W $vfCpT!DX %0&%pnDvaC `vfEvaE%v%i%puaPfP`f@$DRXwfPfGeg@b!w%C `!!ab&&v#if@weDVXaTRb%RCaP`!`v%pCiI@aCpY$`&sr q dfCv&iRb%C C GnX S Wh P C T YX S &pSfPpDDRXIPT 6G C tW G !w`fXfCG bC $eI@VGDXpCvvaC!`qv&aS`X&%gaEC q bW G E Y WC hW boX GC Y @ Su P Chk Y PX P bhC WX C b E G Y@ $%RC$HRh9!v(D)F !au"I@G%u eH YCE G S Y Thk P T vv`I@&vaCi9aCRbVb$}FpTVW q ChkG $svpCb TWt C b C YCAhX G@u T Wh ThC T q q Hc0!wpbvD`e!IpCRbfP!vpVG(FB 4F 8 Y@ r `q G@u G C bWC S b E G P e!fT Q$v&vaC"I@Gu 9cbC syq PW bX Chk Wt P G DpX`ieq $HHVh C GnX S Wh PC Th QfCv!!&v&Rb%$k C ! pbvD`X bC YCAh Wh pCaYD%Ci VCVopTpPIP&!`$%%pCef@v%u 9aCcbIP!hyfG `s DVDDVXaTjbpTD(`aXI@epq bW E C G T S@ bWC S b GE Wh T C T F q q W PX Y G Y@ r C GnX Wh PC Th !VC%!!S 0Rb%yDk T Y@ YCAh W hk P T C GnX Wh PuWWX PX P W r $ IPpT`} Rb%`DX h $lpTVW`@ ! VC%!!&S `iRC$HRhspq PW bX q q q T G YXE h Y G@ Y Gh Eu W Y P@ Y T S Y qC T WXW P DpX`iA$`s`f`aCVY`f@fG`aX`vv!gP`fXea@%VCDX! `fXpTpbvpCpPDipC%Vn&%pC@ `!IA `cbC Chk YXo PW bWX $g`(e@ 4DpXaY$` q `(dpCfY!!VC"Qs(q DH4`a@iRb%w%pnDvaQ!uDvu wDpCD(phQsHD!Y9bm@U Y h X T h u G @ A W r Y h X T Y A P C C S Y X u b h E W X W G C h k P X P W r q C X X W W W h ECA %DB@ bC aPbD&`vEVURTD$&aPpbv``v!x` &$v&%pCDX%aCD!pCD(phsr E@ SX P@hk YCAhX hhC Y@ C bWC S YAh E Y Wh uWWX PX P W qC XXW W Wh P X P@E G PW@ T YA PC C S YXu b D!Y9bm@U gcbC `DvfCe}`a@iRb%d&van`%pwC 4}}v s& vwss~x}| D&VyD%9!qz"v!v x y x { {x { e I{") z"!"!sv x x z Vv`9v!zIDdqy DVyD%9v!x zv`iegRg{"xzvB!e!wz y x { { z| x x { {{ { yx | z y `mD%e(v~{!v6{zexcxH"p~zsRgqz {"$aea { z g"Vs %v"|vyz&mzD"gR{Bv&``j{mz{eBm (!x z" y | x { { j}jlV) q dau"I@Gu G Y `(saC!%%W mYpC q`4fXYgP q& ph mG C b E h X Y e@ sb W C G P ChkG D}vaCb TWt C b C YCAhXG@u T Wh ThC T q q Y@ r i(cg!ypbvD`He%fxipCRbfP!vfGeFB DF `Dq fCVopTaPfP&!De%fT h vC G T S@ G@u W W C b G E G P X P @ E G Y Y u P W E C b W P T F q q W P X Y G Y @ ! puef@v%u 6Rb%C `DvfCQ%pC%!IRTDX!&phfGi8 es`f` `fXpTVbRT$HDpXI@epr 8 7 2 1 # 9465%430)('"&%$#"!
(g1 , . . . , gn ) B
U0
I O X1
q
x : Spm(l) (X[1/])an /X Spec(l)
Spec(A )
X1
U = X 1 X2 U X1
k
X
(Xj / /())
X
U X UX
x
Spec(A ) Spec(A ) X
((Xi Xj )/ /()) ((Xi Xj )[1/])an
X
U1
x : Spm(l) (X/ /())
Mdl(X)
J U2
U X2
Spec(~ l) X
x : Spec(l)
(X[1/])an
Mdl(X)
X
Z = X -X
X
/ Spec(A )
(Xi [1/])
(Xi )i X / ((Xi Xj )[1/])an
/ Spec(l ) k /X U X
S = Spm(A )
(X[1/])an I X
an
(X/ /())
U1
U1
(Xj [1/])
X1
k
U0
/X
k S = Spm(A)
e : U2 J O X1 U2
Spf(l )
X
X1
X
x : Spec(l )
U0
an
/ (X[1/])an
X
X1
k
X / X/ /()
U
/ (X/ (X, O)[1/] /(), O)[1/] han (X[1/]) / (Xi [1/])an (Xi / /())
x0
U1 X 1 X2 I A / U1 J .OU1 = I.OU1
x0
Spec(A ) l A
X
/X j: X Z (X - X)/k /X l x x X A X1 X
Spf(B )
((Xi Xj )/ /())
k x : Spec(l) (X[1/]) / Spec(A ) Spec(l ) x x : Spec(l )
X2 X
X Mdl(X)
(U1 , I ) = (g1 , . . . , gn )
k X
X1
x
X2 = Spm(A)
k
X2
X = A1 A
(Xi / /())
U2 I.OU1
k
X X
X2
C W P Y G C TWt T C YXW G Chk uWWX bW b !`fXpTbIEaTpYRb%aC0I@w!VCH!VWw&pSD0f@d$d`g$`a@pq PC XXW W@ hhC C W TWh Y G C S SX %!Y9i!v!!Q`fX%pu0f@d&&`vE pCfYpTv%u YE bhC W r !%Hs4q ChkG $vaCb jbDX f q T ! C bWC SCA G Y bh b YhX $v&vDvfnvaCs!DpXs`( Ch $k Vbea@s}q T P Wr Chk PW PX P $DRX$HRhdbC Vb`X T C G @ h @ G @ u C D s $k W b W X S T h G Ch f q )`v&VCfSaT!RbT "`9i`%%!IvdIk`vhhWATC aC%f@PRb C dpCfPfA!QirP DpXaYDD {z" q Y !h ! W C Y G T T s P @ h C W 4}}v s& !x z y x { e6{e~x""l(!zHs6` g~x() y { y y y y| x "9Dd 9"`{ ij{yvv`eaV4z eg(e m sRc`yss TX @ W r wd hy dfY``As}q bWC S T E @ W r $v&vaCfYeI@G&sHq ECA PC G@u %DB@ % e!fT G aCesr bXW W dq $ Dc Y@ TW u C YX T S P CWh C PX `fCtw&aS`WjU&%RCw%pnpP$H h@CWW d q q Ch Y@ S Y G TX `%` @ qe DVD0$gke9vaCf@gfY`DA VCH!VWd&aS`WjU&%RC&I@!!Ie@ 6&iVCt!i$`pXP TWt T C YX T S P G C C T G P TW u bW P W P E T P G DVXpTfhGfWgVC`X bC F RC$HspqeHev&Rb%0%!Y&0gPRb%`D%! PX W r G@WtXE PC PC XXW W@ YCAhX PC u Y &aS`X C Go S PW Ph P G FP G C qbW Gt E uWWX YX Go S PW Wh P !fC0vpCvCcUH`pXVCq%Vhq%$eg@pYVb`XD!g`fC%RCHvC QcbC sgq Chk G $w%pCb C P G T S@ YCAhX PC Go S PW VCfopTpPIP&!`aPpbv``%!VC0vpCvC G pC`W DX bX W uWWX bW b qC T Y b Y@ CW T D!Q$`a@pq`!I`ftaTpuvVuaTe"AU h Vb`X f "D0H"vfeI""v z { { z z || x| z y yx | `zeDxw`g!}l9"ep6Vygv~{!v6{zexq geV"QwU y w x { z !"| `xfyey9"epfy m"} e` z w ` eVy Ry z" { z z x d x { d z { zi&vgv!x He z { Hssa|d`Ha%gR{BviY"`xYxD m Q4g j}Vyf q RCp`H 6G Pn bX Vb!%aCpY$DXT%l4q T bW E EC ECA GnX S Wh dp q q C S S G G Y@ bW boX W %DB@ VC%!!&gDYp `f`i&&vfCfCs`&$vfCpT!D`X G@u G bW b Eu W `%!IT $eg@"I@Gs q Cu Y Y@ G W b GhWW h G YhX bW@ v&aSvC&fCpTpbeQI@`fXpT"I@`@ 4 %VCfTDHd$eDB@ TWt C b C bC YCAhX G@u T Wh TX E W r H9Hc&!vRb%`Dse!fVbRTIP`gspqd ! C T Y@ EW Yu IPaTe} CHvaCv!D@ G pu`@ `X 6`X bXW W G YhX `( C P @ Su Ph P Wh P !bfEaTpYRbQ&%gaEPRU`pXRb%C Chk G $l%pCb C GnX S Wh T C T FP@E G P@ T W E P FbC C W ! VC%!!waCcbIP!h)VG(vDvVC`wjb`a@bu 0CRT}%v(vs(q h DX Chk Y bWX S PW G@ PhX DwvaCpY$`&`fXVG`sH`i q b TX hhC bC C WC bW T q Y TW@ S C S S G C bWX Su P Y@ Y T YC Chk G DsfP%pCb C ! bC aP$fW`(Qv!sv(vvi$vfC`X f `aCvfne9&%" &I@!aCpY$D%!pCVCaTRb`pC%VnpW%!9@ qW T bW `fXpTbIE!Da@aYDDXE CWh PC EC C Ch YuWut b TX G bWC S Y P Y rqPh PC E p q q C S S G G Y@ P b T T bW PW@ lRb%(T%lq !$IkpT%vDl$VWDHsVClD%&vugE`XlcbC s%DRXaP%!mVUTYp `f`0&%VCsVClel%aC$VWDcXIP!l$`pXP ` b P Yh PCu Y PW b P C bWC S b Eu W G T S@ bW E G P vC pCg%%aS`XR`a@aYiC q ! $v&vaC"I@Gh 4fCVopTaPfP&!DpCaY$vCVCscbC Chk Y P C YCAhX P Ch $pCpb`pX0sRb%`DRb%C $k Y PX P bhC W rq W b GhWW h G YhX bW@@ TWt C b C G@u T W %RC$HRh!v(9sDVXaT"I@e@ 4x%VCfTyDH D`DB}g(ci!(e!fih Vb`X 6qg($c!lRb%C FvfCpT$vpCpPl$eg@bu T TWt C b C P G bW PC bW Y T YC C GC Y @ Su Ph P h G@u ECA %VCaWvwlq v&aS`Xd9vgpEmPcUH`pXQ!}e!IT G %DB@ bC aSvCs`9vfgpEmPcUH`pXi!v! Pu Y P@ Su Ph P hhC PW@ @ W Hesr qd phfCpTfPDgd$`a@bfCVG%!IRTpu0aS`h9bpv&aS`Xi9vgaEwh Y P TX E bW u GCh P Y C YXW CW CW W@ GC Y @ Su P W b0!c9bm@U Rb%C C C XXW W Wh P X P@E h@ YCWC S Y P bhC WX C CW W@ Ct@ TXA Wh Y@ Y E G S Y Th q P P ``%`%v&eg@ipCx!v(vDpF !9biDefWRTIP`Dv&` %CDf@&%pCwpCRbVbs `aCbIEpTaYcb9Rb%C W P E G Chk G GnX S Wh T C T DVXpTfhGfWT 60$vaCb VC%!!&aCcbIP!h fG(F T q fCv!!&!DpX`( GnX S bh b YhX EWX @ W r PW@ C D!xs}q e ! Ch b@ Su P EW Yu $IkpT"9%gpE&CHvaCv!D@ G pTpPHe9pC`sDC%pC%!Df DRXP bC He !&``&fST 6G pC`wD4F ! fCv&S Ph@ bXW W r q EW Yu @ W PW@ C Ct@ bXW WX C GnX Wh YhXr D(q Chk G $y%pCb jbDX f 4DX%pCP !$v&vfCVGv`@fC`v%!$VWDHd%VC&vaC9C 4}}v s& T q W E h bWC S TE GhXEu b TX Y T S Y z { y z x { %mv`veg{jxy}{ R9"{zDp6vv(!yzy!x z y| "{"g~{(Da| z { H"| x { z z (!{"9v6{v` qw v{}$"xxH!x !x g(vp%D Re` vH"g9gz x z {| { y x z geV" y { z x v&%9lmz{eBm"v(!x z { v" D0v{}$p"&!lgz x z {| x geV"l `9}e&!qz { msRc`y s s y z {| x q d%&pS%CsfCpTpbeHg$v&vfCVG%pCph"HicbC C Cu Y Y@ CWh bWC S G Y b@W P F`v&aSvC&aubfupT`BARU`pXicbC Cu Y Y@ Ph P CWh P RTQ$$v&vfCe$vH`pX`iD!I`ftpTxaubfupT`"AU pC!DpX`( P Chk bWC S G@tu PW bX qC T Y Y@ bh b YhX Ch Tt GX b CE@ P $Ik`DVXDHDpX9v`p%C Wh bWC S GXE Y Y@ TW u Y G@ bhC W r q q q C S YXu b h E T CEWChk PW E CWh PC EC i$v&%VCfG`v%aCiesVYt!aPDVX`s!v(s} s`f`9&van`%ppCb%aCVY!vvDRuH`X0Rb%T%l4q !pbv``h 4 C YCAhX W W P GE G P `fXRTHVhvVWT cbC bWC TAu C T YX S G C Y@ P bC E@ S P@hk C T Y b Y@ CW E@ S $v!Iv&RSIPa`&fCF %pu`vpuwvwaCbD&`XmEVURTD$!fDVtaTpuvVuaTe"AU h 4wb`HDXE VUpT`$0pbvD`&cbC $akaPDX `fCeI@VAaTpYD&cbC !DVXaT"fhe"I@Q$i`a@Deg@e!QaCjbRhf@0`g@vpCD(phsr P@hk YCAhX Wh P Ch Y q G T b WXW P C W b@ G@A G Chk C Yt Y@ h T P G PW@ Y PX P W q6HHc!9vgaEPU h 4iaP$VWDHx%!fCVG%phVbeHDVXaTeI@D`aXpYvpC`pDVC`f@fApTaY9cbC DVXaT"fhe"I@$akaPDX TWt C b C @ Su W b TX PC GC To@ W bXW G CAh b Y WX F G T b P C W b@ G@A G Ch Y D%DIkaTv%`ggPDfW`(VVy`g|qH"pvv PCh YuWut b TX y | z z y D6v9Vv e(yze6{eVyam6zvg!"Ye&Dd & z z z x| dx { H"`fy9v q H"e g(vp%D RH{z"`g| { z {| z {| x y z | z{ " (H"D"mz z z { z y | HD%v"{"g~{(Da| Ry e(vx`aV"a|mg`|0`&{eDaegiVe`&vI{(gz&z~{xV z w { y z z y | { y y | w z z| x { z { vB&v"|vsv&v`"gR{Bv9``j{mv{""m (!x ze pm yR cic4w bW@ T P W bXW G Y b@ bW@ T 9aCD`"VAph`fXpTe0I@ldBsvaC!el}F hy $`BfAphP aCVY!!pXaYDfWT 4!vaCphe)Rb%!mGB%%fCe%fX%puvVuaTDpX)VCyaCfYpTv%!pDH)`H)$vpCxphl!I`ftaTypuvVuaTe"AU h 4%&aSvC aP$fW`(C Th b G Y b@W PC qP G@E G P b Y Y P G YEu YhX P@ bW P W P CW C T Y b Y@ CW Pu Y b TX P q 9vgaEmPU VCG g@vaCeQ`F v&pSDX9%gpEmPU d`D`a@p!DVC`f@fApTaYQ`W @ Su Y bXW WX GC Y @ Su Wh uWWX bW b q G T b WX Rb%C !`fXpTeVh`"&f@d$akaPDVX$Q`icb $k !$v&vfuv0a@%`ftpT%! %$%!%v%auseg@``a@`VCHe&&`l P C W b@ G@A G Ch Y G GhW WXW PC Th C bWC S Gu Wh YCW Pu FbWCuE Y C Yt Y@ G PW@ C S SX
( @F !F (F & C AUIG& IGDH
%Q! 6 H
4 2URQ6 ( ) C 4F
(
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
X
I = I1 + I2 I1 + I 2 = O X
{p2 } = (Z1 )X (Z2 )X (pi )X I1 + I 2 {p1 } {p2 } =
Y X (YX ) ZX ZX X = Spm(A) k Spec(A ) I J ZX X
P(Y )
ColimU Flt(p) (U, O)
k(p) = OX,p /mp
|f | f U X X
mp OX,p
k p P(U )
Y X p P(X) / M(X) m : P(X)
X
k P(X) P(X)
p
p, q P(X)
f
X
p
pX p (YX ) ZX =
= inf{|f | ; f (U, O), U Flt(p) et f ((U, O) OX,p )-1 (f )}.
mp k
k
k
X
X = Spf(A ) ZX J A I +J
f
X
OX,p
p1
I 1 , I2 O X Ii = (Ii .OX ).(I .OX )-1
x
k
p q M(X)
f
OX,p = ColimU Flt(p) (U, O) OX,p (U, O) (U, O) (U, O)
OX,p
OX,p f OX,p f (U, O) f U Flt(p) |k| D(f |) D(|f ) k p2
p
=0
P(X)
p P(X)
X
p2
X
p P(X)
p
X
p1 (Zi )X i {1, 2} Ii O X X
Spec(A /I)
ZX
X
X
X
X red(x) p P(Y ) / {p} P(Y ) = m(p)
pX
M(X)
mp OX,p OX,p
k
p = P(Y )
~ k
Flt (p)
k
X/I1
I
ZX
p P(X)
red : X
X/k
X
{q}
X/I2
mp
Y X
U Flt(p) Flt (p) = ColimU Flt(p) (U, O) OX,p =
(Z1 )X (Z2 )X =
pX
X
p1
P(U )
Ii
P(X) = LimXMdl(X) X / P(X)
{p} P(Y ) = YX X
p2
(Zi )X
q
.
f p>0 0<< f p k U
p
i {1, 2} P(X) / P(X) ZX
YX
P(X)
X
X
P(X)
{p1 } X X YX k A
(Zi )X
d q Dp VDc CWWX d q Gh Y GF Y@ bh D!`VDc9fCaS`X0I@!%pbeaCpY!`@ 0q Chk bWX S TEC DaCpY$`&v%l d q `s VDc Chk Thu WC WX DVb!!vQ`!F bh b YhX !DpXDH C S SX q G YCA T P &DlH`fCVopTgPv`fWgcbC Ch Th $akfP Chk bW T $D%VCA mG4q `(w%%$`aXDDRXpuvVuaT`aXdCx$9RC$HRhP s %%fCeg@$$`pXphpP!mjUvwpuvVuaT`aXdIP`aXpYdVCG YhX PCuAh Y bW P P b Y Y P E Chk PX W r qP T YA bW P P PC TE P b Y Y T b P Ch Y $akaPDX q IdfTfTT F F bh b YhX !`aXg`(IdfTT F b vC IdT P bW@ T P P b Y Y T b P G EW YhE Y Y@ PW bWX dpC$e"fAphg%puvVuaTDpXqIP`aXpYVCwCHvaCpYv%aueDpXaY$` q%%fC`d!dVYD`B!v( DakIP Rhw%pD&`aXv&w`fXpTb`XwVCd`daPvfCe"iRb%&pCVbphxpCpbbC4FpTDVXaT DH`pX`i P G PC TXA@ bhC Ch Th Y P PC YX S Yu S PW EW P G PW@ Yh G@A PC T P E Y Y hk PW bX d q DVDc Y@ TWtu efCv! dW b C C bW Y E Y T P G Y G@ Yn PW E W HDpXg!&pC$%pCaYhpuaCjbRh&I@lgP`fXedaCv!IRTH`Xsr d q DVDc W b G Y G bWC S G@t DVXaT"I@vaCf@g$v&vfCe$vu @ WX P@ bW@W bW E CW P G Eh T P bW PX SXE PC Y P Y@ou P C Wt T C S ThhC CW YCW W Y Th DFd `HiD`vaC$`XgVCfopTb!puaYVYi%pC$`p@$H&`v!ivpCpPDa@`vRC!"6@IP`X&%Vn!!%!0h v9bg@9pCRbVbs q q H F eqvau`cX"9g$`pXP !ggPD%VCvxW Xg!i%fCVWaS`%fCqp Y@ PC Y @ S bW C bW TE C E PC P TWt T PC YXW P G q F F bh b YhX q `pX`(qs q D F Chk Y PX P bhC WX YCW W Y Th 9$ivpCD(ph!v(Q`F %9bg@0aCRbVbs q PW@ b He vC F vC W bWCA G Y $vDvfnvaCRCP bC F $%pCb ChkG gP`fXei%`DpXaY!%Hs4q ! Y G@ YCAh b bhC W r C pCfYeg@pYVbaoe$v&vDvfnvaC!sr T T Y@ bWC SCA G Y Wh Cht W q b vC C bW E CXE PC PC YXW P G bWC S P Y @ S Thk GC Y W gaP$vfCTxW vg%!gaS`%fCiD%&vpCbfEaTpYRbdpCDcX"9$v%puh Vb`X f T dq $ Dc ChkG $ %pCb EWX T `jbDX q W vC pCYaDX 4VWv`g@ib&!DfGT DakIP `fCVopTIP$fA!Rb%C Y C TE Y CWh C S@ h Th Y@ G T T P P G o Y @ A h h C C S f W G X %fCeI@aTe"!%!9&`f`(C ChkG D%pCb Cht W r isHq F G YCA T P DVCVopTaP%`fWRb%C C S SX &&`l qRTIP`XgwvfCT0pCfYeI@e%pEh T E W o T G@ P W ECA `B@ C CE G G !0v`I@Qf@i aC!%pC9pCpbjb$k YW Y Th Chk Y PX P bhC W r $vpCD(ph!v(sq C C E Y C h W YCA G Y C Tt P C fWTD@adW%C vDvfCvaCd!Vb``@ %mG!q ! W Eh Y G C Go Y@ P C TE Y CW DVXaTb%au0I@d!sfC`a@`vpuwVWv`g@wh bC Vb`X T q Y`vCv`@I&vaC9VCxaCRbVb$0Vb!%auY vpCD(phf@%CDHv(svfCVT%RuHvRb%C @ Y E G S Y G Thk Th Y PX P G E YhX bhC W r q W G PWC P Chk PW bWX $d`aXpY$D {z" f q`ggEvgpCgea@VbRhpudfC`X q u YC E Y b G P Y G FY r sHq aujbfTe$vfWT 6g$vfCpT!D`!F G@tu G bW boX WX gPv`A vC Ca!%pCi$ea@IPe@yHq YC b Y W b bW T W Chk bWX S EC $aCpY$`&)T%l q bC fP` q T@ bWC Su Y EWX @ W r $v&vvaEDX&`spq b vC FYG vgP`fXi Vb`X T q Chk G $xfP%pCb PW@ ` bC $vfC`X q bW T !x$v&%`vfnvaCY h C bWC SCA G W q Y`( hX Chk Y bWX S C b W $vaCpY$`&&!&phsfP VGT phw`fXpTefhe"9Vbt $VYD`d`r f Y P W b@ G@A CWh TW u Chk TXA YhX q G YCA T P Chk bWX S TEC q G YCA T P `fCfoRTgPvDVWiRb%C $aCpY$`&v%l`fCfoRTgPvD$fWiRb%C !DVXaTbIEpTaYcbpC&I@IPe C W P Y G T@ bWC S Y EWX @ W r $%%ugE`X`Qs(q Chk Thu WX $iVb!%QD!F aCvfCaTvu!fWq$%%pCbfEaTpYRbicbC Yh Y T bWC S P P 8 7 2 1 # 9465%430)('"&%$#"!
D(|f ) Flt(p) / f D(f |) Flt(p)
q
k(p) a p . a-1 p = 1 a k(p) - {0} g (U, O) a U Flt(p) 1 2 |k | 0 < 1 < e p < 2 D(1 |g) Flt(p) D(g|2 ) Flt(p) D(2 |g|1 ) = D(2 |g) D(g|1 ) Flt(p) -1 |g|D(2 |g|1 ) | 2 |g|D(2 |g|1 ) | -1 a p . a-1 p 2 -1 1 2 1 1 -1 -1 a p a p. a a p. a p 1 p 1 p =1 OX,p X Xred P OX,p [T ] f P f P k(p)[T ] f P f p<1 P (aT ) P (T ) a k P (f ) p > 0 |k | 0 < < min(1, P (f ) p ) P (f ) P (S + T ) - P (S) - P (S)T T2 2 T =0 R(S, T ) OX,p [S, T ]
!
f
~ ~ ~ R f0 P ~ |f 0 | 1
D(|f )
M 1
~ | f n | 1 ~ ~ x U |P (fn )(x)| 2 ~ ~ |P (fn )| 2n 2 M n=0 ~ ~ ~ P (f n ) fn+1 (U, O)
.
(U, O)
~ fn (U, O)
~ ~ |P (f0 )| <
~ ~ x U |P (f0 )(x)| >
~ |fn+1 | 1
2M U
2
p
~ |f n | 1
f
D(f |)
~ ~ ~ P (fn+1 ) = P
f0 OX,p U
~ ~ ~ ~ P (S + T ) = P (S) + P (S)T + R(S, T )T 2.
P (S + T ) = P (S) + P (S)T + R(S, T )T 2.
f
~ ~ P (fn ) ~ fn - ~ ~ P (f n )
~ R
p
~ ~ P (fn ) ~ ~ fn+1 = fn - . ~ (f n ) P ~
~ ~ Pn (f n ) ~ ~ P (f n )
f
~ (fn )nN
~ ~ P (fn ) ~ ~ = R fn , - ~ ~ P (f n )
~ ~ Pn (f n ) ~ ~ R fn , - ~ ~ P (f n )
. 2 2n M
~ ~ |P (fn )(x)|
M
R(S, T )
R (S, T ) T U Flt(p) f0 P
~ ~ P (fn ) ~ ~ P (f n )
M
W To CWWXThk C %VCw`!$dl}q
2
W b Y @ S G bW boX W DVXaT"g@`cX"9I@iD%VCaT!`Q`X
~ ~ |P (fn+1 )| M.
22n .M
2
=
22n+1 .M 2
. U . xU ~ fn R p n
d q $ VDc bqes VDc$ea@fPfTVGaT!s C d q bW bh W
bWC TAu C T YX S h W b Y E CWh T C hk T bW TA D%!I$%dpSIPa`&!DVXaT"a@fPaT`aXb`@wpCRbfP!h4fGT $pCfY!&iD%VC%pCY Chk T DiaCfYeq G@u G W `%!IT vC C b E bXW W r !gpu"I@Gu G aCe&sDq Chk Y PX P bhC $pvpCD(phl!%H W r q GCh P Y C YXW CW TWh S C XXW W@ Yon G CW ECA s`fCVG%!IRTpu9&aS`xh iVC&!Y9&aC%`ftem@U h %DB@ Chk PX W $xRC$HRhP sr 4}} v s& zx z x !e&!"| {gv9V!vviv{ z yx z gg"Vs V"gD{ sxsH"mv y y z z x z D9g!vepv(vD D z { x z {| v" %v{}$e!x zDb}vvR !x v" 0 z { ` j}jlV) `aumq pY $`yRv cuS ys&%S VCG b W & ` cbP C C sS X C q Chk CAh Y EC q Th Y C S" S GC EWX PC G D0`DpX)Tl `pCVb!pu&v#fCfGv0`!dcb0fCfGs}q YhX `( P Th pCVb!puY P Yon G@ PC T G E G C YX S T P pC %`ftei!aCjb&fSVTDX9I@&dpD&`pXfPRb%C F W E PWXE Y@ T@ GhW WXW P `fXpTbahpYRb`viegIPe qd`icbC Ch Th $pkIP Rb%C P C!s`&fS!"9pe!IT 9$9pCaYDD&S vpCe9!Rb%C G@ Th@ S G@u G Chk bWX Yh bh@ C P Chk T G FCh YuWut P Chk Yh h b bW PX $jb`@&VC}`DIkaTv%`Rb%C $aP`cX"DpXd$ea@D(h q hh@tu bWC Su Y bW P T Yh bh@ C W b@ G@A C hh@CWW@ PC hhC P G Ph b bW 4!e$vg$v&vvaEDXDDRXfPfG!F %pC!`!`fXpTeVh`"w!e`g%!!v!gVCi`aXqf $DRXP b vC CSS &DXlq bWC S T E @ W r C YXW G C W b@ G@A C h@CWW $v&%pCfYeI@GQwsq &pSDiI@&!wDVXaT"fhe"&!e%e@ 6G PW bX H`pX`i q `4i`l%VC&vaC}e!I(vpCx(fG!fF`C 6q$`$IkpT%vDqRb%C pTP %pC!`w!Rb%C PW@ GhW WXW Y T S Y G@u T Gh P Wh @ T q TFCh YuWut P Yh bh@ C P Chk bWX S TEC $wpCaYDD&(v%l q b vC aPDVX`@ YG RT$$v&%VCfGTD@saCpY$DQsHq $fW`(wfCi$vfCpT!DD!6F`9VS!"94`%!IT iRb%C $akaPDX P Chk bWC S E bWX S W r b TX G bW boX WX G@ Th@ S G@u G P Ch Y q ! C b TX Wh P T@ bW boX W r bW Y@@Th $VWDH4pTfWe$vfCpT!Ds}q i$vfCpTpbee $k PW@ b TX Wh TW u @u G C YCA T Ct@ e $fW`(djbt! Ge!IT !0RCgPvD$fWDe9VS T q C T YX S W PX T W r C pSIPa`&h 40pCD(pfP!ws4q ! CW W Ct@ TXA Wh TX E WX C Gn S bh b YhX r GhW WXW Y T S Y G@u T W 9b@DefWRTIP`DVbpTfPDg`HF ! VC%!!Xw!`aXiDHq`svfC&vaC `%!Ih Vb`X T q Chk b X S EC $aCpY$W`&4Tl(q IPe q T@ Chk W T $Dfb %VCA mGq F TA WXW PC Y T YC E P Y@ `C!IiDqRb%%VCaWvwCv}pTevE @ bW Y@ qP`Hq$vfCpTpbeHe@ W GCWW b Y C T@ SX q G YCA v`fXpTea@sVW`9`!C `fCfoRTgPvDVWT Rb%$k ! D%&v`%Vn%pCYbC PC Th C bWC SCA G Wh EWX ThC H`!&pCRbfP!vpmG4q fF`C 6qT}FwDdvfCVTvfEaTRb`%cbpVG4F q GhW WXW Y Gh Y@ WC PC T T q h `X Chk Y bWX S C t P $vaCpY$Di!jbTDe@ vqmGq eDVXaTb%auYp@P vC PW@ W Eh b Vb`X f 4}}v s&f T ze&Dd &)"g`"vyv !v!e` y { y { z x zx d q | { }"Dqz)y vD Dc"Hs6` &e(xVDe6{e~x""D"g(H" RgBe&`{v(bg y z y x y g"Velv{v(vDpe&!lz y x z {| x { z y x x ` x x "g}"Red `"{"jx""`"g}"!pz b W &v t u Q s D%)fCme D@G %Q4r g CS @ W j}jlV) qC bWX Su P `%puaYDD&v!Rb%C W PX Y `fXTpVbRT$HDpXi@ q `%Vn%pC)$k C!VWv`g@`wRb%C He ! De9SV T q CA G Y Th C TE Y CWh EWX P PW@ C Ct@ @`(%Vb!%aupuD(ph0aubu WXF Th Y PX P @ &Dl4q C S SX aPDVX`i@s DpCVbRhpCpbbC!&pCVbfSfT0sRb%C T `Dpt%`W`vE pCbVRhf@G(%vVCSDXcbC Y G@ W r q T P E C T G CWh P qC YCA X T P Fb G E P CS X &&S`lsq @ X FbC C W qu YC E bWC SCA G Y G T WXE b @W b @ S E PC W`%v(vs4DgaE%g%aCY $v&vDvnfvaCfxfC9pCfYahpYRbP`vC!0$W`vaCDfWTe9xCffGT`@0cbymG q E YhE Y G CA E@ EC DCW%pCaYv%au0I@w`vng`Tl}q C$sjbT!%!vsr hk h u WC W d q DVDc PW b YX T S P G @ WX `fXpTea@DVW&gVC`!F
/ f k (p) pX k (p) ~ D(f , 1) ~) = D(f -1 |1) Flt(p) ~ D(1|f q k (p) A X k (p) (U, O )/(U, O ) 1 q (X, O)
d q e"VDc u&pSY%Cgb$VWTDHg!hDpXs X b b W d q DVDc
S
!
# #' 1 2 2 )('""v$(H)}$v$6#)HD$&y 6H
~ P ~ P (S + T ) = (S + T ) T
Vb!!`!F Thu WX
Y `@ bvC e $`vI@&vaCs Y@ bW@ G S Y W d q `BVDc fCaS`X0I@ Gh Y G C
~ fn T
~ R ~ ~ ~ (S, T )T 2 . P (S + T ) = P (S) + 2.R(S, T )T + T ~ ~ P (fn ) - ~ ~ P (f n )
VCaS`X0f@i$%VCaT!`Q`X Gh Y G bW boX W
1in
k (p) A = {f k(p), f p 1}
(U, O)
~ ~ ~ P (fn+1 ) = P
UX
~ ~ |P (fn+1 )(x)|
fi A
xU
~ ~ P (fn ) ~ fn - ~ ~ P (f n )
f OX,p
~ ( fn ) ~ ~ (f ) = 0 P
p
k (p)
~ R T
A [t1 , . . . , tn ]/(f0 t1 - f1 , . . . , f0 tn - fn ). ~ ~ P (fn ) ~ ~ 2R fn , - ~ (f n ) P ~ U Flt(p) ~ ~ P (fn ) ~ fn , - ~ (f n ) P ~ X = Spm(A) A X Spf(A ) f 1 ~ ~ P (fn ) ~ ~ ~ ~ = P (f n ) - 2 R f n , - ~ ~ P (f n )
k (p) = OX,p /mp
X
k (p) = OX,p /mp
p
~ f -1
f
k
/ O X,p
~ f
q = (qX )XMdl(X) p
p M(X) p M(D(f0 |f1 , . . . , fn ))
Flt(p) ~ (D(1, f ), O) X
f
~ f
A
p
~ ~ P (fn ) ~ (f n ) P ~
>0
~ ~ P (fn ) ~ (f n ) P ~
1 k (p)
f
OX,p
k(p)
k(p) = OX,p /mp
2
(x) M.
k
(x) M
.
k (p)/k (p)
p
(f0 , . . . , fn ) A / k (p) A P(X) X ~ ~ P (fn ) ~ ~ P (f n )
U Flt(p)
2 2n M
2 M
2n
1
/ 6 k (p) q
2
~ R + T
, 2
f -1 k (p)
~ ~ |fn+1 - fn | 2n 2 M ~ ~ |P (f )| = 0 f P
.
q 1 < q2
k (p) A
. 2
p
~ ~ P (fn ) ~ fn , - ~ ~ P (f n )
f k (p)
f0
p M(D(f0 |f1 , . . . , fn ) f -1 k (p) ~ f f ~ ~ f (D(f , 1), O)
p
q2
qX
k (p) = k (p) ~ ~ P (fn ) ~ ~ P (f n ) k(p)
f 0 , f1 , . . . , fn fi p
UX = Spf(A ) X Spec(A ) X
q1 q1 = q 2 k (p) k (p) A X
2
k (p) . p
Ch Y q W b@ Yu CWh P C T YX S $pkgP`X `fXpTe"VAaTv!9dRb%C pSIPa`&&C q Y@ uWWX bWC S G@E G PC Th `0`!$v&%VC`v!VXRb%p$k PW Gu o@ h@ T C C T YX S W PX T W r vfCfTv(ex!h eCaEfP`@w&pSfPpDQh 4dRC$HpfP sq Cu Y bWC S G@E G W YC %&pS%Cl$v&%VC`v!VX`fXRTgPv&S UfST d!e!IT G `B@ G C G@u ECA bW Yu E GhX $vaCvDXVC!!SU VCG aCeQ`%%!I`ftpTqpubfuaTe"AU ! bXW WXFPC T Y P Y@ C C T YX &RSIPa`&S Wh WWX bW b qP G bW Yu T PC G E C G Y bWC SC b T PC G C W bXW GFCh You G@ bu SXut WC C S SX iu`!qD`a@pDvVCfGvfCpT$%pC%(V!y!fCVG%Ci%VufTvaCl$%%&VSaTDfWyRb%iaujbpTaPfPfTi!xDVXaTeI@"`DIkaTvDVt`iVCaTpYvDvD9vi&&`l qde9VS!e&4`%!IQ`pX%C ev!fX`%eh 4v&aS`XwpuvVu&DXfCG `d`ftpT%!`X DX G@ Th@ S G@u T W P W G@E G h@CWW@ W GC Y b G S E Y@ CW Pu W G @ W vDX)F YhX C C W qCh You G@ PW P h@ Y Ght Y P h@CWW G Y G@ FY Ght Y P DHb}s`DIkaTvDVt`0%RC`0%VCfT`%pu0cbC e`@ gP`fXe vfCVT`vau0Rb%C Tf y { z { y | z x vgI~{``j{mv{""m D(vV}{eDaegRgevwVy z m"}v9vp6evs!ewv~{!`pv|pvgI~{lz)y{ R%VCfT`%pu44|aRCaPfPfTiz viz"ww z w z {| zx z yx y Y Ght Y G x { Dgxg`e H"4!"mvgI~{lz x z {| zx z y y0"{mDRd { z m!"ms!e|{ vCsRCaPfPfTqgz xz"Uv~{!`avsz "gVeg(e R{ %y%VCfT`%puz vlzeQ` ` zx z x W G { w yx y y x y WC Y Ght Y x { w z | z y | x z {| z { z z { {"9!xqv9Vyepv(vDs~x}"c `&94H"9!x v9``j{0v{""m }!x v" 0 myR cgcH4w q G T b P C W b@ G@A G Ch Y G PW P Wh C Y@t bW@ T P W P T G C Y@ CW Yt C DVC`f@fApTaYqRb%C !`fXpT"HVh`B&I@w$pkgP`fXvpCw!a`DdpC$`BfAphQDVXpTaPhaEfPI@!fCpTpbew!`a@DW u j}jlV) qC T YX S T W W To Tt P T Gh DpSIPa`&`pXfPh 4vfCgVb`e@ % VG6FW WXW PC T Y C Yo SC S G Chk bC P Y E Wh PC C Eh@t C Yo SC S G Chk uWWX bW b E Y P bWC S T GE PC Th `Rb%0pCVb`aX!9!aC0v&0fC$vHpDXwRb%0g`D&!0aC0v&&VC$`!$eg@p4cqVpTbcCRhdD%&vaCVY`f@vdRb%p$k
~ k(p)
XMdl(X), pX UX
) IDH D20( gR0( ( 1 C C ( ) ) )F )
'#6 )F &
bWC TAu C T YX S W PX T WXFbC C W q Y EW E C T@ bW@W b T@ S $v!Ivd&RSIPa`&h 4RC$HpfPD%v(vs!`aCfhGDX!dRuIPei$eHvpC$fWe9mG qT E YG %fYeI@GgP`fXe@ Rb%s"a@Vbph%auC q PC b G P Y Chk bW TA qP G bW PC C TWt C b C bWC SCh Tt GX $w$vfCmG(vVCfGvfCpT$%RCaP%wbHc0!s$v&vDIk`DVXDHDpXb PpC%`ftem@U RU`pXxRC!aC$eg@pYVbpCVb&VSfT`v&&`v`!iVbaTY%u %VCfGslqgdVb!pu!Y&@U xcbC $pkIP Yon G Ph P P P C bW Gt T GXE CWh C S SXE EWX E P G Th Y C XXW W Wh P Ch Th C Y@ P PC GCF C C W Ch T Tt GX b G YhX G SXE P vaue%RucbfCVG%%b}ivC$fk`m@U VCD`fX`(`aXI@yDHspCbfn&`vqRb%C aC%`ftem@U Yon G @ q !s9i`eHVWpTfPD`A C CW W@ Ct@ TX W h jbDX q &RSIPa`&0VCwD`vaC$VW`&`aXvfpT`l s`f`&van`%pVCweH`evwRb%C T C T YX S G bW@W b T@ S PW Yu PWX q q q C S YXu b G Y@ G@WtXE P VCfopTpPIP&!`@ G T S pbvD`X 6G f YCAh VC`vvxfCT!pXpPDqDvaCbDXxfCx$x$gke9vaC`)F &pSfPpDiVC0` %%fCfoRTgPvDVW$DRXP GhXE Wh E P@ Thk Yh EW G Chk Ch Y@ S Y WX C T YX S G YhX r qP G YCA T bW
XMdl(X), pX UX
gP$v!Ivq%&pSfPpD!v!gVCDVXaTjb`vdaCRbvC`pX$lH`aXpY$` jb!%au!Y&@U Rb%C $gvpCD(phf@%CiDH bWC TAu PC T YX S hhC P GF W TWXE b E Ph P Chk PW bWX q Th Y C XXW W Wh P Chk Y PX P G E YhX bhC W r !%H spq C T YX S Wh T PWXE q T F W Y P@ Y T YC G Y bWX S P Y pSIPa`&`pXfPT 9pCfYahpYRb`vfF`C 6q `fXpTpbvpCpPD&aCvfnaWv!f@%pCaYDD&xpCRb%aCymG q !dVWv`g@vfCgRb%C ! C C TE Y CWh W To P C Ct@ `e9fST q C E Y CWh W YCA G Y bhC WXF W G PWC fWTDa@%C %`vfCvaC!%HQ`vfCVT%RuHv fC`vQqRb%C GhXE Wh P C S SX &Dlq gP%VC`B0 ! fWT`g@dh 4l$vfC"pX fWTDa@0I@s$k Yh G@A C C E Y CW bW TA Y C E Y G Ch b vC C SW GX W Ct@ G P Chk Y PX P bhC WX &DVDHh 4``9VST 6scbC $%RC$HRhw!v(`F C S SX &&`l q W Eh Y G C G P C E Y CW C C SW GX W `fXpTbH!%auI@q!qfC&fSRTfWT`g@qh bq&`f`(h jbDX vfCVT%RuHvcbC $DpXaY$` T q W G PWC P ChkPW bWX qd e`fW @RU`pXfCevaCDDXRb%C h@CW Ph P G PW@ hW bW E P G@u G Chk T bh W `%!IT $TIEpCfPfTVGaT!Q`X Chk bWX S EC $pCaY$`&4Tl(q egRb%wpCaP%`VWQ`pXF PW@ PC YCA T W P Chk uWWX bW b $d`!$`a@p q !RCgPvDVWT 6Rb%C e ! RCaP`I@GQ@ q ` pCgPv`fWwdv&!`pfG4F C YCA G P PW@ C P E PW@ YCA T Wh bC S@ T C S SX &Dlq He PW@ pCpPDf@va@P DpX`qvC P GE PW bXW b Vb`pX!%v(vs q e9VS!e&}`%!IT (`v!VXRb%C `%` @ T PFbC C W G@ Th@ S G@u G@E G P h@CWW 4}}v s& x !e| "D Di$v9{ Ry a"" ya!6xe(99I|"9"vI!"gajx(i e&Dd & z { { x y| y { y y y y e%D{ v~{vpeg%Vdz x y y x z {| "g(H" RwageVs %v{}$e!x z" 0m yR c`yss { u qPh PC TE b G P Y G ECA@ W E Ph P G C T Y b W %`pXpP!mjUvgea@Vbph%auVCs%DB`fXpTb%pCPRUDRX&I@VW&aSvaCisr Vb`X T j}jlV) q Y G@ bW TAW vgP`fXei$vfC`XE pbv`` X q YCAh ChkG D%pCb q ChkG $ %pCb W EW Th G C bW h 4CHvpCRbIPvC dWg@pY$vC W TWXE GF q q W PX Y G Y@ r G@u h DVX aTjb`vI@!Dp DVDDVXaTcb$HDpX&I@gepqe!IT $fk bW W C $eg@b vC ! gPvaC"g@v%`0%!wfuT!RXaP`qv`fXpT"g@QVW`9`!0Rb%C Yh b YuWut PC E P@ GCWW b Y C T@ SX Wh P b!Y9bm@U C C XXW W Wh P xcbC Chk Y PX P EWX bhC W r C Yh bh@ $xvpCD(ph`!iv(sq !iDpX!`ibC vfCev!fXxcb`fXpTRb%$@ 4}}v s& WC G@E G PC W PChk y xvmz zIDRd)yygg"Vs { { y x || x v"Dxw %v{v(vD ze!x {| zv`iegRg{"xzvB!e!x z| x Dej{imz{eBm (!x z" ( m qclsy { { V(Vyf) q YhX `( Chk T bW TA Y G EFCh YuWut P Chk Pn bX G C q q DaCVYi$vfCvaCf@%C!`$IkpT%vDicbC $dpC%pe( ibDp `f` C S S G G Y@ r &&vfCifCy`$q RTqD%&vfCvpClb(pTpCRbfP!v0DVXaT"a@fPaT`aXb`@VCfGvaCiW u q e !``9VST 6G H`pX`i P bWC S Gh P C P ThC W b Y E G b C PW@ C Ct@ PW bX 8 7 2 1 # 9465%430)('"&%$#"!
k (p)
VarRig/k
^ Ax
mp OX,p
X
k
P(X)
X = Spm(A) ^ OX,x
fi (p)
Colim
OX,p ~ f OX,p P OX,p [T ] ~ k(p) = ColimU Flt(p) (U, O )/(U, O ) P ~ f g ~ Q (U, O )/(U, O ) g ~ g (U, O ) f OX,p
U Flt(p) pX U k
OX,p X Mdl(X)
|k |
f: Y
X ^ k k X k k X SmAfnd/k Afnd/k
U = D(f0 |f1 , . . . , fn )
f mp f k (p) - k (p)
(U, O)/
X
/X
fi p
f0
(U, O)
/ Y X Y ^ : Y df = f 1 - 1 f X p M(U )
p OX,p
< < f i0
()
k
k(p)
^
k
X
(1)
p M(X)
(2)
x
Colim
/
p
XMdl(X), pX UX
(U , O)
k
k
U
p M(U ) V = D(f0 |) D(|fi0 )
Colim
A
f
-1
(U, O)/
x
U X V Flt(p)
~ k(p) = OX,p /OX,p
OX,p OX,p k (p)
(U , O)
f
x OX,p
(U , O) /(U , O)
p
()
OY k
f0
d
x
p
(U, O) ^ ColimX Mdl(X)/X (UX X , O)
fi
U V =
/
XMdl(X)
f OX,p - OX,p f -1 k(p) f -1 -1 f OX,p
~ Colim k(pX ),
OX,p ~ ColimX k(pX )
p
~ ~ P k(p)[T ] Q (U, O ) ((U, O ), (U, O )) g P ~ X) OX,p /OX,p ColimX k(p
X = Spm(A) f 0 , f1 , . . . , fn A 1 i0 n f 0 p < f i0
X
fi (p)/f0 (p) k (p) 1in
(2)
2 I /I
k
/
X
U Flt(p)
Colim (U, O) /(U, O)
pX
(X, pX U) (1) UX
p M(X)
/ f d : OY Y = X{T1 , . . . , Tn } x Spm(A) I x SmRig/k f p f
p
{ d X 9 V B V 9 G#n { EEy{7 ) }y wy w o r p r z{|teBVsssp d5df t QHI U C HW RYXQ #e0Q x H xe I q C I q h RfPB#PVfc Q U S IQ S h QI b U c QI U D I V#t7BUITF7R`QHYWqfe RBfcV#CBU#C t #CdPWVD7fbsU C H I h q H W D e Q Fe Q b H Q U S I Q W C q q U W H U Q q D I Q e c U e H I h q B7P#5BBP75#C 7ffBPPVD t TREBD c h FI C ERYEVe0Q XQ h Q b Q b D h Q h De e ECVnEEVdPWEC BUReeQtde D QHW c D Q I Qec U x I H W 7wXdfB#7RYXQ Q F F C H Q Q E7 )D zv))7|zB d e79 7 P ! 5 w s E 9 p #P| 9 #27 5 w }y wy w o r p r sz{|teBVsssp k W H D e U W W Q Ie hcc C I C W Qc W I Cc h H D Qe H g C `Rh BYtXRfd#P#XftR#dE7fP#@D h d5df t HW YXQ Q D C bEfeYWVDXQdebcqPX!XdV7PvPqI YXqV#27 F h e W H D Q F Qc U Q W H Q e W e h H W Q e U S Q D h BERI D Ce W D Q Fe b Q b H W d Q b Q h F c Q U S H D Q #dPV7fTse0Q g t e EEfe Bs#Xde c
0 / mx /m2 x
e H h H U R9wF a F C Q #wFFhIEEhfbGB`eBRBU# e D U H DI C W D Ce W I Q F Fe U h Q x U Sec q q g f d Q H h Q Q He U W V#dP7dE!XBffBBEh E0d t e P#B!PdBY@h a t Qc h x U C Q I U Qe I x q U W D Ce W D Q Fe b Q b c Qe I C H `f#X@#`PBXdR7VBYEfPV7f2v7fP#PXQ 7EVPXQ Q h qW W U CW D U HW V#YBYXQ `f7x7Q77R#E@XQ Qc h H D F x I C He h D g E t g d e QRfXVdef@#dRH9XffcBBqEh csbsEEfe vP77fB7PvPW RE7V7vEC Ie h x D c D Ce h e q Q Q h F c e W H D Q F Qc U Q W H Q e Q I C D Q U D Q I Qec U x I H W Q ce e W H D Q F Qc U Q W H Q e XtXdfB#7Re0vdqP77fB7P|YW DXQ`YRffceW0Q Q W We H I hVq I7#fcV!XP!2RYdB)XBB#dR9#fE5Be0Q B7P#5BBPBYXdBB75@D h Q h q F Q I Q H He U S c Q D D Ce H h I H De C q D U H W Q U S I Q W C q q U W I W D Qe g H U Q q WEXsVEh 5t h Q W D t d#`f#EPRBVBqTd|XBfe c H De C q Qc H D h W We D x b c h Fe h F c h x b Q b U He g h b Q F F C Q H C VRVVi vs!#2RBD Q b c h F I C D U h Q We h Qc Q U S Q H I C W Q TREBEXR2fwV#PPEPb W D CH V#PEC zv))7|zB D h t
rangk(x) n-d
9v
w r 9 sB d 0t { E YyE { @#d! 2 7 E 7 G B 'YY # Tt T9y##9 9)
rangk(x) dimk(x) X/k A k(x) d dimk(x) X/k A k(x) = d X x //A X = Spm(A) =n-d Ns A k f 1 , . . . , fm d x
~ }y wy w o u o r p o m z|{zxvvtsqnl k H D e U W Q F j I C x i H Qc h 2h VYs7REXBPrfD h a t g f d #0d e
OZ s
Q b e H h H U F F C Qc D he I H D U x D D C b H D h u R9wE7Td#BPRyVTBBEBw#ERH vt
Q W h g I U C q H D h i h Q I gec Qc U b C PEsBE5rE#pTPVdfdBBF
D U HW V`YXQ
A B
18 38 6 8 6 & $ " & 3 1 ( $ & $ " 59@97%5420)'%#!
f
OY
a
a W Q be e I W x H xe I XBf#dRXRfPEh Q U S QIH D C F D V#TRPEGEC
k
Z
g
s
' X
/Y
Ns = Is /Is2
U h Q We h Q b Q H h Q Q He U W Q D b Q W C q We b D BX7PBR#7PdBPBBU vP#5YTEC
c h F I C D U h Q We h Q b H TP#B!X7PBQ
Is
W Qc U b C 0dV!F c h x b b Q x F I Q D Ce W I Q F Fe Q D XBfe v7R7EfY7fBBU
p : k{T1 , . . . , Tn }
(f1 , . . . , fm ) (x) (T1 , . . . , Tn ) (f1 , . . . , fm ) (x) (T1 , . . . , Tn )
I = p-1 (0)
Qc U b C fBF
U h xe C W W h He C Ewd7YREdEPW
/ s f
(dT1 , . . . , dTn )
/ g
f
/0.
Spm(p)
A
a
Q b 0Q h
x
k(x)
A.f1 A.fm
mx
I/I 2
x
x
I/I 2 A k(x)
/ k(x).dT1 k(x).dTn
" f "
j Ti
' / A.dT1 A.dTn
n-d x Spm(A) Spm(k{T1 , . . . , Tn })
i,j
x
/ mx /m2 x
k
/ k(x)/k = 0.
/ A/k
k(x)
/0
fj (x) Ti
Q b HW sYXQ
Q D D Qe g C h Q e I H h F hc e W H D Q F Qc U Q W H Q e BBXdB###!PR9TvfP`7XdBXY27PW
n-d
X
dimk(x) (X/k x) d
k
X
X
k
X
n-d
X
x
A/k A k(x)
x
d
d
d
U = Spm(A)
n-d
U
X
p
-1
X
x
I/I 2
x
ij
(0)
A
d XT
2 @b7 & 0 B #{ R{ d 9T #
g
7XT7
7T2
9 EE 7 E2 !9 dE 9v` a P7 B t7 & B T7d d R EE { #v vT ` )!&Y )z{|w v@vo W X }y wy o u u k W Q W Wec W Q F We i q I C F U `0YRdtXPfVBP#@Eh V $ t d t d RdEffERE7`ft7Pff7BXEB75@D h Q Ie hcc C I C Qc I Q Wec h I x D x H U Q q VG B r R U Qc h H 0 tT R S T9!E R E fx 9 R d & B Q W We 9T Q P|{zw HY|@@o 7 I }y wy 9 p A 9 8 w 7 YRffc EE { # t PEV`77#BVq QW h g Q b H DQ FQ D h i I h Qc g h H W H W Q Q b U He H hc q hc Q U S H D Q F Qce h He U b x D fBenYXVRR9ByvBTX!Xdf#BBXb 7Q D h t 9BTe0Q g H hc q HW QI C D U RE7V7Q EC U h C U h Q D D b Q F We i q I C F Q Xfc EXBVh vGPfVBP#dc x F I Q H De C q H U C H I U C q e W H D Q F Qc U Q W H Q e W H hc q H W !R7fE5ERBE52PX!XdV7PysPfV4e0Q Q U S QIH D C BPR##F D h 7 'Y2 Y C B & B #9 E B p 9T9 t 7d d H hc 0 D P7GF EC T 9T fVq 9 EE { Ev 9T 9v ||{zw 6Yz@@Vo 7 } }y wy 9 p A 9 8 k Q b U He H hc q Q b D Ce H C D hc I Qc Q q q h I I h q Q D Q F F C TBVPdPfBsBv#dRB!7f75BV`7V7E7vD h t PdYRd!srQ R97R#dPRXQ BVdB22ffPYXqc x He W Wec hc Q b e H hc Q I D Ce W I Q D U Ie D x b Q b Qce H U H W Q )d t Ie hc H D h D Q H De h F H W Q H h Hc U W x I dEf7#EV7RdEYXRdBY0P`Q t QR#fEeyEPBBve0Q H De C "We b Q F F C W Q D U H W Q U S IQIH D C F D C b Bv7RPE4E%h D h t B4EdPf#C EyYBBREEPB@YXQ Q b Q h De We U h Q F We i q I C F C We D U H W D Ce W Uc D #dPVfVfe c D D Ce W D Q Fe b Q b H W #dPVX!fB!YXQ QBnRPEXQ 6t 7Q U S QIH D C F e 4 Q U S eWW U h H DQ BPPVE7fe c t a Q U S We U q Q W Wec H W BRfVB`YRffYXQ Q be e I x H xe I EfP`P7dRh Q U S QIH D C B2RPEF h fc x F I Q H De C q H U C H D Q Q U S Q I H D C F e 4 R7d#BEPvXsB`RP#!|XQ 5t
OX,f (x) / OY,y f -1 (x) f Spm(u) f ^ OX,f (x) f : Y u k(x) k f : Y /X / OY,y ^ f /X k y k k k Y1 yY y Y /0 k Y1 dY Y1 y f f
Q H h Q Q He U W PEBsRVYh t Q x I Q D C W I F Fe Q D U H D h We U b D 7FR7#dePRXQ!fwVBRBVfe e H Q "I U W Q F We i q I C F D U h R0eBPwPfBVP#!yBTyD h t H Q W D CH EPEC t `EdPf#C h Q b Q h De We U Q U S IQIH D C F Q b hI UW B7RPE`BY|c t c hx b EXe c W D CH C VERBD H7Q QBS7PH U Wc Q H D Qe ! #Xd#C 3t 0Q h Q W h g Q b Q I gec H W PEVwRBfdYXQ Q S He U h D BUQRVYW4csWVEhBbIXYVBYVE75D h t g QW C q q UW D C b H UQ q XQ h RBdfYXQ Q I gec H W Q U S cQ V#7RH Q b Qc ge W We F b h H D Q F Q U C Q I D U Q H We Q fBfYRf#77RI E7XR'B@PYf7tc t PdBYPP7!B#BVyvVRREB7Q XPRE#h 7c Q He U W Q H H Q I U C q Q i U h Q b U He H h c I Q I H D C F Q b He W
I/I 2
Q b c hx b Q D U h VBwD h t vEXe c VERC t W D CH a Q I U q D Ce W D Q Fe b Q b W Q W Wec W Q b C D RBBEfY7dB2XPPf0EBT9h
I e
Q H h Q Q He U PEBsRVYW e H Q "I U W Q F We i q I C F D U I h q Q x D D C b W I Cc h H W dPXeBY`YBBREvBV`XBBEBEfe0Q #dP7d e t H 7Q D Ce W I Q F F W0##YW Q b H D C H Q Q U S IQW C q q UW H UQ q V#XYVBYr75@D h zv))7|zB VT X# Tf f w{ R! 7Rt Td dt 2
dX Ns s 0 Y = Spm(B) / Ns / s X/k X = Spm(A) e: A / Y /k
2 { Rw{ d t17 R 0 ) )z{|w eBBssp ( }y wy o r p r # { E@P v07 7T )d t Q i q I C F C We H D Q F Q U Se D C D h H W Q U h i C D C Q `BBRE#Y#X7BBEBE7YXnE#9B#s #$ t d t d 7RE7VPvsQ XPVsb Q FjI Cx iH U b U Q I q hc Q QEEfrdth t #dPVX!f`qR0e#RBYXTE#9B#nEPQ F F h I he b Qc I D Ce W D Q Fe b Q b e H Q " C I q H W Q U h i C D C D C W H D Ce W D Q Fe b Q b e H Q " C I q Qc U b C F D W I Cc h H W Q Q h F EfPV7f|dPXe'PBdV!@BU Efe0EETde Q D D Qe g C h BBXdB### D C #YW D h I Q b H W Q Q D D Qe g C h Q e I H h F hc Q U S Q W C q q U W D c e D V`e02BBXdVE##!PR9TnwV#!YVBYEC YW D t D hI Q b HW BER!`e0Q Q e I H h F h RP9Tfc #$ t d t d !XP#7BRrdBEe#EC &#$ t d t d 7REXBPftEVXR7t#fE5tXPVERXrPRffe0Q Q F j I C x i H Qc I h q W I U C " U % t Q F j I C x i H Qc I h q W x F I Q W H De C q W Q W W U C H D Q Q W Wec H W W I Cc Efh D h I Q b e H Q " C I q Qc U b C F D U H W BvR0e#RBdV!BsYXQ e ! t EVTBse0Q Q b C D h D U HW Q U S I Q W C q q U W I W D Qe g H U Q q V#sXYVBYBYTXdB75D h zv))7|zB
Spm(A) (f1 , . . . , fm ) (T1 , . . . , Tn ) (f1 , . . . , fm ) (T1 , . . . , Tn ) s: Y n-d /X n-d X = Spm(A) n-d A A/k A k d n-d
# q
V zr V9 2# s tB R { d X Tf P #X 7 E f d 'YY B #0 7 qt !E9 V T9!E!P7 7d d
X/k X = Spm(A) k d
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
xX
B1 /k B1 B B/k dimy (Y1 ) dimy (Y ) dimk(y) (Y /k OY k(y)) Y Y1 Y1 = Y Z
^ X ker(Ai /k Ai Bi Bi /k ) B i = B A Ai ker(A/k A B B/k ) (a1 , . . . , ar ) r = d X - dY I1 (f1 , . . . , fr ) I1 = I B1 = A/I1 Y1 = Spm(B1 ) B1 Y Y1 2 / A/k A B1 / B /k I1 /I1 1
dY
d
(f1 , . . . , fm )
s
f
g
/ A/k A B
Y
s
/X
( T
f
/ B/k
p : k{T1 , . . . , Tn }
Y
/0.
/0.
//B
k //B
f1 , . . . , fr I
//A
(Xi = Spm(Ai ))i
OY
0
/ Ns
/ s f
/ g
/0. (fi ) = ai p-1 (0) =
t g0d e d 9
HW YXQ Q U IQW C q UW H Q V#S7P#5qBBPwU75q D h t YWBBREC#YWe BU2e0Q EnB7@RR7Q D h t dER72e0Q Q F e i qI F C D HW Q U S Q F F hII Qc h H x H W Q F We i q I C F Qc Q U PfVBP#TdTBS IQI D C F Q hI W t Q S Q C UW D C b H UQ q VU#rI7PW#5qBqBPwV#s754D h t !dBEfY7P7wBBXRBPfnfrPW de Q b Qe D D Ce W D Q H Q Q D U W j I q h Q Ie h hc Q b H U c 7RPHEbTUYc D Qc h H x H XQ2dER7eW0Q QBXQPRECIB#C t U S I IH D F U HQ Q U S IQW C q q UW H UQ q BXYBVYB75D t Q U S W D CIH D h ge g e BV#PREC t Q H he b x F Fe H W e R9Xfe0Q g fe ee g dfe DEfP7dcBf e t fV#b`YXQsRdcBY0PI!Q t Ce H h e q F Ie hc D C H W H h H U W x cE#X`HdECPW VsdEP2e0Q h x e c U D He C W H W c hx b EXe c ! H D C q H CH D eW D feE5nUERXQ YVfe t WI C R#dcEh DEfYWVfcDde cBXBfe nYXQ Ce U Q b c h x b c H W e bt e e e Q U S W D CIH D fe V#V#PREC g fde t ge D Ce H h ec q F g de g e EfP7dV!fe c I U C q h D Qc h H x H W B##C fwYXQ e ! zv))7|zB
C |t
Q U S BTQ #PVs0Q 4 t U CI q e
n
V9 d 9 @{d7 9 w# 'Y ) 99 E@E9
! !B s { 9 EE {
)d Q t Q U S He U b x b D BdB0EC x D D C b H D h u BVEwH vt
/K /0. E =0
H U CH I U C q e h H D hHx e 4 BEPBE5vI R7q0Q 3t Q U S H Q Q I gec H D Q F Qc h Cc H W Bw!PVdfs#X7f7fse0Q H 7Q
/0 0 /E
C B Y p 9T9 E P F 'Y ) E#9 p ! Y {9 7!s 7 7 d 7 { d X B& #@T 9v |{zw @vtsqTl }y wy o u o r p o m k W Qc h x W W `XfRH7XQFPfeBiVqPI#C!FW0dPXV#h VY9VY0Psf!vD h Qc H D h D I Q D C H D e U W H h Hc U W x I Qc h t x #PVe0s!Xdc U CI q HW Q Q F FQ H D Q F x I C h D 777R#!#C Q U BS a Q U S He U b x b D W U W W Q e D Ce W W U We b hc B2dBXBG#C VPP0b 7G#dPPRfvTQ
Tor1 Y (K , k(y)) = 0 O f f f A 0 B y Y V = Spm(B) Y f = 0 /K / Y X Y ^ / s f f : Y A{T1 , . . . , Tn }/(f1 , . . . , fn ) K E OY k(y) = 0 / g /X y
WXPdBPn7R#dEwnD h t Q He U W U Q b W I Cc h h H 7Q Q b U h i C D Q T9Brdc t e H Q " C I q H W Q c e D Q q i H Q b H W R0e#RBe0vde B`e`YXQ Q U S He U b x b D Q I gec H W BdB02EC RBdfzYXQ Q F F 4 EC #t Q H I U C Q H h Q Q He U W Q D U Q I C D Q H W V d Q U S He PPB#`P#B`PdBP`VBRE7V7YXQ g ' t e BdEC #C RXeBPe0Q D e H Q "I U W H W
n B.dTi i=1 Ns / s f / B/A K = Ns /E /0.
WQHI U C W QH h XRYVE0PEBQ QfdcXBqEDEC QF7fcnUrdeRHYIEhVE0YfBEC c Q q h FQ b Q q Q b D C Q W hc I U QFEC 45t F QBb xFPX#fE5ERB#q I Q H De C q H U C H I U C Qb2#dR97fBB|XRBh D Ce H h ec q q h W j I q H U H U ECRqIB#Cq Q U S x D D C b H D h BrVBEERH u
(I/I 2 ) B k(y) -B k(y) y Spec(B) 0 /K
e e W D Qc ge W I De H W PW PVde t fBfY7Q dYXQ
QPdeBUPWThfc g WXjRIBqrQ!FPeBiBqRIECFECPf #XdREvEC g # t e RBPf! H W We H D Qe H g C D Q U Sec q q h D BdBVEC d Q He U W hc H D Q b x Q F We i q I C F Qc Q U S Q I H D C F Q Ie hce Fe W H D Q F U I h 7e sYBBRE2f2B2RPEsRdEffdP7B#PEyD
- A k(x) (I/I 2 ) A k(x)
h Q e H D Q b e VdPXfe XW Q F We i q I C F Q !PBBRE@fc
c Q I U H h D Q F We i q I C F C We D U D C b h XPVP9PfBVP#!#YBrVEB!D h t Q b Q h F W Uc q Qc ge W I De H W Br#Tde c VfB`Q t dBfPRXQ dYXQ a H U C H I U C q Q U S He U b x b #PBE5@V#ndB0D h t fBF Qc U b C Q U S H D h H D Q H hc q H W B'X!fV!YXQ
I/I 2 / B.dT1 B.dTn A / B/A I 2 A k(x) /0.
c h x b H hc q H W XBfe c 9B!YXQ
`Q!FPeBiBqRIECEPf7 W F C We H Q H DQ b 7e x Q F F EC 4
Q b Q H h Q Q He U W Q D U h sREB72PdBP2BBD h t E#9Bsb U h i C D Q a W Q I g j c XRBXEfh `YBBRE@BU qt7fe PBq Q b Q F We i q I C F D b H D Q C I
I k
t D Ce H h H D Q W x I q Q D U He We C i #dR9#XY0PVsBBdYEBD h Q U S I Q W C q q U W H U Q q D Q F F Qc U b Qe H I h q Q I je F Q I q hc I U B7P#5BBPB75EC !XdvfPPR7f!XPVwVEC
p : A{T1 , . . . , Tn }
# t g d e a W Qc U b C 0dBBF W Q b C D 0#BT9h zv))7|zB
A
18 38 6 8 6 & $ " & 3 1 ( $ & $ " 59@97%5420)'%#!
B
0
f I A k(x)
/ I(B A k(x))/I 2 (B A k(x))
/B
I / I(B A k(x)) I 2 (B A k(x))
/ (B A k(x)).dT1 (B A k(x)).dTn
/ I(B A k(x))/I 2 (B A k(x)) .
B/A A k(x)
f //B
/ B k(x)/k(x) A
x Spm(A) / I A k(x)
/ B k(x)/k(x) A
y Spm(B)
V ' t g d e a W Qc U b C 0dBBF Q b QHI U C sRYB# QPHEhBQ`QReBUPWsQBB!@#C |#`PPEPsb H D U h D U S QHI CW Q H 7Q H D h D Q H De h F W D Cc Q q #EV7RdEEf75Bq t Q H I U C Q H h Q Q He U W Q D U H W RYVEsREB7sPdBYsBVe0Q Q b D Ce H h ec q q h I h B@EfP7dVBEVq g Et Y 2PdB0sPdBPTfc d Q b Q He U b x b Q He U W h t H U CH I U C q Q U S H DQ ERBE5B7fe c V $ t d t d RdEffERE7dc EVXRBfdc 77fXdc 0dV!BRYVER#7PdBPBBsYXBBS Q Ie hcc C I C Q I h q W Q I ge H D Q F Qc h C W Qc U b C F Q b Q H I U C Q H h Q Q He U W Q D U H W Q e U
B - B k(y) N = ker((I/I 2 ) n B.dTi ) i=1 / B.dT1 B.dTn K = coker(N (I/I 2 )) /0
/ K B k(y)
y Spm(B) B/A E
U = Spm(A) (f1 , . . . , fn ) (T1 , . . . , Tn )
k
Ns OY k(y) K OY k(y) yY
/ Ns
Tor1 (B/A , k(y)) = 0 B
f (V )
B
^ e Y X Y
^ Ax
I OY ^
f
/ By ^
XY
I OY X Y,e ^
x X (f ) OX k(x) = f -1 (x) = 0
k
X = Spm(A)
^ Y Y X Y OY X Y,e ^
Y = Spm(B) k(y) = k(x) = k
2 I = I
f
f = 0
y
y
Q F We i q I C F PfBVP#!sQ t De h H I Q fYX D U I U C BB#q H W Q Qe H I h q Q H H Q Q U S W D C W C q q U W H YXrfPPV`PP7rBtVEP#5BBPQ Vfe VEde t Q e b D c W D C Q b Qe H I h q Q D U I h q x I b D Q D `dRYEV`BBtVrPBV7#B7Q a H77RBfdPf#X7fBBBP'B!EnRBfd!#YW D Q F Q I gec Q Ie h H D Q F xc q q U W D U H Q F b h H Q Q I gec He C Qc U b C W U C W Q fBF V#Yfc Q U S h i BEV I U C q Q U S Wc Q H Wc Q D D Ce H h I W Q De h F C b W Q b I h B#@B7R`7BB#dR920BdETE!X!Vq Q b Qc ge W We F b h H D Q F Q U C Q I D U I U C I H H U Q q fBdPRf#w#X7RI E0P'B27Q ERP!XD h t dPXe'PBvPRVYXQ e H Q " C I q e W W U h H W Q U S He U b x b D e H Q " C I q H W V#rBBXEC dPXe#RBYXQ Q F F EC 4
rn i (0) f (Yi = Spm(Bi ))i (I/I 2 ) B Bi Bi .dT1 Bi .dTn {dT1 , . . . , dTn } i 0 / I/I 2 I/I 2
a W Qc b 0dBUBCF Q b H I U C Q H h Q Q He U W hc h sQRYB#sPEBsRVYD h t Q C PHEBDGD h t bGDEfeP9H7QPXxRBqBDBU Q C H h D W I Q H ! W Q b C H C e#C t 0#BDT9h2DEPW H 7Q Q U BS IXYWBqVYW`HU7Q5D h t QRfhRPW0XVDYXQwQffc7Q qUSWVDECRIPHD#C t QPHDhRWnUYWXQ!XPIf`eW0DEfPdebV#wE0Yh zv))7|zB Q C q U q Ie W Q x H W c H D F Q e hc H Q Ce H D C Q b D C Q W C B d TT 9T p
e
9 9 & B E R wET 9{ s 9Twd ` { 7 E 7 d7 !9 7P0 P #w{ ERz 2 { d 0 `B z7 C B { E EE { #T 9T 9) |{|w eBBssp }y wy o r p r d5df t Q U S H Q Qc h H x H W B`fYXQ Q U S He h U b W I Cc h Qc U C x b H h Hc U W x I BE@BtEfsfBE7XRdBY0PsQ t PP7Q EvEfPRX!fsBBU Q H I U C D Ce W I Q F Fe Q D $ t t 7REXBPwdrh 5t 7fPXPPXrYXQ D Qe W x H I h H W H D Q F Qc h x H W 7XdE#7re0Q Q U S H DQ Bs#Xde c t PP7Q E#dPRX!f!BBYXQ Q H I U C D Ce W I Q F Fe Q D U H W x I I h Qc RR7!d C d d Q F j I C x i H Qc I
f Yi f -1 (Xi ) (c) f i : Yi f: Y / Xi (Yi )iI /X (Xi )iI
pr2
eRH9hHUwFFEC7rEEfB2dVER7BfPVEC 4 zv))7|zB Q F F h I he b Qc W D C I x be W D E!7wE d 7
h g f
h
9 EE { E2 !9 dE 9v |{zw eBBssp }y wy o r p r )d t H D Q F x I C h D D Qe I x i H Q C D H W 777R#!v#C 7fPXBP7Be0Q Q U S H DQ B7fe c t e H Q " I U W D C b H W U h Q D D c Q F F 0BBh vEC 4 t Q F We i q I C F !PBBREQ t dP0eVYV#ne0Q W D CH EPEC t PdP7e R#dEy7PW x He H D Q b c W I Cc h h I Q W D h b Q D D Qe g C h Q e I H h F hc Q b Q W W hc h H Q Q VEB@VB7fBE7Ev7fRPTy@B@PP#f7c 757D t Q U S IQIH D C F Q B7RPEb W I Cc h H U W EfPc t Q U S Wc Q V#fXPH W H D Q F xc x W Q b H D Qe ! Qc ge W I De H W RX!XdXXXd#C Ht fBfY7Q fnYXQ Q F We i q I C F Qc Q U S Q I H D C !PBBREdV#RPEF
0 / I/I 2
0Q h
D U HW BYXQ h t
QYBBREEBV7Q sV#fe h c W I C c h H W Q t F We i q I C F C b D c Q U S I EfYX|c I U C B#q WQ FIQH WQIH U b XPXP2XRPEh vD t
fi (0, . . . , 0) = 0
$ t t Q F F Qc Qc Ie Q H h Q Q He U W g d d 7f`dfEC PEBsRBPh DEfePHhRHD7QPWXxRIBqQVDBUQBDBD#CbPWnD h t g e C Q e Q U S Ie Q H W Q I c Ie D I U g de BsfEC PYXRVfe dB BEC t 9HBPWXxRIsf@ C Vt H h c U Qc Q b Q F We i q I C F C H U s!PBBREERh a I h q Qx D D C b D C b HW EVs7VBEsEYXQ Q I g j c RB7#dEh Q b Q De e I c I U W Q x i D nBfEfREC BYnX9C 7Q I h Vq
^ By Spm(A{T1 , . . . , Tn }) Spm(p) A[[T1 , . . . , Tn ]]/(f1 , . . . , fn ) A[[T1 , . . . , Tn ]]
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
^ Ax
p : A{T1 , . . . , Tn }
/B
1 i n Ti fi
I = p-1 (0)
t
I =p
I = (f1 , . . . , fn ) 2 I1 = (f1 , . . . , fn ) I1 /I1 A{T1 , . . . , Tn }
X = Spm(A)
-1
Y = Spm(B)
(Bi .dT1 . . . Bi .dTr ) ((I/I 2 ) B Bi ) B / I/I 2 I = I1
idf
/ BdT1 B.dTn
/ B.dT1 B.dTn
^ Y T X
Yi
Y
Y
e
p : A{T1 , . . . , Tn }
(c)
/ B n k Xi ^ k
f
/X
pr2
( T
k
/ X T X ^
g
* X
/X
f 1 , . . . , fn I
Y
pr2
pr2
k
/ Bi .dT1 Bi .dTn Y X / Xi / f / f /B id f f = pr2 (id f ) /0. /0 (fi ) = dTi B I = I1 + I 2 {dT1 , . . . , dTr } B
t x H e c h x D e Q I j e D I Q b h c I Q I H D C F Q H W Q I Ie h I U C fEC BE5q $ t g d Q Wece H U D hc Q I U Y YfdR@#C 77BEC Rf7Bf`PXdVPXf7RP#!!sPYXRqc t W D h b W H D Qe Q C H W Q c U S H Q e D x b H W VVR7fn7!YX|de qqVVYXQ Q U S I Qec U e H I h q D Q Q I H D C F e 4 BXdfBPPvXRP#!|XQ 5t W Q I U e I x D W Q I C D Q b H D C W W H D Qe Q C W Qc W U C H H D C b I U Q H 0PBXQdR7Bde0FP#Bwr##YR7fn770dVERErB7RXQ BYXQ D U HW Q U BS V#HXde c t D C b DQ I h WQxI C h F H D C qsXXP#e" T#YW Q b W H D Qe Q C W Q b W Q F I C D W Qc Q U S H Q Qc ge W I De H W R7fn770tXP#BXfBfBfY7Q de0Q Q U S BQ #PV57XQ 4 U CI q e t I q Q I C F D b H EhVt7xREe" hTV#CQ bff7sf7P#BtfBdP7Q fse0Q Q Qcc Q Qc h x Q F I C D Q b H Q Qc ge W I De H W Q U d Q b He U b x b BS g t Y VXTD h t f7sYXQ Qc h x H W Q I De C W b Q F C D h H PWRXQ dD#YQBRIEB2fcs7Q W h b Qc g W I De H W VDEBdVdePRXQ fe0Q eW D YVfe t Q U S He B#C EC D Q F 4 FEC Ut XdeR7DBsB#CRE7VD7sEPWnd`0dfc7Q qsV#YBVU ! t 07VdR7x #X7RdEf7s#Y2EfPdVE7sX7 I Q H D Q U I U q Q I C Q H D C Qc W Qc U S W D C W C q q W Q x e I H D Q F Q Ie hc H D C W W D Ce He b D C W Q Q U SWI BRR#C t I U C BE5q e g fe H Q e D x b D Qe g H W 7qBV77fBe0Q ge k Q U S D Q I I x I I q I Q I D C F D Ccc W U BrQ7V7RRBU0PEhV7RPHEWV#dfhV#C t g I U C BE5q Q U S W D CW C q q V#EP#5BBU ! t I h q W xI C" h F D C EVtXQ7REe#TH#YW Q`RPH9T#fsBbRD7fn7XQU77Q RV`sREB!c VV b Q e I h F C hc Q W H Qe Q C W b Q H Q b WW U h Q b Q FI C D h t t0dE#Xxv#tXRIB7fPXxfPWVE`WXFREBD`bDEPW W Qc h U C W Q U Qe I De W U h Q b Q I C Q H C W Q b W H D Qe Q C W Q t XBR7fn7Xfc B W QH Q 0PD7#PIXQ E7#X7PfRPYXfR7PXfc t d D C H D Q F Q H e I H W W Qe I x W W Q 0PV7fPXxd7QXPRPHYPVEh`XQREBD`HEPW W Q I U Qe I De H D F Q H e I W W W U W F I C Q b D C k Q U S cQ TBq7PH cQxI D U Q X7RvVrBvD h
M >1 Di,j f wi tr+1 = tr - [Jf (tr )-1 ](f (tr )).
n i=1
He C dEPW
Qc U F I C hc I h fBwREVq g EPQ b XPRB0P`7RePBVVD h t D CH Q H D Q I I U x I Q F j H W i W D U He D x b a Qcc Q U S hc I U C q Q Ie h I He g I h Qcc Q U be W x I Q F I C D Q D b Qe D U F D c Q U Q b C D h Q I g j c df7V##fBE5`PfPdBRwff7BPXR2P#B2BBU v!dBVwEC 2BS B#BT92RB7#dEh
det(Jf (T)).g(T) = 1 +
i=1
Q U S W Qcc Q B0df7RH W Q H D Q I D C H D Q F Q H e I H W W Qe I x W W Q b Q H We Q c U S Q Ie b H D Q Q XR#XER7Q #77RPRe0dR7PXB2PYfvfe q2Pf#Xde PI I U W Qc h H x H W W D h b Qc ge W I De He C BsfBdP7Q f#YW D Ce He b D C h #dRBVE7Tfc e 4 7XQ t g BYsfYXQ D C Eb Q U S W D CW C q q ! BEP#5BBU Ct I h W Q x De F I Q H x b H D Q F Q U Se D U W Q H D Q I D C H D Q F Q H e I H W Qe I x W W Q EVq XXBf!R7Rt#X7BBBXR#XER7Q #7XPRPYW XfPXYXb H Q Q D D Qe g C h Q e I H h F h 7BB7fB#ERP9T`c 0Q h
Jf = (f1 , . . . , fn ) (T1 , . . . , Tn ) Di,j A{T1 , . . . , Tn , Sj , . . . , Sn } f f (T + S) = f (T) + [Jf (T)](S) + Di,j f Jf C g, w1 , . . . , wn k{T1 , . . . , Tn }
1ijn
$ # t g d e H Q Q Ie I x H U Q q RdR075vD t W I U Q H W Q b Q F F C Q I j b e W D C D c Q U W Q x D e F I Q H x b D b Q F j H W i W Q F j e U Q b D h B7RXQ XB`E7rR7BfPVE7EC `BS g X7VdR7P7Vfe q`XPYP27f7BU a hIQH C D 7PEBD h t EBT92RBXEfh Q b C D h Q I g j c Q D BVU He ! dEC U R t f7V7EEfYRVRBVEV7V7E7D t #V t t #dRP#5ERBwfQ 7RB2VPRVtd#C ERQ nEfPTdpRBBh vb Qc h I x D x D Ce W W U We b Q D U I h q Q D Q F F C h g $ D Ce He W C q C I q hc Q b U Q I q hc e W W U h Ie D C H Q b D Ce H h Fe C I q q d d Q b C i H x F hc Q b D Ce H h ec q q h Q D U H W Q Q F F Qc U b BVPyf@%EfP7dBVBBYX@!XdQ 7RB4h t BP%E@!REV@BVB#dEfe PREBYXQ UQI q I U W i h D h Q b Q F I C D Q D U Q D C H Q Qc h e I H D C D H W Q b Q F I C D hc Q U S W I P#BGf@BPEC zv))7|zB Q b Q F I C D hc Q U S D C b Q W C q q U BvREBfBvVEBvP#5BBPW D h t 7RPEGXdRy'i qfe fe RPe0Q I Q I H D C F D Qe I h D c Qc h e I H H W & B Y 0{ p 9T9 E P F 7d B 9 { s{ { ERs @d 7p T 9 #
C = A{T1 , . . . , Tn }/(f1 , . . . , fn ) S = (S1 , . . . , Sn ) Pi A[T1 , . . . , Tn ] (f1 , . . . , fn ) A{T1 , . . . , Tn }n A{T1 , . . . , Tn }/(P1 , . . . , Pn ) k (P1 , . . . , Pn ) (T1 , . . . , Tn ) |fi - Pi | A A A A{T1 , . . . , Tn }/(f1 , . . . , fn )
y0 7 'YY 7p Ps #9
C B Eyd y 9T
a k Q I g j c PBXEfh hc Q b Q H D e U W D Ce H h H D Q W x I ssPh BYEfPR7PXRBq h c W I Cch H D Qe H g C EfXdRED h t XPXfPPsBB`!# Q x F I Q Qe H I h q Q D U Q F F C H D h D Q H D C c Q D D Ce H h I Q De h F C b D VXPE7q7BB#dR9`BdETEBU H D Q F Q I gec H D h I b D Q D X!XPVdfs#ERBV7#B7Q W H D Q F xc x W Q b W D C W We We C 4 Qc ge W I De D C b H W R77f72X`V#YRfPEBi )t dBfPRXQ dEwYXQ D D Q He ! VX2#C )t
Bi
i i fr+1 , . . . , fn I
18 38 6 8 6 & $ " & 3 1 ( $ & $ " 59@97%5420)'%#!
Spm(A{T1 , . . . , Tn } h0 |h1 , . . . , hm ) A
Bi
i i A{T1 , . . . , Tr }{Tr+1, . . . , Tn , S1 , . . . , Sm }/(fr+1 , . . . , fn , h0 .S1 - h1 , . . . , h0 .Sm - hm ).
i i (fr+1 , . . . , fn , h0 .S1 - h1 , . . . , h0 .Sm - hm ) (Tr+1 , . . . , Tn , S1 , . . . , Sm )
)d W D h b Qc ge W I De D Qe g H W VEBwfBfY7Q f4XdBe0Q
t x ERBe0Rfd#P#sQ t dERrRB7#dEh a U C I q H W Q Q Ie hcc C I C Qc h H x Q I g j c
7 9v vI z{|w vvvo W ~ y wy o u u ksHh eVUYWHHYI#CqFfs!Xdsf!vD h D D h e Q F F Qc Qc h Q U S QIH D C F e 4 BsRPEv0Q 3t Q D U HW VBe0Q
Bi
Q D D Qe g C h Q e I H h F hc Q U S H D Q F Qce h Q e I VB7fBE7EwRP9Tf2V#`77fd7EsR7x
Bi
e : Spm(B)
A{T1 , . . . , Tr }
/ Spm(A)
Yi
k
g 1 |fi (t0 )| 1in 2M 3 n tr tr (R ) 1 |fi (tr )| 2r 3 1in 2 M r =0 |wi | < M |wi (tr ).fi (tr )| < 1 1+ 1 det(Jf (tr )) Jf (tr ) Jf (tr )-1 (f (tr )) = (e1 (tr ), . . . , en (tr )) = e(tr ) 1 tr+1 2r M 2 2
D h
k
n
f (tr+1 ) = f (tr - Jf (tr )-1 f (tr )) = f (tr ) - Jf (tr )Jf (tr )-1 f (tr ) + R A t0 = (t10 , . . . , tn0 ) (R )n fi (tr ).wi (tr )
n
fi (T).wi (T).
Si .Sj .Di,j (T, S), f
Jf (T)
B
Jf (tr )-1
1ijn
/B
M
ei (tr )ej (tr )Dij (tr , e(tr )) f (I/I 2 ) B Bi R R g M C >0 M M A
f = (f1 , . . . , fn ) T = (T1 , . . . , Tn )
r
A
/R
Di =
B
M
g
t
d e
H QH D hH hIH D C 7Q g P#RR## Q D BBU H D Qe 7fEC ! g
t
d e
W07xVdPIXx DEPW g t e WV#deRbECXfwBS 7RH Q e H C D C He D W Qc Q U c Q Q H We Q c PY7de Q b H D Q F De H D C H D Q b D Q q x rX!VBdREs7757b H Q Q U BS d a Q D C XxBBDEBbH#ERH ut D h vfh7xnYWXQ Qc H Q I g j c RBXEfh QIH C D Q U S IQW C q q UW H UQ PRBBnXYBVYXq D h t !Xdc 7VEBXx GTVEV7Q Q R Q F F Q U b x D C D c W D C D t 0PV7Efh a Bv7tXBVP#!#YEPW W Q I g j c Q U S H D h H D Q W Q i q I C F C We H D C H Q Q U S H DQ B#Xde c t XYBBRE#YXB#YW W Q F We i q I C F C We W Q b H D C
det(Jf ) Di,j f Spm(C) Com(Jf ) 1+ 1+ Ti / Spm(C ) C C / Spm(C) et f Spm(C ) >0 A A / Spm(C) B / Spm(C ) C
W Q x W C q F C W Qc Q U S He U b x b D Q 07P#5!#0dsBdB07vD t Q b Q Q D D C QH D hW C `7BB#sPR#5q F C Q D U Q e H D Q b VdP7e EV t d YBBREsfc 7F Q F We i q I C F Q Q F h E72BB! Q t Q b Q Q D D C QH D hW C q F C Q D U TVBE7nPR#5EnVB Q e H D Q b VdPXfe g t e dPXeBYnPfVBP#Tdc $ t d t d Pfd#P#nf2h Q Ie hcc C I C Qc I d e H Q "I U W Q F We i q I C F Q
C = Spm(C) C A = A{T1 , . . . , Tn } (f1 , . . . , fn ) / / C = A{T1 , . . . , Tn } . (f1 , . . . , fn ) Spm(C ) Spm(C ) Spm(C)
g H D Q F x We x I q W Uc q h D X!XY0PVVfBsn#C
| |<1
#V t
d e
e H Q "I U W Q F We i q I C F D W I Cc h He U b x b D Q R0eBPPfBVP#!TVU R#dEtdB07TD h t g
Q eI x UQ D RIdR0H75qvEC QqERH`QbDE0YsBXEfYRVRfEVq dBsQ Vt h x C Q W hc Q b H U g x b U b D Ce W W U We b hc I h W Uc q QbnW0QFPI#CBDTWXfcnIBUPWTWXQVPI#gT0Q7FnW0dc h DEC Q)Q D t Q D C W F Q H D Ce H h U S c Q b Q De h I Q D U W I #dR9V0x 4Bf#R4VBRXQ ER7Q # Q I D C t g #f e d QXjPHYWiPW2fc QPXQX7XRB`ERx h Bt F Q H D b x xI q Q q hH c I WVEhBb D Q Q W hc b4PPW#f7hfc W D C H C H D Q F Q U S C I qe EPEBD #X7BPVd7Xx Q U S He h U b Qc U C x b x H e H Q "I U W BdvsdVE02Pde R0eBPh
C= |ti - ti0 | < 1 Ti
QPI#CVDXQh D h t
t
Q U S Wc Q B7RH
W D h b We VEBh t
H 7Q
W Q I g j c XRB7#dEh
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
1ijn
ei (tr )ej (tr )Dij (tr , e(tr )). f
1 22r .M 2
2
1 22r+1 .M 3
W h Qc EXsd C Bt
I h q W QxI C h F H D C Vt07REe" TEPW
W Q D e U W x Hec h x De W Qc Q W C q q U W XRHh VYW0PddE#7Vd0d`P#5BBPvD h t H D Qe #Xd#C ! t P7QBX0PV5R7x BV#dR9BwXfn#D h R Q H D b x x I q Q q h H c Q b W D Ce H h H C D W Qc Q b I h QbsQYBiBRIEDBvePWVdE7fP#@D h t F We q C F U De h H D Qe H g C Q U S He U b x b D V#dB0EC a HQ q fcBBU DV7Q ERI7Q E U WI Q D C Q He U W hc Q U S D C b He U b x b PdBP`BEdB0@D h t VXPRB7XRsb Q D Q I I U x I hc Q Q b W H D Qe Q C W Q b W Q F I C D W Qc Q U S Q I H D C F e R7fn77X0P#BXf`BrRPEq7XQ 4
fi (t1 , . . . , tn ) = 0 (tr )rN tr+1 n
t = (t1 , . . . , tn ) (R )n |fi (t1 , . . . , tn )| = 0 /R A Ti ti C n f = (f1 , . . . , fn ) A{T1 , . . . , Tn } C =
W VDECRHBQ C D H t d7YREh Qe C W W
e U qBS Q F F !#C 4 t
W Q I g j c XRBXEfh
a
A{T1 , . . . , Tn }/(f1 , . . . , fn )
|fi - fi |
1 , 2.M 3
1 |det(Jf ) - det(Jf )| < , M
|Di,j - Di,j | < M f f
et |Com(Jf ) - Com(Jf )| <
.M =
A
a
W Qc Q U S I Q I H D C F W D Ccc h W U Xf`B7RPEV#dfV#C t g 07RP70Ys0!REVXb W QxIH DQ W QW Q b W Q FI C D WQ
QBS2BDB#bDRH ut U x D C H h W D hBb`dcVdePWRIXQ feeW0Q Qg D H Q U S I Q I H D C F Q b U b D Q H D Q D Qe g H U B7RPEs7PXvXdV!BYW QBDBXdeVE##27fRIPhTfc2QVU#S t Q t e D Q g C h Q e H F h I VUYW2QfYXQ c hHx HW Q U S bI C g b W D CIH D BvE5h vV#PREC t XVBP#EPfEPW W Q i q I C F C We H D C F U F Q I q U W Qc H W Q Q e I H h F Q D b Q F I C D hc Vw7RBBPsdYX`7fRP`BVU vsREB!vC
Spm(C ) g.det(Jf ) Spm(A) C
g
t
C
c t
W D h b Qc ge W I De H W VBsfBfY7Q fYXQ
n i=1
h D EC
W D h b Qc ge W I De H W B`dVdPRXQ fYXQ
fi (T).wi (T)
|g.det(Jf ) - g.det(Jf )| < 1,
C
C
|fi - fi | < |wi |-1
i=1
n
fi (T).wi (T) = 1 +
i=1
n
(fi - fi )(T).wi (T)
mod (f1 , . . . , fn ).
Jf
1 2.M 3 fi (t1 , . . . , tn ) = 0
Q He U W RBPh t
a W Qc h H x W Q I g j c 0dER70PV7Efh Q b e H Q "I U W Q F We i q I C F D U He D I U C e BdPXeBYsPfBVP#!BdBRBEv7XQ 4 W H D Q F xc x W Q b D C b He D I U C Q H D Q b x x I q Q q h H c Q b Q H D Q I I U x 77f7X2EdBRBE`R70XRB25x w2P7RRB0PI h GD h t W D h VEBb Q b Q W W hc h !YREvfc W D CH EPEC t dBfPRXQ dXdBYX`PdER2sfPXYc Qc ge W I De D Qe g H W Q Q He C I b Q b Qe I x W h Q F F EC 4
ti0
|fi (t10 , . . . , tn0 )| = |(fi - fi )(t10 , . . . , tn0 )| |fi - fi |
ti A{Ti , . . . , Tn }/(f1 , . . . , fn )
t
Q U S IQIH D C F Q b hI UW c B7RP#!`TYzde
1 |fi - fi | 2.M 3
ti0
A{T1 , . . . , Tn } (f1 , . . . , fn )
f = 0
tr+1 = tr - [Jf (tr )-1 ](f (tr ))
C
/ / C = A{T1 , . . . , Tn } . (f1 , . . . , fn )
|fi (t10 , . . . , tn0 )| = |(fi - fi )(t10 , . . . , tn0 )|
det(Jf ) Di,j f
Com(Jf )
det(Jf ).g.(1 + )-1 = 1 +
n
= det(Jf ) - det(Jf ) +
i=1
i (fi - fi ).wi
0Q h
fi (T).wi (T)(1 + )-1 M, C r+1 C
g
t
d e
d e
t H 7Q d k Q U S Wc Q TB7PH Q b H Q WXQbECBDThWHYIXQ U#7BbrEb`PHYf7vc t C UQ D C Q We Q h D v#C e PW V t d t d EfPdY#PVh t sRBV D Ce He W C q C I q hc I Q b QI g hc Q Q c H C Q F W D C H C D H Q Q U Se I x D x H De C q D RIBUXBi @sBUEPH!FEC7rEPEB2!BPXB7#`fE5VU He dEC ! t dRYEV7fB7V#PREC Qe H I h q Q F je U Q b hc W D C I H D t EBXx ssdRYEVrPXd7RBRP#!|YVfD h x D C D c Q b Q e H I h q Q I j e F Q I q h c x I H D C F e W e h QVU#SvcXQPH H 7Q Q H We Q c PYf)fe H U CH I U C q Q ERVE5BU S Q I H D C F e 4 rPR##57XQ a e W Qc h H x H W Q Q I g j c h Q H H 4 Qe D Q I g j c YrdERYXRB7#dErRY7Q &t dVVRB7#dEh Q D U HW BBte0Q I Qec U e H I h q 7ffBPPVvD
OY ,q1 OY ,qr ColimV1 Flt(q1 ),...,Vr Flt(qr ) (V1 Vr , O) qi P(Y - Z) U Flt(e(q)) V
Qbrfct#X7P7 P@BtYXQ Q hHx H DQ FQH QI D U HW e W I Qec U e H I h q D Q H Yt7fdVPPV@7Q
W Q F We i q I C F C We W Q e H XYBBRE#YXb g fe 7Q g e 0P7dRBERB0dBXBvD h W x H xe I q C I q W Q b He U b x b t W Q U Se I x D x H D C XBPXB7###YW W Qc Q U S He h U b H Xf`BEvQ V t d t d Pfd#P# Q Ie hcc C I C U b Qc U C x e x H xe I q C I q sfBE7Xb g de RfPVERBwh t g e VE5t7XPf7fe # I U C q H D Q F Q Ie hc H D Q D C HI U P7Q E C t Q b H De C We d#e" b Q b x FI Q D U HW !R7vBYXQ I h Vq Q b Q h F ETf e t Q U S Q b j x I q e U S Q Q b He U b x b BBX0PVB`rdB0D t h W Q b e H Q " C I q Q He Fec hc H W 0nQ PXe'PBPd!fffwYXQ We h t Q PXeBY!e0Q eH Q"I UW HW D Ce H h ec q q c Q U EfP9XffBBEh nBS a Q D' h IH D c Q FI C h F x i X !REnT7BRW U b Q b U He H hc q h TVRR9Bc Q U S D W Uc q e We C i V#GVrVfBYEB H Q Q U S D W U c q H Q c Q e H D Q W W BGVVdV`77fP7PP0Q B(P7Q e HQ Q YW )GD t h Q be D C 0Q f7VE W D h Bb Q b Q h F WI C !#Tfe c Efc t W D h VEBb Q b Q h F EEfe c tH U D W D CH V#PC t c Q FI C h F x i W U X!REsXBR@b W e I h H I U C D b Q U S e I x D x Q I g h c H W e Q U S IQW C q q UW D C b H UQ q R)&P7Q Eva BU vBPXB7#RBV2fYXQ B7P#5BBPV#75nD h XQ EC c Q U S W D eWW U BtVVYRVEh H 7Q fPf#BB75nEC RBfdPBXEfh Ie We C i H U Q q D Q I gec Q I g j c Q D U HW BBYXQ Q U S He h Qc H V#d fQ g v %#d#C Vph vEfP9XdP9Bq $ Ie b U h D i Q b D Ce H h e H hc Q b Q b c Qe H D Q W W Q Qc j b C F D U x D D C b H D h 7fP7PRXdXGV!VBEwERH u Q b Q F j I C x i H Qc I 7REXBPf`h vt BPw9BssB I U W H hc q H Q e D Qc j b C F D U He We C i D f7BGBsPEB EC t e YW Q U S Wc Q V#fXPH W Q b C D h W Q De h F C b W Q b Q H We 0#BT9`XVdE#`XPY7Q c t W D CH EPEC t Q U S e I x D x H D e C q D He #C ! g fe e BPXB7#s#fE5GBU t H Q Q U S cQ B|7RH Q b Qe H I h q hc H D Qe H D C dRYEV27fP# Q U S cQ B)XPH Q b Q De h F C b D rBfTE@BU H Q Q b Qe H I h q Q D BfPPVBBU Q De h F C b D U Q H We Q BfT#@BPYf7c t H D Qe #Xd#C ! g e t V#YRXBb a V#dRYXYRE@B70dXPR##VEfdEVEC t W D CH EPEC W U C W W Q e W D Ce H I Q W W h U Q b W Qc I Q I H D C F W D Ccc h W U t Q b H De C We b H W Q e U d#e" YXqBS D Ce H h U S EfPVXx vb a x F I Q Qc H W R7sfYXQ Q be e I x EfPwPH x e I hc Q b I Qec U De W U Qec Q W Uc q wr7fdB#BfYyXdfwfc VfBQ )t Q b Q De h F C b D U H W BwBfTEBYXQ Q U S Ie hc H W V#rdEf7rYXQ fRh a a c t EBTh Q b C D D U D C Qe D Q I g j c B!V#b fBVPV7Efh W I Cc h W D C H EfsV#PC t g PfBVU Q Ie h He D 0Q h RdER7XdX!fx Q Ie h H D Q F xc x Qc h H Q D U HW 4 BV`YX Q t a Q I g j c RB7 #dEh Q D BBU XQ h Q b Qc ge W We F b H I U C D H W fBfYR!BEh YXQ #TBU e0Q Q U S IQW C q q UW D C b H UQ BsXYBVYV#Xq D W Qc ge W We F b h W H D Q F Q U C Q I Q b W Qc ge W We F b h W H I U C W Qc I h EC XfBfYR!BE`R7XPI #XRnwXfBdPRf#sRP7Q Es0dsEVq H Q I Q hc q F Q I Q H He ! Q U S D C c Q U S Q F D ic C q D XEB7RnPPVU "t BV#7BnEfE5BU a H Q Q Ie h He D U Q F D ic C q D 7wRdERdBBw#d#4VU XQ h Q F I C hc Q P#Tb t Q t e RdER7XdXwfR72RB7#dEh Q Ie h H D Q F xc x Qc h H x Q I g j c Q D U HW BVe0Q Q U S Wc Q V#fXPH Q b e W e I h W H D Q F Q U C Q I W Q b H fR)77RI E7XRXrQ Q b a Q b Qc ge W We F b h H I U C D U H W sfBfYRf#YXQ #vVe0Q W Uc q Qc h H x Q I g j c VfBsQ Bt fR7`PBXEfh e We I h H D Q F Q U C Q I D U Q H We Q R)77RI E7XRvVsPYf|c t Q D U HW BVe0 Q 4 a
U p P(X) Vi V j = F X Spec(E)an Spec(A) (Spec(Eij ))j Spec(Ei ) = Spec(E A Ai ) Eij (Ai [T ]/Pij (T ))[1/Pij (T ).Qij (T )] Pij X Y Spm(B) Spec(E)an E = (A[T ]/P (T ))[1/P (T )] A = A[T ]/P (T ) P E A k Spm(B) Y = Spm(E) A Y P (T ) = 0 Spm(B) / Spm(A) = X e : Spm(E) = Y ( YX X P(X) V F P(U ) e-1 (U ) V e-1 (F ) ( YX ) ( YX YX YX X / (YX X)-1 (pX ) X )-1 (pX ) (YX X)-1 (FX ) V i=j E = A[T1 , . . . , Tn , U ]/(P1 , . . . , Pn , Jac(P1 , . . . , Pn ).U - 1). FX qi Spm(B) X V X U = (X - T ) e-1 (p) = {q1 , . . . , qr } E Y A P(V ) YX Y V X)-1 (FX ) X -1 e (p) ( YX ) - V e-1 (F ) X X YX YX YX YX /X
C
W D h b Qc ge W I D VEBfBdP7Q fe XQ h
#72f7PdP`B!BY|c zv))7|zB C W h Qc I Q He h I H Q b H U W 9 f 77 P !{9 T!2
uij fij
V T !9 dE E! t E E d7 !9 TP0 P 'Y{ #R5 ` a t d7 !9 7P0 P E 9 R d 'Y ) E 9 # 9sR {d 7t7s nd 7{ R s #9 99 { ER| d 7t7s C B s V B 9Td 7 | 9 # 7 {9 ` 9)t&Y EE { #!f V #0y0 C B 9v d 0 E w ! I z{|w 6Yz@sr w y wy 9 p A p p 9T t D Ce He W C q C Ie W Q H h q F C ae W h U S W Q be e I W x H xe I k g t d V #dRP#5ERI d#C 0P#VE7 Y#VXEfRXP7dRh a Q I H D Q W Qc h H x W Q F We i q I C F W Q b W Q H D h H I C q Fe W x H xe I q C I q W Q U Sc Q U S Iec g h H x I U C q H D Q b x x I q Q F F Qc Qc h I Q U Sec q q h PR#XsXfwXYBBREwX20PY#fsXRfPVERB20BfXBwdfBERBE5w7BX0PVn!Xdd7BdVB%D h )d t a Q W D Q U h Q D D W U C W Qc W D h PV7b 0BBEh VEP4fVBb Q U S Wj V#Xb Wc Q H Q b Ie We C i H U Q D c Q U S He h U b W I Cc h Qc U C x b H h Hc U W x I 7PBndPf#B@Xq #C B@E R#dE4dB#0n9RdBPXRQ t
k q P(Y ) U Flt(p) p P(X) (Xi )iI e X V Flt(q) q k f: Y fi (Yij )jJi /X OX,x / OY,y k
q
18 38 6 8 6 & $ " & 3 1 ( $ & $ " 59@97%5420)'%#!
|f - f | < A[T1 , . . . , Tn ] A{T1 , . . . , Tn }
X = Spm(A) Y = Spm(B) (P1 , . . . , Pn ) Jac(P1 , . . . , Pn ) = det (T1 , . . . , Tn )
Yij
uij
/ Yij
' Xi
fij
B = A{T1 , . . . , Tn }/(P1 , . . . , Pn )
Q q i H Q b Q I g j c 5erPV7Efh
hc W D C I x be W D 4 fVER7PVEC Ct
H Q W Q F D ic C q W Q b H D C 70!#d#XBtEPW
H Q
(YX X )-1 (pX ) T X
F X Mdl(X)
Qe D fBV
Pi
Z
A (Spec(Ai ))i Ai
OY ,qi
OX,p q P(Y )
ColimU Flt(p) (U X Y, O)
B
A
OX,p A E.
^ Y i = Y X Xi
Vi Flt(qi )
qi P(Y ) U
D t t h
p Vs Y Ve V s = V s Y =
Ve Z =
p M(X) qi P(Y - Z)
V Flt(q)
{qi } P(Y - Z)
{q1 , . . . , qr }
FX p P(X)
Qij
Ve
W Q H D h W C q F C W Q W W Q H U C H Q U S Q Ie h H D Q F xc q q U W x H xe I q C I q hc 0PR#5E7sXP0P#PBRdER7XdBVBYRfPVERB`f0Q h XPB`VXPV!D h t cQH D U hI b DQI q Q U S cQH H B57PQ EV7R## H D h DQH D C a Q b H h q F C e W h U S H I U C D U Q HWe Q c E!# PE#rYXQ #4BwRede d t d t d #dRP#5ERBTwBwQ XPVnB75wfdBER7x g e RfPB#PVnh D Ce He W C q C I q hc Q b U Q I q hc H D h b D Q q Qec g h H x H xe I q C I q hc I t Q I b D Q I q Q b W I Cc h H U W P7RBsR#dE!BYqc t b PXd#q Q x F I Q H De C a H U CH I U C ERVE5q Q U S cQ BsXPH Q HWe c H h q F C e W h U S H W ReQ fe E!# PE#sYXQ Q F F 4 EC )t VY277VRdBdP`EVq I U W H D Q F x D h Hc U Fe W W h a H D Qc U D D W Q 7fBBh XBD W Q WI C 0dc Efc t Q b C D W U C W U b D Ce He D x b Q b c h x b c Q b W I U Q H h I x D x W Q #BT9h EPyy#dRB! EXe B7R97BXEXb H D Qe ! 7fEC 5t Q U S cQ Q VyD h t Ph BYR7fBTvf0PV#C RP7Q E`7R!!RfBPReEsB#C vt Q H D e U W Q I je D h F hc Q b Q b j C I q D W H I U C Wc Q H Q b Q Ie U I H W D C I U H Q Bs7RH a Q U S Wc Q B27RH Q b WH h q F C eW h U S WHI U C UQ BE!# PE#`YXQ #y7 Bb H Q H D Qe 7fEC ! t Q H De C "We b Q F F C W D Q D Ce He W C q F C x PfEefTEPGX#dRP#5E7Xb Q D U He D I U C VBtdBRBE EfPXPh t D Ce H Q W W D CW C HQ Q EP#5q )vD t B7VXRtRdBY0Pt7RP#!Xs`YVXPdETtYXvc x i I Q i Q I H h Hc U W x I I Q I H D C F x b Q b x We h H D h D Q H De h F H W Q t H Q QI b DQI q H UQ q D P7RB75EC
Z U = t-1 (Y ) f -1 (U ) V U Y Ve Y t = t|U T =Z V T W = n DX ( |fi ) i=1 Z U V /U W / X, |k | ^ T = Y X Z Y T fi maxi=1,...,n (|fi (v)|) > f: Y f: Y y M(Y ) f: Y OX,f (y) /X Y /X Z V s U Z V
a e H h H U F F C Q F F h I he b D U dP9w#`!ER#B`Q #PR#qYW )D t dER7X!XP7 RBs7F Bfe0Q BXYBVYW U CIH D C e HQ Q Qc h H x H D Q F Q H Q I D U Q F Q e Uc H W Q U S IQW C q q U a a Qc h H x H D Q F Q H Q I D b H h q F C e W h U S H I U C D U H W dER777R RBU vEVE7 PEVtP7Q E@Be0Q Q U S HQ Q b C D B7EBTh D U HW BtYXQ Q F F H UQ q 775D h t Q U BS H Q IQW C q q UW D C b H UQ q D XYBVYVEXn#C d t d t d #dRP#5ERB2th #t D Ce He W C q C I q hc I W Qc h H x W Q F We i q I C F W Qc I U C q Q Ie hcc C I C Xfs0YBBREXftVE5Pfd#P# a U b D C W Uc D C h I U C I q Q W I Cc h H U W EfeYd7VE7sfc XQ #PBBb R#dEYc t QH C D D C H RBEQ Q U S cQ BV7RH Q H h q F C eW h U Bb EV# PEVS a a H I U D He We C i YXQ #CnBUdPf#BnD t QW C q P#5TD h t XRE!# PE#XEfPsXP7dRh W Q H h q F C e W h U S W Q be e I W x H xe I W Q b I h 0EVq Q b Qc ge W We F b h H D Q F Q U C Q BdVdPPEX!XPI #0PI h DBU W x FI Q W Q b H XR7XQ W Qc h H x W Q F We i q I C F W Q b W h Qc I Q He h I H Q 0dERwXPfBVP#!!0w#7d7PdP@b He C Q F F #YW X vQ t a a H U W c W Q h F C e W h U S W Q be e I W x H xe I P`de 0PH#VqE7 Y#V0f#dRT0PfRh WQ b I h XBvEVq Q b Qc ge W We F b h H D Q F Q U C Q D U H W BydVdPPEv#X7RI E0PI B@YXQ e H D Q F x We x I PW 77PXRBq W U dc 5t b2#VfeYeEC r#X7f7fc RdBRPYV#B2VEB`Pc t Q Q h D W U h H D Q F Qc h C Q Ie U I H W D C Q b D C b H U W D Ce H Q W hc b D Q H x e U #dRXPn4VXP7zBS D Ce H Q W Q D U W Uc q U h Q H We Q c EfP0YVBVfBPYf7Bde Q U SWI V#REC t 7BB# YXQ #!VU Q b Q Q D D C HI U C D He dEC ! zv))7|zB
X k f s : U Z (Xi )iI Xj X k Xj Z = Z j / Xj Yj / Y X (U Z) ^ ^ Y X Xi X Z Z Xi Xi Zj = s-1 (Yj ) -1 Yj = Yj f (Xj ) /Y Zj j / Xi /U U Z = X / X. k t ^ Y X U (Yj )jJ Z /U
B pV 5 2 !9 T E E0tS 9 70 9 9 p B d {pV#R| 7 T TEE!7 { 9 EE 7 !ETf V E00 C B vI |{|w teVBssp } y wy o r p r 9T 9) d X # ` 2 t BV e D a Q F We i q I C F D b Q e U W Q H I U C D Ce W I Q F Fe Q D U D Q I Q We I C H h Q W H U Q q W h F x i W Q b e D e W h U S Q F We i q I C F D U S H D h F I YBBREBU v2fe BPsPP7Q EvEfY7f2BB7XYREPE`PX#XBRwqBV Y#V2!PBBREBU |#ERT9h a e We I h Q b Q F j I C x i H U D D C D Qe U b Q be e I D Ce W I R) 7REXBP'VE XdBg EfP'EfY7Q BfRBR5BBG7ffX!XdR#fdVw7PW EfPYXBPPQ 4 BV Q U Se H x i H C q i i Q D U Q xec H D Q F Q Fe H De Qc g F Q D Ce H W Q U S Q H H e D a HW H e dEC ! t PfBVP#!z7RBU sR pV 7f75BE!RBE5EC Q F We i q I C F c Q H D I Qc Q q q h h I I U C q D Q F We i q I C F Qc Q U S e h c H PfBVP#!sf2B|ERI de YW t YXQ a Q U Se I g x c h Qe I H x F C x D Q Q F F V#RB7#dEfPREXE!7t!#C 4 t fBVe0Q Qe D H W H U CH I U C q Q U S cQH WQ b C D ERzVE5tV#XP|XEBTh YBBREF QI g h RBVfc Q b Q F We i q I C D BU He dEC ! t fI e0sPEh BPr#dR9Rff7V7EfYz7VET7tY75sD h t dER#BV|YXQ Qe h H W Q Q H D e U W D Ce H h Wec h I x D x hc e W I Q b D h F Q b Q W H U Q q Qc h H x H Q e D H W Q F We i I C F Q PfVBqP#!fc H Q Qc h H x H W 7dER2YXQ eW Q U S Q FI PV#TRh t t D C e H e W C q C I q A 6 $ 55 z %@ d d d #dRP#5ERBvh t t V Q W e c e H t V t B Q b U Q I q h H D Q F Q W U Q I U Q i c t g d pe d YdfPBU g d B`Q 7RB!c 77PVXPBXBfh t Wc Q D D Ce h I W Q De h F b Q b D C D U x I D U H W Q Qc h H x Q F We i q I C F D U I h q c Q D D Ce H h I Q De h F C b D b Q h F c Q U S H D h F I t fXBB#dRH90BdETEC!#deBV7RQBVrYXdERYBBREGBEV7BB#dR9wBfT#BU v!ETfe !BrTRh t d B t Q F F Qc Qc Q Wece H t Q Q b g V de#sfBYDX!XPddcV!frYWXPqVdvBUS V t d V !Xdsf`YdfPBU t d B b pe d#v@#dR9PYVEXfBk H D C q Qc I U W H Q F Q He e q Fe Q C q Q I e Uc e g d H De C q U b D Ce H h I H W D C F x b hc W D h b QxW C CI q VEB47P#5qERBQ 7RBqTVVb@VCPR#fnYXQ 7W4 7fe d7RX BU |#v)Q 7fB7Pc t d BnB7P EC PEVh U Q I hc W D h He U b I H De H W U Qe Qc I Q D U S H Q D Q Qc g F Q W W D h b I UW g UI h q W U C D e U S H C q D U IQ b W x C Q W F F C W W U D W U C Q U e U q Q c q F C W c q H W Q e x H D Q W x I q H D Q F E I E7Q # % U e C H U t(t B V#BtBdDe#%Vw 7RHe x vwXIE)2XQEPwV#CB2V#BDnBSRfWVBnfQBE7wVfUBYX Ue QnPXY0PB2X!V5Rh zc W%P@C t D Ce He W q C EfPdYC#PI Q 7RBvfcT7RfBYWdsYXQ d t d t d #dRP#5ERBv!B!Q 7RBvh Q b U Q I q h Q b Q x Ie q De H W D Ce He W C q C I q hc Q b I q 5 6 $ A d )d t Q b Qc ge W We F b h H D Q F Q U C Q I D U W I Cc B `fBfYR!BE#X7RI E0PBtEfh HW YXQ H Q 7Q57D t Q U S Qcc Q Q D Q H I h q Q D U Q H We Q c feBVdeRYEVBVPY7)de Q U S Qcc Q Brff7PH Q b B `df7RH W Q D h F C b Q b Qc e F h Q D U W e YW t Q t e EV#YXQ H h q F C H W Q U H h Qc I Q Wece H b H U W c Q I Uc D C I U Qc h H x H W Q c BSeE`d7PfffPU vP|fe RBfV#B#C Bt fYXzde 0BdeETE2dfcdEsBBHYXQ #t BY5eBVPRVYWXQ I UW D eWW U h H I UW D HW BPeVVsYXQ Q b Qc ge W We F b h H I U C D U H W fBdPRf#YXQ #TBYXQ W Uc Q U S He U b x D BdB0bnEC Q F FEC 4 t VfBqQ D Ce H U I H W D C I #dRVRPYV#wEh zt H Q XQ h Q H Q I b F F C W D D Ce He W C q F C x b Q D U D C b h R0PfenQEP%7Q EfPdY#XBBBTE%D h a t W D h b U DQ D C HW B7RH##YXQ Q U S cQ B|7RH W D h Bb b Q Q#VDfPfe#C BU7Q UERIPHB75@#C VPRXb 7v#dPRVRfB!h h eW D I C H U Q q D W U W W Q e D Ce W W U We b hc I t H D Qe H D C 7fP# W I Cc Efh e t PW VV H D Qe H D C #XdRE W I Cc R#dEh e t PW B
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
qi P(Y - Z) qi P(Z) Vs
{qi }
T W V W = V T = f 1 , . . . , fn V v V
U X Y
OX,f (y)
X
/ OY,y
k
iI0 P(Ui ) = P(X)
s
We
{qi }
X
xX
ZX
U p ^ U X Y = W e Ws U X Y U P(X) I0 I
f -1 (x) = Y X x y M(Y )
s
s
k
t
t
f : Y
37 Y
37 Y
f
f
ZU X
X
/X
^ U X Y We V e Ws V s We (Ui )iI iI0 P(Ui ) = P(X)
^ Y X Z
(Ui )iI0
U
/ OY,y
Ve V s
/X
/Z
X
t PXRI Vt qWQ g g 0d e d t QW C q D P#5EC g qWQ PXRI H cQ FI C Q g 7RE a t a ! W I Q e H D b H Q c q a BU h Fx i T7BRW qWQ h Fx i PXRI T7BRW D U HW BYXQ g t PXRI qWQ e 3t 7fP7Q vr7dVBU D He dEC ! X0V" F A ( "8 t t W I Cc h h EfD h t QW C q P#5vD h Q U S cQ B7RH I U CIH H UQ D HQ 7Q ERP@75q EC )Q D t Ie We C i W I Cc h H U Q D fYEBTEfv75q EC XQ h t Q H We Q c ! PY7f e St He dEC ! # t d t d EfPdBV7%y@VEfPRV0d D h D Ce He D x b hc Q b W D Ce H h H C D W Qc Q b I h B% A 6 $ t zv P a 0! P d7 !7 a d 7d dE T{ 9d Qc g h H e F Q H D Q F Qc h g Cc H D Q F Q H e I H fBERYW 7PW 7XdEVEfE #X7PfRPYW q Qc g h H e F Q W H D Q F Q H e I H dBERYW 7P77RPReW Pt 9 V 97p V ER W x He ec qe Hc U 0PddVdRBF d 7 f W Q i D hI # 0BVERBg d d7 X s T7 P X # 7 ' R V X | 09!d ` e@ 7d 7p 9tR V 9 #P 9 V9 v e 7 P7 P7 { a E D B @ E Cv P B 0 2A 5 p { #R5 Qc g h H e F Q W H D Q F Qc h g Cc fBeW !XYs7XdEVEfE q P a 0 ) ` sR B E 2 QfBeW !XYW c g hH e FQ 9 E 9 q #7 ) I z{|w 6Yz@@o 7 y wy 9 p A 9 8 V t t k g 'V BV BV EfPdBV7x 3 5 PPVEd#C Rh VYW D Ce He D e W W U h Ie QH D e U " D Ce He D x b hc He h #dRB%d D h t fdBBPeR7PEX4dE#7x v@YXQ Qcc U D Q U Se H We I x H h I h Qc h b H W Q U S h I Q W C q q U D I Q ec q Fe W I U BRXYBVYW EC XVddBfYvBEC t Q H D h We F I Ce D PRfREdBBU Q D U Q VBB D h t g fe RPEBVEB7Q PRRYXQ Qc h e I H D C D D C b H Q H j I We b H W Q D Ce H h Uc Bb EfPVfh V#vV#PXYBVYV#BBBER#VGVEh hc Q U S W D C I Q W C q q U W W U C D Q i q h I h I h q Q W D
t XftET7VPW a W Qc h H x W h F x i WQ Xb Q U S Q FI t BRT9h g (t d B d 7REXBi % 9 Q FjI Cx c Qc h H q W Q Q W Wec H W fe 7W g dERx t PXRI PPdYXQ 9f
5 sR
9d
T9
B 7
#@{d
D Ce H h H D Q W x I q Q D U H Q F b EfPR7PXRBsBB7!BEh
p
Q I g j c RB7#dEh
h t BBEV7vEfPXe#RBc Q U Se D C D h D Ce H Q " C I q h
Y = Spm(B)
H Q Qc h H 7`dERx
H UQ q D X'EC
f
P 'Y dE P E
Q x W C q F C hc H W Q Q F We i q I C F Q I H C D Q U S I Q W C q q U 7P#5E7!YX`YBBRE`PRB`BXYBVYW $t t Q I h c c C I Q I t fce RIPH #CdR9Vfh PvtBVwYXQ Q h e D e H h Uc hW Q b e D U F HW e RPwE7vdPP70%D h zv))7|zB C c h e I H W h Qc Q H I h x d d RdeEffEREC7@fcwEh sR Y Y EP EF |%e 9 27T97d 9 dw 7 V B E & 7 d !9 TRX XT ET 7 9 ) I I z{|w 6Yz@sr y wy 9 p A p p 9f { #P EE { #v @T k E t d t d !XdvEfP7dBV27f72VBvEC Q F F Qc U b D Ce H h ec q q h Q F je U Q b Q D U e e t Q U S x I H D C F e W De h h BsRPEvPVdEvD h t REX2#PPVq xI Q b Q b H U CHI h D 7Q QBbWXQRIBV0dc VfBsQ Vt 2f77R PVe0Q g W Q W Uc q Q b Qc h H x H D Q F Q H Q I D U H W WI C Efc H DQIH D C DQI W Q Q D D C D Ce He W C q F C x b Q D U D C b h EfPdP#5!#0wBB2VET4D h t 7RP#VXP0BB#
Yi = Spf(Bi )
f -1 (U ) = E F E/U zZ
QBUSHD7fe c t VERfEBfYXEPW Q W D C H Qc De W W Q b H D C H 7Q 0Q h
E = f -1 (U ) V
18 38 6 8 6 & $ " & 3 1 ( $ & $ " 59@97%5420)'%#!
Z
F = f -1 (U ) W E/U
t
d5df HW YXQ
E
1
Y (Yi = Spm(Bi ))i / Xi = Spf(A ) i
U
/X (Xi = Spm(Ai ))i Bi
E
k
e
p
e
A{T1 , . . . , Tn }
a
(f1 , . . . , fm ) (S1 , . . . , Sm )
Q t
W D h VBb
a Qe C x H hc Q D Qc e S x H W Q W Qc h H x Wc Q F I C W h F x i dRIE#7Ph7T!`RH7fh VU#0e0Xf7P#s#T7BRW WQ Xb Qe I C x H h h fP#E7R9XTfc $ e dI#C Q e b D Q H C I b Q j I C x i H t Q c h H t q W Q Q W W e c H W e W H Q H h c q H W 0d7BiRRr QrF7RE7VPnD g fx PXRI PRfftYXQ a P7tfVtYXQ a Wc Q I C W F x i 7FREt#hXBRW Q b Q We i I C 2FPfVBqP#FBU |#XdXV2wfdR2RP7 75e0|c D U S I Qc Q q q h I Q b Qce H U Q I H H U Q q H W Q 7 v & B # Eed 7 Y Pn'Y 9T v R B& 7 #Ted 7 Rn 2T 7w#9 9v` a 7 v & B Eed R ! 9T q R BC EE { 7 P@ ! 9T )r`&Y I |{|x6Yz@sr X y wy w 9 p A p p a t Q e I Q Q D Q x W We c H Q W U C 7f7RXv7Q`7PPhYWXtVEPWRWXQb e7@#dRP#5ERB!h D Ce He W C q C I q a a c Q F I C x H xc q F C 7RE`P7dV!# 0Q h Q I g j c RBXEfh W Qc Ie We C i H U Q q D XffYEV`XyEC E t d t d 7fwfEh t Q F F Qc Qc I hc Q f`b g Bf#h Q U Se b W D h b Qc ge W I D VEBwfBfY7Q fe
A f Et/X k X f: Y A B A{T1 , . . . , Tn }{S1 , . . . , Sm }/f1 , . . . , fm f: Y k /X A /X k B, fi
A [T1 , . . . , Tn , S1 , . . . , Sm ]/(f1 , . . . , fm )
)d
t Ef7fe # W I Cc h H D Q D C
/ Spm(A{T1 , . . . , Tr })
U
Y
B
X
k
/ Spm(A) = X
f (Yi ) Xi
A [T1 , . . . , Tn , S1 , . . . , Sm ]
A i
k
9
0Q h
f
9 B s# 7
v (U, O) vd = u b1 .a1 + + bn an ti = v -bi .ti
# 9864210)' &%$" ! 7 5 3 # ( ! # # !
(1, . . . , 1)
x
U
/U
f (X, O)
u
Di
a a Stf = X[T1 , . . . , Tn ]/(T1 1 . . . Tn n - f ) X,a
n+1
k
/X
1in
X
x
Di
k
a = (a1 , . . . , an ) f (X, O) = u.ta1 . . . tan n 1 X
u, t1 , . . . , tn (U, O)
k (a1 , . . . , an ) (N - {0})n
(U, O)
Di
X
n
u = 1 a u.ta1 . . . tan = t1a1 . . . tn n = n 1
x
ti = 0
k
x
k
a a Stf = X{T1 , . . . , Tn }/(T1 1 . . . Tn n - f ) . X,a
X
x
X
x
~ k
Et/X
ai
f: Y
X
d = p.g.c.d.(a1 , . . . , an ) (b1 , . . . , bn ) Zn A X X /X x X
k
(a1 , . . . , an ) =
k
d = f an
g
Vt
W x H xc q F C W Qc I U W He U b De Q F We i q I C F Qc I Q I x be W D C XP7dV!#0dVYdBVf`YBBREsftXPXYV# Q b H UW TPc t Q b Q h De We @EdPf#C T9B!YXQ U h H hc q H W Q U S Ie B!d#C @PYXRde 7BfPXPBEC Q H W Q I c I Q De F I Q H I U t Q b $ Qc h H x H D Q F Q H Q I Q fR777R R2fc Q b W UWWQ b U h QI g I U C q HQ F b h H YRX4EwRBVB#ErQ D C e W I C H W D h W I h H h c q Q U S W e U q Q W W e c H W 'V t g #dPR#PRX fVsBPBB2PRffYXQ g d Q U S aH D Q e V#r#Xde c t BwXVVBX'#dR0Y7R#fe @VEfPVEwX@PBXEfh yf#XBRfVBPPf!YXQ Q b W Q i D h I g W Q b D Ce H Q W I Q H D c Q b W D Ce H D C W Q b Q I g j c c Qc h x Q U S We U q Q W Wec H W Q I g j c RB7#dEh h g
x[R, R
-1
D Ce H hc Q I hc h EfPfXPyfy'D h t
D C b Q e H D Q b sV#sVR7e XW g V V t e PfBVP#!sQ t I U C BE5q Q b Q h F c H W sEETde YXQ Q U S I Q W C q q U W W Uc q Q B7P#5BBPVfBsb d Q F We i q I C F H UQ q XGD h t W Qcc Q F I C W Qe I x W Q b U h Q D D c Q i q I C F C We H W XfdXP#0dR7PTEXBBEh @BBRE#Yse0Q Q I jec U x I H W R7ffBEXP`e0Q Q F F EC 4 t fBsYXQ H hc q H W V Vt g d e W x H xc q F C XP7dV!# W Q c I W H e U b e Q F W i q I C F c Q U h c Q I C q h Q I H C 0drBUP2BBVDdYeBBREwQf!BS7`BU#TI7RPD#!FD h t 7`9B`e0Q D Q H hc q HW Q U S Ie V#rd#C PYXRde RBfV#`BEC 5t PPfy7fBg Q H W Q I c Q I Uc D C I U Q W Wec D Qe DQ QI g 7sRBV H W Q e U W Q i D I g W Q D Ce H Q I Q H D c Qc h x H W YX|BSXBVEhRVXb#dRXPW7Pfe sdE#Xe0Q t Q t e I h q Qx D D C b HW EVs7VBEYXQ Q b h t b!#VfPWf#C hnPRffwHYXD7REsPVDfh Q Q h De e U Q W Wec W Q U Q H g C e W e Q F We i q I C F Qc Q U S I Q I H D C F Q b W I Cc h H U W YBBREnfTBw7RP#!sR#dE!Pc t VYW I U a QXBdR7RxVfe nfeEC 7teBS x De F I Q H b D c Q DQ U W Q U Se Cc C q C H W Q I g j c 0Bf#EfE5ER`XRB7#dEh Q b H D Q b x Q F We i q I C !Xfe YBBREF D BUh #C #7WVEh st g t d t d BIT7RycThq D W h Q D Q U S h FQI h I QBST7P#C5BBPWn75#C U IQW q q U H UQ q D Q U SWI BRR#C t # t t D e He D x d d #daC RBb hc D h b Q F C WVEB2F!# W D C HQ V#deRXPW Q We U H Qe D Q q H b H D Q F Q U Se Cc C q C H Q I jec U x I Q I g j c PHYfQzcde qS7Q2dVVs5ie2Q77BE#d##P`R7fdB#7R`PBXEfh Q D BBU IQ c q FQI QH e 0Q h Q S IQW q q UW H Q q D BUtXYCVBYUX@EC Q b g e QBPfe t YW0PI eRI)#Q#VfPf#C BEVq i DW q Q W e h h t D e W e D U I h 7#fhV!XP`PYdHBU ! Qcc U D Q U Se H We I x H h I h Qc h x D Q Qcce h I ffBBPYfR7REXsdE#7v7`fdfh PH Q b Qcc Q H W Q fdXwYXTQ 7RBh zv))7|zB U QI q D U S He h U b H D Q EC qd Xde
X{R, R-1}
e U S D Ce H h ec q Fe W hc B%#dR975fBdP0Q h
U
f
V BV BV EfPdY#PI t t D Ce He W C q C "
/ StU a
-1
9 B & B T9 7 E! ! 9T B D B E 8v R B X7 s@ ) p 9 P T 7 9 d V9! ` a 7 # p B R C B D E B q P B 07 s@ 7T9 7 E! ! 9T 7 E 5 d p 9 R fr5n&Y ` P 77P 9 v B ) |z{|HY|@E sr ~ Xy wy w 9 p A p p tEred7 7 7d d Prd ! T9 k Q H D e U W D Ce He W C q C I q h Ph BYvEfPdY#PVfc Q2VU XRHERPE5q!fDEP7QREtETXBPXQ t b Q D Q W D h H I C Fe H C W Wc F I C W h F x i W W 4 H D Q F Qc q Fe W W I Cc h h I Q H C D 77fBdPEf7PEBD t H 7Q h a a CqVdXYX`QBfR7V7xEE7sQ t Q c h H e F Q W H D Q F Q c h g C c H W e Uc Q H W Q U Se I x D W h c Q FI C h F x i 7RET7BRW Q Qc ge W I De H W fc fBfY7Q dYXQ H Q fBgeW !XY7XdEVEfEYXQ IBPPWRffseW0Q U W Q Wec H e H D Q Qc h I D U PW 7F7f7xV7xE2WVdc vt g W D Ce H hc U D D h W Qc I h q e D x VEfPfVVsXfssBV7b d#Gh H De C q U Q q i eH a Q b QfBERYW 7PWr77fcEfE`e0Q c g h H e F Q H D Q F Q h g Cc H W c F I C h F x i W Q Q H D h We F I Ce D 7QRET7BR!fc R#EPP#VBU c H Q I U C q b DQI q D c Q U SWI BE547RBy#C wBRR#C a a t W Q FI C W h F x i c7REET7VPW Q b Q U Se D C D h Q F We i q I C F C We D U h D h F x i rV#B#BX`PfVBP#EPfvBEC T7VPW D U HW Ve0Q a a e Q F PW F7!Q vt WQ e eI x x I XbEfPWXRHfePh Q b Q U Se D C D h Q F We i q I C F C We D U h D c Q F I C h F x i BBBEBE7wPfVBP#EPfBny#C 7RETXBPW HW YXQ e ! t H 7Q H D Q F Qc q Fe W h I Q H C D D 77fBdPRXPEB@#C a
(a1 , . . . , an ) f (X, O) X k xX k k f =U ai U o X = Spf(k ) x k X
a a Stf = X{T1, . . . , Tn }/(T1 1 . . . Tn n - f ). X,a
QW C q D P#5@#C
H Q Q be e I x H xe I 7`f#dRrPfRh
DEfePWVDEC72Qb`QBUSPePIW#hVqhi qvfe qRR#C C U W D c U SWI V t g d e e Q F F PW !XsQ Q D U HW VBe0Q
X k
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
Ti
X = Spf(k {U, U -1 })
X
k
ti
o = (0, . . . , 0)
p.g.c.d.(a1 , . . . , an ) = 1 k
A
zi
X
~ k[[T1 , . . . , Tn ]]
St a
a a (k {T1 , . . . , Tn }/T1 1 . . . Tn n - ) / 1 , . . . , Tn ) /(T
f
(A )x / x /m ti 1in
k
f (X, O)
U
/ k(x)[[z1 , . . . , zm ]] ~
f
f
t1 = = t n = 0 x
u = 1 a a k {T1 , . . . , Tn }/T1 1 . . . Tn n - / St f : X
u = u.R-a1 t1 = R.t1
I U C BE5q
H Q
a W0QPIVg7jEfch Q b Q F We i q I C F Qc W D C I x be W D 4 sYBBREsfER7YEC 3t QW C q P#5%D h t ERXB7#2#7'E!VXPfTPP#V%D h c h I x D x W h U h H D h D Q H De h F Q W W h q t RdEf7e0PfBVP#!s722VRR9B!h Q Ie hc H W Q Q F We i q I C F Q Q b Q b U He H hc q
.t1a1
a . . . tn n
u
=
a a k {U, U -1 , T1 , . . . , Tn }/U.T1 1 . . . Tn n -
k
f StX,a .
-1
T1 = = T n = 0
k
p.g.c.d.(a1 , . . . , an ) = 1 k U
U Sta
Stf a
-1
Ti
k
Stf a
x n+1
a
zi
/ A {R, R-1 }
pour 1 i n.
[Stf ]/ X,a /() St a
/ (A )x / x /m
ti = t i
/ StU -1
U h Q W Wec H W Q U D Q H g C e W De TPPd`YXGXP#YVfh
a hT7VPW EPBVYRXBEh 2RBVh t Fx i W U CW U b W UWWQ b U Q b QI g Q F We i q I C F Qc Q U S W D C I H D !PBBREd!B`VERPEC t BPW I U H Q
] Ti ti f
I U BPW
U
u
f : X{R, R-1 }
-1
-1 StU a
HW YXQ
a
Spec(A /(t1 , . . . , tn )[R]/Ra1 - uv -1 )
Qb`EdPf#C Q h De We Qe D e U d#C 7Q BS
~ k
A /(t1 , . . . , tn )
a a k{U, U -1 , T1 , . . . , Tn }/U.T1 1 . . . , Tn n / 1 , . . . , Tn ) /(T
Spec(A /(t1 , . . . , tn )[R, R-1 ])
Spec(A /(t1 , . . . , tn )) / StU -1 f : X{R, R-1}
a
/ Spec(k[U, U -1 ]) ~
o[U, U
]
/ (A )x {R, R-1}/ x /m U X = Spm(A ) u, t1 , . . . , tn A [Stf ] X,a u.R-a1 . / A / St a x[R, R-1 ] ~ v k ~ k(x)[[z1 , . . . , zr ]] Stf /(),a X/ i 2 x[R, R-1 ] Stf ,a X X (a1 , . . . , an ) x X A k
d e
$ 'V t
d e
Q F We i q I C F Qc I h !PBBREfEVq Q b Q h F @#Tfe c Q b Qc j b C F H U C H I U C q H D h D Q I q D dX!!ERwVE5VXPV%7Q Q U Se I xD x H De C q D U He D x BfR7V7Ed#BBb D EC x F I Q H De C q D U H !R7`fE5yBsQ W H Qc q F C W x Uc fBE`0BdEh VRE7!BEfPV7R7VBXBB#`ERx #X7BPVd7Xx W q I C Q b D Ce W D Q H Q Q D U Q x D D C b H D h H H D Q F Q U S C I qe t Ie hc H D h D Q H De h F H W Q Q H Q Ie b D Ce H h ec q F dEf7EV7R#fTYX`PXRd#dR97fBd e t WQ i qI C F CW XVBP#EPfe H D Q F Qcc Q I U H h D H D C W Wc Q U be W x I W q I C W Qc H D C 77ff7RBPVEPtfXBY0PVREtXf##b W x F I Q W H De C q Q b Qc g F Q W D Q D U H D Q F Q U Se D C D h Q e H D Q b 0PXd#rfB7PV7@B2#X7BBEBE7`VdP7e 7W D 7Q Q b Q I g hc Q U S H RBVtsBzQ c Q D D Ce H h I He h D x F I Q H De C q D U He U b D Q U S I Q U S I h F Q I Q b W I Cc h H U W g 7BVEfP9|d`XQ R7qfE5sBzdBVfe V#qXBRET7RsvEfzP#c t YXQ c HW e W H D Q F Qc U Q W H Q e W Qc h e I H H W Y77fBXY7vYsfe RPYXQ D Ce W D Q H c Q U S H D Q EfY7P7Q `B#Xde c t g V t d t d EfPdY#PVq 7fdXY7Bi D Ce He W C q C I q hc I h D Qec x W D Q t t HYXQ g W qW Q Y0PI Q U S We U BPBBq g t PXRI qWQ W D h b W Cc H D Q F Qc h I x H De H W Br#fs77fEXPd`YXQ g qWQ PXRI W qI C VRE7TQ t EfPVfh PY7 D Ce H h Uc QHHQ t Y0PI t PXRI rPE5BERIVq g HI C q q h I h qW Q Q b x H xc q F C Q RfBE7fc g t PXRI qWQ W D CH V#PC t BXPEV!b g I UQH U h i Q qWQ Q b D Ce H h Uc EfP9dEh b Q t Vt t U h D D D HW 0QBBh BUsYXQ g qWQ PXRI Q F F Q c Q c I vt I U C BE5q Q b Qc g h I h q x W Qe D D Ce W D Q H Q Q D U H W fBEREV7PndVV4#dPV7RTBBsYXQ d d 7fdsEh g Q U S He W D V B`EhP4#C ' t d t d BT7RnEh )t Q U S I h F Q I hc I Q I g hc Q b W H De C q W Q PBfrRd#Xfc W D CH V#PEC zv))7|zB 4B 2 7B& 7 & 9 w Tf B X7 s@ #T7 P R 7 { #P 7p T ! EE 7 #nd 7T ) |HY|@E sr y y w 9 p A p p k 7d V t d t d EfPdBDVxB!cfEC XBUPXBD7#RHd#XQsh !sP7dRB#PBsRY77fBwRE7PB75@D h D Ce He b h Ie W Q Se I x x W De C q W b Q be c x H xe I q C I q Q H H Q I Qc U F I C Q I H U Q q g 7 s 7 R D V z T7B 9 { ERv T 9 T{ P q 2 E` {#P` B 9 E V t 9 E 9 P9 T97E R#r{7 R z7 B& 5 sR 7R B i Q DW Q b x H xe I q C I 9Td {ER{ { fe VYe sP7dRBERBq 9 9 #r{ TP 0 7d & B hc Q e I VdR7x w EE 7 #4d 74 9T 9) {zw 6Yz@@Vo 7 wy y 9 p A 9 8 t #dRVV2VBV`7V7E7vD D Ce He D x b Q D U I h q Q D Q F F C h
c b y
( ! # A (
w x
! ! v f 5 ! ! u s( # 5 Y Y ( p" p63 " A # A@t&97 # r3 q(
h f h f ip ig
3!
c e
3( c a Y "( 9db`X
EWV
t W De C W Q g W C D I U C q Q xe I q C I q q h W Uc q hc Qc g F Q W Qe Cc C q C H Q H H Q Q U Se I g x c h Qe I H x F C x Vd#Y0BB#r7fPVERBBdB`dVw7P`d#EfE5EP`RY7 BPV7EfsfPREXE t D Q We H C F W Qc I U C q Q F F 7dPEXftVE5rEC 4 8 F e BPfe BrfE#d##Pf``EfPrB#EfVEh BBRPf#C EfPXP`RY7VEh i Q DW Q b Qe Cc C q C H hc Q b Q be e I Q U Cc h D c He U b C I H De D D Ce H Q W Q H H Q W D t B#BT9`PBXEfh g Q b C D h Q I g j c Q D U I U W e D Q q i H Q b W h F x i W U h W Q xe C W W h W Q be e I W Q U Se H ic h D h W x H xe I W Q b W h Qc H W Q e Q Qc g h I g F C D x b H D Q F Q U C Q BVBP|BVs5e`ETXBP07fPP#XEfP0BfRdEV0PfRh 0EX2dYX|X7 fBBw#B7BX!XPI #0PI a D U H D Q H H Q F b h W Q be e I W x H xe I B2XPY7Es0f#dRrXRfPEh W Qc Q U S H D Q F Qc h x h I Q W C q q U W 0dB277f7@RXYBVYD h t 0PEV7PsXP#5BBP2ER7PsXEfP`0P7dRh `7RE W x I h q x W W x W C q q U W H D C I Q W W Q b e e I W x H x e I H Q Wc Q F I C a W h F x i W W Qc W U % We H Q " C I e W h U S W x W C q q U W H D C I Q W Q W h g Q D U I U W e D Q q i H Q b W h F x i W W Qc W U C H D h D Q H De h F Q b Ie H I h ETXBP0dzVEC Gt dP0e#RBq PEVXP#5BBPER7PPEBVzBPBVe|#T7BRXf|V#PH 7PfTtfPPq U
k
( 0 ' & ' S ( ' S H I 2))D4)!TRQPFGFED)A@28)5421)%#! H & 7 ( C B ' 9 ( 7 6 ' 3 ( 0 ( ' & $ "
a d)d t Q b Qc g h H e F Q W H D Q F Qc h g Cc Qc j b C F D U H W sdBERYW 7PX!XdEVEfEsf7BBYXQ Q U S H DQ BXde c t W D h b U DQH D C VB7R## HW YXQ Q b I Qec U De U Qec Qc Q U S We U BXdfB#BdPW 7fdfBPBBq Q b Qc ge W We F b h H D Q F Q H hc D U W I Cc h H W dVdPPET77R90x VwEfnYXQ c Q F I C x H xc q F C 7REvP7dB#Q t e H Q " C I dPXe#RBq He U b x I H D Q F Q Ie h W W Q x D D C U h F I C D W H D Q F Q We C I I U Q W e b D H Q I Qec U x 7XdfBEXPI 0Q h g BBXR77RfRP0XVEVD EREV#X7Pf#P7BXYe BU a h Fx i T7VPW D U W I Cc h H D Qe H g C BEfr#XdRED t I U W Qe We C i H D h H x Qcc Q U be W x I Q F I C D Q D Qcc U D Q U Se H We I x H h I h Q b H D Qcc Q ffBrV#PYPXP#RE7wB7fdXBQ h g BP2dPf#B2fdXBY0PsREBwBBU Q U S QW C q q UW V#!YVBYGD h zv))7|zB h F x i W U h h h D C Ie Q b W x He I hc U De W W Q b D Ce H Uc C W x I hc Q U Sec q q h TXBP9 VERdBXRRB#BdP`XEfPfEPXRnV#fBBD h t eY d7 !7 Td 'dE`dE XT T9 E w Xy wy s|{zw 6Yz@E sr 9 p A p p t h h D C e Q b Wx eI c U eW WQ b D eH c CW xI c Q eH U H D e UW H h c UWxI " 5968 p VERfIXRHRhBEVdDPXB@EfCPBdU#Y0P!hsYWdefcPB#Eh BP9HBPXR`Q )d t dEf7e0v0Q 3t fBYXQ Ie hc H W Q e 4 H hc q H W t Q U S IQIH D C F x DQ F hI HW Q D Bs7RP#!TBX2e0G#C g d BV t #d !XP#7Bi % 9 fEC 0PB`VTPdP9BnBR7RRnfsh zt 9B2YXQ Ie W Q I g I h q Q b U He H hc q Q b Q I j He I Qc I H hc q H W $ d Q FjI Cx
c h H D C e I C i Q F We i q I C F Qc Q U S I U C I q Q b D C b H U W EREXREBYBBRE!d!BXQ #PB!VEsPc t
Q F We i q I C F Q I Q e H D Q be H U Q q sYBBREs77VdP7BfXD h
A 5
18 38 6 8 6 & $ " & ( 8 ( 5997%553 q 08
u.R-a1 T1
I U C B#q
I U VYW
H 7Q
I U BPW
e Hh H U F F C Qc D he I H Qc W D h R9wE7sd#BPR`dBb I U VYW Qe D Q e U d#C 7BS
U
F
1 8 3 " " " @55XG
L
^ ^ f k K : Y k K
~ k[U, U -1 ]
k(p) = k(q)
^ (f k K)(y) = x
E
L1 , . . . , L n k(p)
/ Spec(A) E = E/ /() Spec(A/)
/ k(x)[R, R-1 ] ~
/ X k K ^
K K(x) = K(y)
q1 , . . . , qn M(Y ) k(p) 1in k(qi ) 1 K k(p) k(qi )
px M(X) z1 .R p k E
~ k[[T1 , . . . , Tn ]]{U, U -1 }
~ k[[T1 ]]{U, U -1 } b
X = Spm(A) Spec(A ) E E/()
f : Y
f : Y
Ti
/X
/X
zi
X
~ k[[T1 ]]
i2
(q1 )K , . . . , (qn )K
/ k(x)[[z1 , . . . , zm ]]{R, R-1 } ~
K k(p) k(qi )
Li
/ k(x)[[z1 ]]{R, R-1 } ~ 5
f -1 (p)
^ x X k K
k p M(X)
X
Spf(A )
^ pK X k K
k
kK X
k (p) k(p) Li
X
A
q M(Y )
red(x)
k (qi ) k(qi ) k (p)
kK
^ x X k K / Spm(L) f A
k (qi ) K Li ^ f k K pK ^ y Y k K f f (q) = p A k(qi )
{
#nd! t
sB sR
& B E f B R# V 7 7T9 p B E f T B !#7 7 9v |w 6Yz@sr y y 9 p A p p t #5ER`X7fh V#0XB#XYBde W C q C H Q b W Q D Qc e U S x W Q b H D Q We U b D
(Et/X, Nis) / (Etqc /X, Nis) , (VarRig/k, Nis) k k X k
2
/ X )iI U ( Vlr /X / U )jJ /U
H Q a a a tW W He 0QPdYW2QBb We q I F e h U W Q W x F C W De Qc q W Qe I C x H h W U C W W Q Xb0Q7PI#tXQBf7fBXfP#E7R9X VEP0dc `QFPfViBP#CFrQ 0QPH#hVqEC7 YW#VSt0QEfPtWXRt fePEh W b e e I x H x I a QH D RCBGDEC Q!FXF!Q ) 0QP #hVqFE7 Y#hVrHEPWrWXB#Ys0d`E`Xfx W H C e W U S D C Q I U C W W Qc H D C b W Qc h H a a W0Qb@QXx!FRIECQBfe7fBqQdREC#7PH7 UEPW4fc D Qc e I x h W C h hIQ C D RXPHVD%EC Q e I x xe I bEfePnRHfPEh Q D U Qx D D C b H D h BB@7BB#!ERH u yzw HY|@@o 7 X y 9 p A 9 8 t e BPe b i Q DW Q Qe D Qcce F h hc Q U fBV2dfdE!fsBS
Q I H D Q W Qc h H x W Q F We i q I C PR7rXfXYBBREF
G B #R @ r{9 Ps 7 0 29# ERd9 { d 7d7 Y2 ERd9 { Es P9 T7d EPf {77 )d x H xe I q C I q hc Q e I P7dRB#PB!sVdPXx Q U S H V#7Q Q D h BT9I a IfR#dE`YXc t e BPfe !P7dRBERB@fr7VdPXx e hc W I Cc h H W Q i Q DW Q b x H xe I q C I q hc H D Q e I Q U S W Qcc Q H W Q b C D B`XfdXPXEBTh Q b W Qe D W Qcce F h W Q b I h q W H D Q F Q D h I W Q b W I Cc h Q H We Q XfBVwXffd0!EV!R7XBT9w0EfPY7c t
k P i l Yi l U l / U )lK,rR l ( Zij
r
Eg a dcVB@fQv!REC%ETQ h c U q c Q FI D C H
I U i Q DW BEC t e BPfe Q b x H x e I q C I q h c Q e I D Ce H h We I C H h Q D U He We C i D EfPPP#PErBBdYEB#C Q U BS `PfRBERB!f`VdR7x a He #C ! t #BT9h Q b C D Qcc Q H H fdXP7Q Q D hI e U S W Q b C D BTRqB0#BT9h RP70dER7XPfBVP#!wwfBV`fdfsBVU Q I H D Q W Qc h H x W Q F We i q I C F Q b Qe D Qcce F h Q D a D U Q I U C W Q b Q F We i q I C V!7RVEPnnYBBREF BU D He ! i Q D W #C )t e BPe 77RI E7XRGVsYRVEwe0Q H DQ FQ U CQI D U eWW U h HW Qcce F h hc Q U S W D C I H D fdfnf2V#VERP#C t fe VYe 77RI E7XR0b i Q DW WH DQ FQ U CQI W Q H Q H D Qe 7fEC ! t Q b i Q DW e BPfe 7XPI #XRvBe0Q H DQ FQ U CQI D U HW Q b W UWWQ Q U S QIH D C F e BsPREq0Q 4 t VPRXb a U i Q DW he BPfe sPfRBERB!f`VdR7x XB#Y0BBRBE72d`B2fdXPXB#BT9XRfPEh Q b x H xe I q C I q hc Q e I W Q I U C W W Q b He U b C I q C Qc Q U S Qcc Q H W Q b C D h W x H xe I Q I H D Q W Qc h H x W Q F We i q I C F Q RP70dER0YBBREsb a Qe D Qcc h Q D U I h q Q x D h I Q I H D C b H U Q q Qc fBVfdfeFTVBsVXBT9TPRV#Xnffc t Q e H D Q b nVR7e XW Qcce F h dff!Evh t #BT9h Q b C D D BU 0Q h a Q F We i q I C F D U Q D D C b Q W PfBVP#!BsVBE2PD h t Q F We i q I C F D U H D e U W h cc U q D C PfVBP#vV#Eh BP #Vg fBBvEPW HQ i Q DW 7e BPfe HX!XQPI #C0P!BU DQ F U QI D H D Qe 7fEC ! t 0dVPEVS #YW 0dER0!PBBREXQ xzv))7|zB W Qc g h I I h U H D C W Qc h H x W Q F We i q I C F W R B R9 | 9P{ d Rd 77 # X7 2@ 7P0 P E I |w v@vo W y y o u u k2Eh eBPIBhRY2Q7fcsd!hD h t ie BPfe 77RI E7XRB#X7RdEf7e0rfBfYR!BEYXQ #t7XPI #XRD H D U W b h b D H W F F Q Qc Q DW H D Q F Q U C Q I D U H D Q F Q Ie hc H W Q Qc ge W We F b h H I U C H D Q F Q U C Q I VB 'YY 07 s@ E{ P r w R 79 p f # f{7 E9 sR B e e d E d! ) & B #R H D Q F Q U C Q 9T {w 9d fe VYe 7XPI #XRI i Q DW 0d7 # { E d! ` Q #7 9) |w HY|@@Vo 7 }y y 9 p A 9 8 C B T 9TE p & B d 7 T E
U k k
lK
I U W Qe D x BP2dBb D Ce H Q W Q D U H v#dRXP2BB7Q HI U P7Q EC D U Q H We Q c VsPYfzde H De C q H U C H D Q e W H D Q F Qc U Q W H Q e W i Q D W d#BEPvXqYr#X7fB7PzYe BPfe 2P7dRBERB`VR7x Q b x H xe I q C I q hc Q e I Q U S Q H D Q b x x I q D Ce He W C q C I BsP7BX0PVv#dRP#5ERBq hc Q b He U b x b D Q TwVXXGD h t dBfPRXQ de Qc ge W I D H 7Q Q U S Wc Q B2fXPH H Q W H I U C W Q h De We RP7Q Er0EdPf#C W0`VE`PY7vc t 7BBfPXRP#XfBPYBBRE#Y@VdBdvfe |#7Q g ' t d t d BT7RffEC #B9Bq Q b D C b Q H We Q Wc Q U be W x I W q I C W Qc I U W Q F We i q I C F C We D U He U b De c U S H V Q U S I h F Q I hc Ie x e F h I D C D H Q H hc e D H W Q c U S We U q Qc ge W I De W I Cc h H W BVwe0sde qRfBVTdBfPRXQ d`EfwYXQ Q F We i q I C F YBBRETQ t XQ h Q U S c Q H Q U Se I x D x V#t7RBfR7BXE a H De C D #fE5q BU He ! W Q be e I W x H xe I dEC Ht XBf#dRXRfPEh Q b Qc h H x Q F We i q I C 4fYBBREF BU D He dEC ! 6 $ q 5 % d5df
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
/X ^ Xk K ^ Y k K
D U He U b De c VdBVf|fe
t 7BBfPXRtVP#B#q c Q U be W x I W q I C I U C H D h i ph Q U S cQ BXPH D Q Q I g hc W D h b Q U Se I x D x H De C q D U H W 72RBV!B2BPXB7#d#Ve0Q
k(q) k(px ) K(x) K(x) px
q M(Y ) x
k(px ) = k(q)
QBbWVUYWRXBEh WQ b U ! e 3t
SmAfnd/k Et/X
U R / X )iI R = ( Yi / X )iI,jJi P = ( Zij
Qb2xFPIXQd#q H De C H DQ b 7e x
R = ( Yi U k
Afnd/k SmRig/k SmAfnd/k Et/X
U Flt(p)
Ul
Vj
( Ul
/ X )iI
/X f : Y k(p) = k(q) p = f (q)
k
V Flt(q) f
( Vlr
/U
R X U = ( Yi X U
/ U )lK
k
X
U Flt(p)
R X U
f
Pi = ( Zij
k
j
/ U )iI
Vj
U
Ul
Vlr
U
Etqc /X Et/X X VarRigqc /k VarRig/k SmRigqc /k SmRig/k
H R7Q a W x H xe I 0P7dRh
k
X
/ Yi )jJi
f (V ) = U
/U
/ (VarRigqc /k, Nis)
R X U
/ Yi X U l
/ X.
/ X )iI,jJi
/ Ul
6Y
*
lK
U
R = ( Yi /X X
OX,p
f
f|V : V
X
R = ( Yi
k / OY,q
U
R = (ui : Ui k (SmRig/k, Nis) / X )iI /X /U
l,r
( Vj
Vlr
VarRig/k Afnd/k SmRig/k Nis
/ X )iI
p M(X)
q M(Y )
/ (SmRigqc /k, Nis) R X U / Ul )rR l U VarRig/k /X k
C
& B E 9TwETT 9 0 t V 9V | 9 E EGd7'0 7G GEEy7 9v zw 6Yz@sr y y 9 p A p p t 7REET7VPW a Wc Q F I C W h F x i W Qc I U W i Q D W XfBPve BPfe `d#EfE5EP`P7EV7qBS Q b Qe Cc C q C H hc Q I b D Q D Q e U I U W Qe Cc C q C H x I q Q D U H W Q Wc Q F I C W h F x i W Q b i Q D W W H D Q F Q U C Q I W Q b Q W W hc hc Q U S H D Q F Qce h Q e I BPsd#EfE5ERRBVBzYX|fX!REqET7VPtwe BPfe R7XPI #XRzXPREV#77fd7E R7x D h B B B i Q DW 0 7 9 7 B # X7 2 @ T&P0 P H D Q F Q U C Q f fe BYe 77RI E7XRI ` T9 7 E d T E sR B d! s 7s #T9 7 ) zw HY|@@o 7 y y 9 p A 9 8 k Q H D e U W D Ce He D x b hc Q Ie h Q b Qce H U D C b H W Q R#Eh BPEfPdBV7f`RdE`2dfPBEYXqc )d t H D h DQI q D 7RB7Q #X7Q dXPBdB0EC Q b H D Q F jc Q I D U He U b x b D H D Q F jc Q I D U x D D C b H D h #X7Q dXPBsVBEERH u t Q b Qc he x q W Q I g hc H W Tf0BYTRBVf2YXQ t Q t e Q U S cQ BXPH x F I Q H De C q D U Q H We Q c PXs#fE5GBPYfde 4 c Qe H D Q W Q D C H hc q H W Q F F Ft Q W x F I Q W H De C q W Q W h Qc H D Q F Q U Se D U I Q I x be W D C x D Q F h I e W De h H W Q b 0PXRfE5Xb #7dsX!XBfVB7R7YE`BX!ERPVfYX!D h t 0EfPI W Q be e g 7fP7PPW 0VEBb 9BsYXQ EC W x H xe I Q b i Q D W H D Q F Q U C Q I D U Q I C D Q D C b H W 0PfRh 2e BPfe X!XPI #0PsBRE7V7V#zYX Q 4 t H D Q F Q U Se D C D h Q e H D Q b |7XBfB#BXsVdP7e 7W Q F We i q I C F PfVBP#sQ
^ (uk K ) x : Spm(l) k
x I I h hc W D h c Q D D Ce H h I He h D x F I Q H De C q U b H D Q F jc Q I D U I U C I H H D Q Q PRXfVEBb g XBB#dR9E@XQ R7d#77Q f7RB7Q ERP#Xde PI
~ Spec(K)
p
x I I h Qc W D h b H D Q F jc Q I D U I U C RPE7sdVB77Q f7RB7Q ERI % a t Q i q I C F C We He C `VBP#EPfdEPW Q U S Qcc Q B2fdXPH h Fx i TXBPW W H Qc q F C W x Uc W q I C Q b D Ce W D Q H Q Q D U I U C I H H U Q q fBE7XBfh VRE72v#dPVXPsBVtXQ #PR75vD h t U b H De C q D fE54BU dEC 3t e BPfe 77RI E7XR4BYXQ He ! i Q D W H DQ FQ U CQI D U HW Q U S I Q I H D C F Q b He W BXPR##2dEEh Xzc t H D Q F xc x D U Q He U b x I H W 7XdX4BTwRVXRYXQ a Q U S IQW C q q UW QI C DQ H UQ q D WH h q F C eW h U S H D C Br7P#5BBPwP#VXrB754EC REV# PEV#YW W Qc H 0drQ e D Ce H h ec q F c Q W W h q Q F F EC 4 t ge g fe EfP9XffBfe TwPP#V4D h t e BPe #X7RI E0PBYXQ i Q DW H DQ FQ U C QI D U HW D Ce H Q v#dR0YW Q U S We U q Qc ge W W C q H W Q e 4 BRfVB`dBfPPYXv7XQ 3t a Q D U I U CIH Q b H UW c t BV7Q ERP`P|fe g (t d B d XP#7Bi % 9 $ fEC Xf#T7BRW Q FjI Cx Ie W Qc h H x W h F x i W Q b Qcc Q Q H D Qc e U S x H W Q W Qc h H X2ff7T2R7fh V#0e0Xfx a Wc Q F I C W h F x i 7RE`#XBRW Q b D Ce H Q W Q D U Q I C D Q U %EfPXP@BVTP#VX%EC W Q b Qe I C x H h hc Q U S x D D C b H D h u XTfP#E7R9XynB@BVE#ERH t
~ k a : U X Spec(~ l) X u: U ^ a : UX Spf(l ) Spec(~ l) / Spec(~ l) /X X / Spf(l ) Spf(l ) (Ui ) /X u u kK ~ k(p) Spf(l ) p ~ K
H D Q F jc Q I D U I U C I H Q b D C b H U W 77Q f7RB7Q ERPrsV#Yqc t a Wc Q F I C W h F x i 7REET7VPW Q b Q F We i q I C GYBBREF BU qBPXB7#RBV'fYX Q Dt PX@fE5q VU D b Q U Se I x D x Q I g hc H W 4 x F I Q H De C D He dEC ! t Q b Q b W h Qc I Q I x be W D C Q b H U W c #72fXPXfPV#2P|fe Qe D D Ce W D Q H Q Q D fBV#dPVXP!BBU 0Q h W Q F We i q I C F U h H I C q q h I I h q Q He C I b H D Q F jc Q I Q b x H xe I q C I q hc Q b j W W C XPfVBP#y`Y#VEVwR#PvrX!XQ dXP!RfPVERBnBXPPq Q U S IQIH D C F I U C q Q U S H DQ B`XPREsBE5TB27fe c t Q b i Q DW e BPfe H D Q F Q U C Q D U Q I C D Q D C b H W Q Q c t 7XPI #XRI BP#VX!VE2YXTfdc a Q b W Q F We i q I C `XYBBREF b U Q e H D Q b VdPXfe XW Qc he x q W Q I g fXVYTRBV@h t Wc Q F I C W h F x i 7REET7VPW Q U Se x D x Q I g hc H D Q F Q U Se D C D h Q e H D Q b BfRI7V7EsRBV!77V#B#BX2VR7e XW Q F We i q I C F PfVBP#sQ t R7dB#0BdEh VP# W H Qc q F C W x Uc W q I C a QB%EfeYVXPHQBBU b D C W DQ Q D He ! dEC Ut I U C q i Q DW B#e BPfe Q b x H x e I q C I q h c I Q e I QHW QI c WQH h q F C eW h U S H D C H Q nRfPVERB27VdR7x Re0Ptde XR#VE Y#VEPW Q F 4 W FEC Ft g 0QIBEP0dBPVErdRVXPVsRYdBS U C W W Q b He U b C I q C Qc Q I b D Q I q Q H He U H D Q F xc x D U Q He U b x I Q F F H e D Q q i H Q b h F x i 77f7B`RBBXR77Q g BV5e2T7VPW a DVeW0Q U H QBPBBq dVVYX2fdfsBXYBVY`75D h t g fe U S We U Qe D H W Q Qcce F h hc Q U S I Q W C q q U W H U Q q e D Ce H h ec q F c W D C I H D g e EfP7dV!fe V#PREC zv))7|zB G z 0 Y 9d @ d X 7 B EE2{7 # X7 2$ 7P0 P @ d B B 07 s$ T7PX R @ d 7
18 38 6 8 6 & $ " & ( 8 ( 5997%553 q 08
F
1 8 3 " " " @55XG
U
x : Spm(l) X /X Spf(l )
~ u k K ~
K
X
R = ((ui ) : (Ui )
R = ((ui ) : (Ui )
K
~ k
r : Spm(l)
(Et/X, Nis) X X X k=K / X / U k k X k
^ ^ u k K : U k K
^ ^ uk K : Uk K
~ Spec(K)
/ X )iI
/ X )iI
u: U
/ (Et/X, Nis) o ^ u k K p = red(x)
pidK ~
R
k
/ X k K ^
/X
/ (Xk K ) . ^
/ (Uk K ) ^
/ X ,
/ U
5U
/ Xk K ^ ^ k K ) (X
u
(Et/X , Nis) .
u
R = (ui : Ui
p
X
X
/ Spm(L)
u
~ k
k
X
(r )
k K
k
x : Spf(l )
^ (uk K )
X
L
X
/ X )iI
X
X
R
SchF/k ^ u k K x : /X R ~ k
Qc Ie e H fsfEC g fe 7Q g e VEfPdV#nX0VBU VdR7x W D Ce He b D C U Q b W Q b Q D c Q e I H De C q H U C W I Cc d#tERH Efh e ! t He C W H dEPQ Q b W Q U Se I x D x W H De C q W Q b Qc g F Q W D Bt0BfR7V7Ed#0sdB7PV7Q c W D CH EPEC t h D W Qc j b C F Q b Q F We i q I C F D U H W #C Xf7Bv!PBBRE'BYXQ e W Uc q PW dBQ t Q b Q b e EBv7R7nfPP@BVV#Te0 Q 4 D C D Q x F I Q Qe H I h q Q D U D C b H W t fdf4RY7yGPddf%c Qcce F h Q H H Q Q b Q He Fec h QH C PEBD D h t Vn7RnPW D H DQ F F hW U I U C q Q H D h H W D C H D Q Q b Qcce F h Q H H Q BE5PERYVE7v7fe yffdyPP XdR7BRBD D Qe I x i H Q C Qe Cc C q C H Q h q W Q D H W d#EfE5ER7EVPX!BU YXQ Q F F 4 EC Ft Q q h H x Q I j e F Q I q h I Q W e c e H Q W Q x F I Q W Qe H I h q Q b Q H D h I Hc C Qcce F h Q D H D Q F I C b 07R7zXfPPVPERRVEfdftBBU 7RE g 5R7R7f7RBrfc XYfdRU I U C B#q W Qe H I h q W 0dRYEV`XQ t He #C ! R t V#Q D C b H Q U S QIH D C F e 4 BnPREs0Q Pt EEVdPEC VU vwVYRXBb ET!PBBREEPfVwe0Q Q b Q h De We D b W U W W Q U h Q F We i q I C F C We D U H W Q U BS a t Q F F Qc Q QRPEVEPP0b 7 t 7fdc IH D C F W U CWW Q e I U W H hc q H W BY2fBsYXQ Q F F 4 !#C )t C H 7Q D 7Q d Ed d Qc h H x H W fR72e0Q W h Q W D EXnsVEh |t H 7Q D Q Qc h H x H W 7Tfse0Q Q U S cQ B7RH D U Q HWe Q c BResfe e !
Y Y p N (Y/X) q (fX )-1 (p) fX -1 Y^ ~ ) = k(q ) ~ k(p q e1 (q) p = [fX X X ](q ) / Y X X ^ Y
Q U S QI C DQ U V#`RE7EC e H h H U F F C Q F F h I he b D U h R9wE7rEEfBVD t h I h q W D h b x i DQ HW VBs9C Xe0Q Q U S W D CIH D V#VERPEC t Q b Q ij Q D 2BXBBU He ! dEC P R t W Q q h H x W I U Q e W U c q D Q Q b j C I q X5B7fPVdVvX`X7RBvD h t EfPf#RRE2VBdV7PBEV#dfh D Ce H e b h I H D C Q D U Ie D Q H g C W D Ccc W U EC t H U CH I U C q Q b D C D HW BEPtB#re EVYXQ WI C R#dc t YRVEdEPr7REXBP#dPVfV#wrBtEP#5BBU ! Q W W U h He C W Q F j I C x i H U b D Ce W Uc D C hc Q U S W D C W C q q t fe RP@EBYXQ Qc h e I H D C D H W D Ce W D Q H e EfPV7R7Q c g fe D Q x e F h I H W X`VERYXQ ge k Q He h We H h REePPW ! H W Q W Q H D e U W W x H xe I q C I q W Q b Q D YXt0Ph BYtXRfPVERBXsBVU c H Q Q b Q U Se I x D x H De C q D U H W BrBPXB7##fE5vBtYXQ e Ct Q Ie h H D Q F xc q F C rRfRX!XdV!# x FI Q Q R7dc eWW U h QH C D YRVERBD t Q b i Q DW Be BPfe H D Q F Q U C Q I D U D Q Q D h I Q 77RI E7XRB7tBT9PW h Q U S cQ Bz7RH Q b e We I h H I U C b D h I W Uc q Q B|PE)#P7Q E@VEVdV2dc QH C D D RB#C I U BEC t H D Q F Qc q Fe W U 77fBdPEC WxH C D H D CIQ XPEB##PXYW Q b WHQ g C W R7e" E0Q t V7#YW U h D C H D C H 7Q Q I U C W H Q H U g W I U Q H D C W Qc H Q Q H D h I Hc C Qe I C x H h Q D U H W 7RVEPBzB7RV#q0dsRPdV7sdRE#7P7tBBe0 Q 4 a t Q b We D e W h U S H Q W H hc q W Qc j b C F W Q b Qe I C x H h h BrBV Y#Vs7`RfB`0dX`X!dRE#7P7c hIQH C D 7RBGD h t XQ EC BS VPPVEh H U D c Q U D eWW U Q b a t e D c Q F I C h F x i W D b e We I h H I U t t We D e W h U S W H hc q W Qc j b C F W Q b I U C I H H U Q D I Qec U e H I h cQ FI C g BVXP#4T7VPBU vR) YXQ #C Q e fB Y#VnRfVXf7TXBXQ #PRnXq EC 7fdVPPVq D 7RE a a h Fx i T7BRW D b H hc q c Q F I C h F x i W U C W W C I W Uc q Qc Q D e W x BU vfB|7P#2TXBPW VEP#REdBwd`B#dPXb Q U S W D Cc Q q q V#VEf75Bh t fBYXQ H hc q H W Q U BS cQ XPH Qc ge W We F b h H D Q F Q H hc x D fBfYRf#77R9XVU g v % VEVph EfP7dPfV22BBBXR!fVq PYf7qde 9Bq Q $ b U h D i Q b D Ce H h e H hc q Q b Q U Se D i Q H hc I h Q H We Q c W H hc H a XQ h b Q Qc j b C F D U x D D C b H D h u dX!TVBB#tERH #t Qc h H x H D Q F Q H Q I D b H h q F C e W h U S Q De h F C b D U H W dER777R RnBU v#VE7 Y#VBfTE@BYXQ Q S I Q W C q q W H Q W Qc g W We F b h W H I U C W H D Q F Q U C Q I W Q b I h BU7P#5BBUPtXfBfeYRf#RP7Q ER77RI E7XRtXEVq H 7Q I Q hc q F Q I H U Q q D 77EB7R75@#C d t d t d EfPdP#5ERB!fh D Ce He W C q C I q hc I t BfR7BXEsRBV g Q U S e I x D x Q I g hW W D b c h FI C HW R2VEhBsTP#BD2YXQ QdX!TfcsP2VfECF b H D Q D C c j b C F Q e W W De U Q b W H D Q F xc x W Q b W Q Q D D C W Q H D h W C q F C W Q b Qcce F h h B2R7XdXsXs0BB#`XRR#5!#20fdf@c Xde E7 t t Q FI C h Q b i P#`cne QBPfe XQ!XPI #0PnVsIVq DW H D FQ U C QI D U h I Q D h I He h W D c e H Q Q XBT9dRnEC YW 757nD t Q e H D Q F xc x U Q D x F I C H 77f7cBXYW BU vb!REreW0Q H D Q F Q U C Q I Qc Q U S I Q W C q q U W H U Q q 77RI E7XRfBrXYBVY2XD h zv))7|zB 'Y R B B { 0 2C TRX P @ E d7'0T #R5 E 07 s@ V 9V 9v ~ wy y o u o r p o m ){|w vvqTl TRX P 7! EEs{ t WfX!REt0dXbXPW V#PH Qve BPe d#EfE5EPs2h ! c Q F I C W Qc j C F W Q g W U C b i Q D W Qe Cc C q C H hc Q b Q be c a a Q H h q F C e h U S Q be e I x H x I PEV# PWEVBf#dRRRh Q D b i BBU ve QBPfe Bd#EfEC5ERsfBV7x@BERIBEEC qURIT9tHEh BP7REXBPQ DW Q b Qe Cc q C H hc Ie D b U q He h U h D S Q F h D e U W Q F j I C x i H t t )d (t B Q F I Cx 7jREXBi % 9 b7Q7R9XFffBE7XW0PdY`rQ!PBBRIEEPfe Y#VUvDBYXQ BdrQ $ Q H D F Q H he b x Fe Qc U C x b Q He W Q b F We i q C F C W ae W h S U H W Q U S He h t WXRPT0Yd WBBd REWX!VEB b` 7PBbVfrfc t e BPfe #XQ7RI E0PrW0dXQ I7PXRBrH7rXfBDVXRHfdc4UE`HXP!# Q He W Q b W Q F e i q I C F Q b D C H D Q We U De W i Q D W W H D F Q U C Q I Qc H D Q W x I q Q W Qe W Q e Fe h D Q H U F F C H 7Q Qc h x q H Q Q U Se I x D Q I g hc Q h W W h Q b W I Q H D W dEfe7XBPW7BPXB7x#RBVww#RP#VqrRBUXPVECXQ zv))7|zB e 9 pV B d & B 9V t 9{ #! d { EnT
Y fX : Y ^ N (Y/X)X Y
8 ( 18 38 6 & &8 6 1 8 zXvBzV59X9Bz!0V"
(e21 ) (Z (X2 )) = Z (X1 ) Z (X)gen Y / X Mdlpqf (Y /X) fX : Y e-1 (p) N (Y /X ) R P Y Y P = (f : Y ^ (YX X )es
Y fX
/X
X
e
Y e : fX
X Mdl(X)
f
-1
q ~ ~ k(p) k(q)
(N (Y/X)) N (Y /X )
N (Y/X)
/ N (Y/X)
/X
Mdlpqf (Y /X) / X Mdlpqf (Y /X)
/X)
/ fY X
p
X
X
e (Z(Y /X )) X Y fX X k Y
Y fX
Mdlpqf (Y /X)
X
(-)es
N (Y/X)
Z (X) Y q (fX )-1 (pX ) Y f: Y / Y X X ^ ) X (
X e
Mdlpqf (Y /X) Mdlpqf (Y /X) /X fY : Y
Z(Y/X)
R
P = (fe : Ye /X ) Ye
e
e21 : X2 Z(Y/X)
e
Y (fX : Y
Y
e1
/X
/ X.
%/
q
Y
q
Y fX
N (Y/X)
R = (ui : Ui
p = (pX )X LimXMdl(X) Z (X)gen P(X) ~ ~ k(p) = k(q) X
Y (fX )-1 (p)
q
/ X1
Et/X
Z(Y /X )
/ X ,e : X P = ( Ye
/ Mdl(X)
/ X )eE
Z (X) k
/ Et/X
Y/X X
/ X ) Mdlpqf (Y /X)/X X / X )iI
Y fX Mdlpqf (Y /X)
Mdlpqf (f )
/ X )eE
p N (Y /X ) Y Z(Y/X) X X Et/X
Y^ f X X X
^ Y N (Y/X)X fX :
/X X
Z(Y/X) k R
f / Mdl(Y )
f
/ Et/X
/X
e Y X
% p z EE2{ 9v q{zw eBBssp wy y o r p r t k d t d 7REXBPvh BPRdEffERE72dc Q F j I C x i H U b H D e U W Q Ie hcc C I C Q t h D h t I U BPW Qe Cc C q C H Q D U He U b D fEEfE5ER`BBdBde Q He W U b Qe Cc C q C H PdY4B`fE#d##Ph V 072 V R r#YW He b H D C Q b WHQ g C W e" #0Q t I U Q H D C U b Qcc Qe H D Q W W Q Q h F B7RV#nrdf7fPXYRXETfe c QH C D PEBnD h t Q He Fec hc Q h W W h q I h q W I Cc h H D Qe H g Rfd!r#RP#VVtEf7fPBEC 7W x Hec x b Q De Qc q Rf7BBf7fBwh t h Q U Se D C D h D Ce H Q e g D Q H W Q Qc g F Q W D Q H XQ BBEBE7#dR0e" B7e0dVw7PVXt7Q 4 t I U W H D Q F Q U Se i q I C F C We H D Qe D W e U BYt#X7BBBREEPfsXd#C 7Q 7vBS Q W Q Q D D C WQH D hW C q F C Q b 0BB#XRR#5!#b W Qcce F h W Q b Qc g F Q W D c Q U Se I x D x Q I g hc Q h W W h I h q Q e H D Q b W Qc g F Q W D 0dff!EXBfB7PV7Q rBfR7V7ERBVrEEPREq VVdPXfe XfB7PV7Q H 4 Qc ge W I De H W 7Q t dVdPRXQ f`YXQ Q H D Q b x D Ce H Q " C I q hc Q U S W Qcc Q P7e EfP0e#RBvfB20df7RH Q b W Q Q D D C WQH D hW C q F C Q s0BB#rXR#EP#nb W Qcce F h Xfdf W Qc h Q U Se D C D h D Ce H Q e g D Q H D C 0dwXQ wBfVEBE7#dR0e" B7r##YW Q b W H D Q F xc x W Q `R7XdX0dc Q b W Q Q D D C W Q H D h W C q F C W Qc H D Q D e W x rXVBE7XRR#5!#rXf7VEfY0b e YW HQ Q )GD t Q RXe" BR#dE`YXQ e H Q e g W I Cc h H W D Ce H h ec q q #dR9XffBBE h t Q b Q Q D D C QH D hW C q F C Q D U HW 7BBE7wPEPE7!VBrYXQ Q b Q Q D D C Q H D h W C q F C Q D b Q U Se I x D x Q I g hc Q U S H !BB#!PR#5ETBVU v!BfR7V7E!RBVfBsQ I U W H hc q H W c U BPsfVrYXQ fe |#S Q U S I Q W C q q U W H U Q q D W Qc j b C B` 7P#5BBPsB75#C Xf7BF W0d7BT9T!PYdBU ! t Qc I Q D h I Q H He Qe I C x H h hc Q b W H Q g C U Q dRE#7P7n2R7e" E7Bb H 7Q H D Qe C H Q Q #Xd#YW 7574D t f777Bf7fBq Qc j b H D Q F Q De Qc HW Q I U QH D C YX@B7RV#4Q 4 t I U Q H D C D U He D x b Q U Se I x D x Q I g hc Q h W W h q Q b I U Q H D C BXPVE VvdBV74BfR7V7EPVV 4EEPREV@V7PVEyQ
(Ui )i (Vi )i (Et/X)br hom((U /X ), (U/X)) = U/X X=X U X u Ui k U /X
^ ^ X Xk X Mdl(X)/Xk X
W Q F We i q I C F W Qc H 0YBBRE0d7Q XQ h Wc Q F I C W h F x i W Q b W Qc h H x W Q F We i q I C F W Q b H D C W W H Q " g C W Q 7REETXBP2BXfR7XPfBVP#!0EPRe&E0dc a eW D YVfe t W Qe I C x H h W Q b Q He Fec C h H W XfREEXP7s0Pd!ffE`c e0Q Q H D Q h W U C W Qe I C x H h P77EY" V#YfP#E7R9Xrh t Q H D h I Hc C Qe I C RPdV7tfP#E a x H h hc H D e U R9XtEh BPW W Q He W W Q b e H Q " C I q Q He Fe 0PdY0Q RXe'PVrRfdc 0dQBV7BRRr BrRHPsdQVER7bPVEC 4 e b DQ iH CI Q b Q eW c W D CIx eW D )d t D hI Q b H D Q F x I C HW Ie hc H D h D Q H De h F H W Q H h Hc U W x I f7PfTe09BPXR`Q t BERsBX!XEYXQ a a Qc U b C F Q C I x Q b I U Q W e D C D H W Q U S W e Bt RBfd2fBF Q I gec Qc U b C Q D U HW VBYXQ Q U S IQW C q q UW H UQ q D B7P#5BBP75EC fBs R7XBXYe b t #Be0Q BRfBU H hc q H W Q c U S x D D C b H D h u e D HW 9BYX)fe |#`BB##ERH BVte0Q BtdR@#C t d t d 7frdth t fBdP7Q fYXQ Q U S He h W D Q F F Qc Qc I Qc ge W I De H W Q U S H D h D Q H De h F W D C W C q q BtVXPfTEP#5BBU ! t Q b I U Q H h I Qe F Q I q Qc I U W D Ce H Q " C I q hc H W tV7PE7f!XPVfVYn#dRXe'PV2tYXQ C Q x W C q F C hc H W XY#!fte0Q B7V#TXPEfYvc Q U S I Q U S I h F Q I Q b W I Cc h H U W t 9BYX5de XdfB7fRYEVnD t PP7Q ET#dPRX!fVBVPPYXQ H hc q H W Q c I Qec U e H I h q Q H I U C D Ce W I Q F Fe Q D U e W W U h H W Q F We i q I C F Qc Q U S H D Q PfBVP#!dBt7fe c t PP7Q EC QHI U a HW YXQ Qc h D C he D Ce W I Q F F Qc h H x H W fEEEfBb #dP7de c fYXQ Q F F 4 H hc q H W EC Tt 9BYXQ h Fx i T7BRW Q zv))7|zB 9 E 5 B p d7 79#0y0 R d7 9 C d d 7 R T B s B 7 # 7 #@T 9T 9v 7 nE7 7 9) z{|w @vvo W w wy y o u u )d t RPEXe0rXP#7BR`Q t 7VRXBXRsdEVVEfP#RR##e0 Q 4 xIH D C Fx b HW Q Q FjI Cx iH Q x i I Q i Q I Qc h D D Ce H e b h I H D C hc H W t QIH D C F e 4 RPE0Q t H Q D Q Qc h H x H W 7!dER`YXQ Q U S H DQ Bs7fe c t Q F We i q I C F C We D !PBBREEPfGBU t PXRI He U b D dBVfe Q F We i q I C F C W c Q U S I Q W C q q U W H U Q q D PfBVP#!#Ye Bs7P#5BBPX!EC I Q D h I Q H He XBT9rPYdBU ! t g qWQ e We I h H I U fR)P7Q EC t PXRI D b Q U Se I x D x Q I g hc H W BU vsBPXB7#sPB!fYXQ qWQ Q U S cQ B|7RH He dEC ! t #Tf@#YVYW Q h Fe D C W I U H D Q F Q U Se i q I C F C We Qe D 77V#BBRE#Y`d#C 7Q g Q U S Wc Q BfXPH HI U C D U H P7Q E@Bt7Q HI U C D YXQ #BU Q HWe Q c t Q U S I h F Q I hc I Re)de V t d BREXPtEh ) R t D Ce He b D C hc h D Ce H e b h I H D C D Q H W EfPdV#0Q @#dREP#7YXQ e 4 7XQ 5t Q U S QIH D C F e 4 B`RP#!|XQ 5t h Q be D C Qcc XQ `Bf7V#2fdXQ Q U Se I x D x H De C q I U C q BPXB7#d#B#h I h Vq Q b Q h F 2EEfe c Q F F 4 EC Ft Q b Q h De We #VfPf#C h U I U W Qc h H x H W BPdERsYXQ Q U S He U b x b D Q BdB0!XD h t g #TdRY7`BVP#!#YV#b Q h Fe Q H H Q Q i q I C F C We D C W D h b Q h Fe D C W I U VEBETf#YsVYW c Q D D Ce H h Ie g H Q H hc q H W XBBEfPRfB79Be0Q Q b I U q x F I C W D h I H Qc H BBr!REYVERR`d7Q I U W H hc q H W BP9BYXQ Q U S cQ B|7RH Qc ge W We F b dBfPPEh V H D Q F Q H hc x D U Q Ie U I H W D C H U Q q D 77R90BRfBPReEr75EC I e b U h D i Q b D C e H h e H h c q Q b Q F j I C x i H Q c H D h W e c e H U t 'V t t Q F F Q g q $ d#C VEVph EfP9XdP9Bs2XP#7BR`dRfffPD g d d !Xdc W D h Bb Qc Ie ffEC RfEef`EPW W H De C "We b H D C H Q Q b H Q W Q I U q W x F I C W D h I H W Qc Q U S c Q 0PBVXREYVP0dBs7PH Q Bb Qc j b C F D U H D Qe H g C D dXyB`7fP#yEC U D Q H D C H I C q q U W C Q b H Q e D Q q iH Q b H I U C c h x be D U H D h H hc x 7R##PE5BBP#7w7|BV2e`P7Q EqEXB904D t Q h Fe D C W I U W e D H Q c Q D D Ce H h Ie Qc ge H U b x I I EETdEPBPzVV|7BB#dR9 dVg fBfPVXRRde 0Q h W x F I Q U Q b Q D Ce D U x I Q D U Q F F C XR7 7Byb EfBBXP4VB!# Q W I De Q h F c Q Ie I x H U Q D W h Q W D P7Q fEEfe RdR0@Xq #C #747VEh H Q D Q x e F h I D C D H W X2VdEBYXQ Q U S cQ B7PH D C b Q H We Q VEBwPYf7|c t Q U S H BrQ Q b H U g Qc H W `B2fYXQ c U S IQW C q q UW H UQ q D fe qXYVBYB75EC I Q D h I Q H He 7BT9!sPYdBU ! t H Q h D D H DQ F F hW U EC V#X!EPnPW I U Qc ge H U b x I Ie Q x F I Q B#C t g fBfPVXRRd`7R7 Qe H I h q Q D U D C b H W Q e U fPPVrBBVEBYX)BS Q b Q D Q I x i b W D CH V#PEC t g 0Q h Q U S cQ Bv7PH rVXPXBEh c He C W H Q Q Ie h I H D C Qc W D C W C q q U H Q Q dEPrPfP#2fVEP#5BBPW )D t Q U S e I x D x H W Q Q H D Q b x x I q Q q h H c W D h b H e U I H W D C H De C q fE5sQ R BfR7V7EYXsR70XRBs5x BdBRPYVE7 t dEfe PRH a EBD Qc h e I D C Q b Q W h g Q b H D Q F Q D h i D b H Q Ie b I U Q H h D U h Q be D C Qcc Q Q F F !PE!Br77#BGBU v20PfsBXP#yV2XQ !Bf7V#df7EC 4 H7FXdE#7`YX!PXdBR7BRY7 D Q Qc h x H W Q Q I je D I Q b Q H H Q t dcEfe PIREVV#!YXQ t Q h e H D C D D C b HW D Q x e F h I D C D H W I Qec U e H I h q 7fdVPPVD t 7#d B t d 7f@f!7@5x Q F F Qc Qc H Q Q q h H D Ce W D Q H #dPVXP Q t 7@VER%#Be0Q gd QIj Q R7fe!FXPIVqfT27B#PEh 2XYdfPBU hc Q b H D Q F U I c I Q Wece H H D e U h BYW Q b QW h g Q b H DQ FQ D h i U BnY#VT277#BVEh Q b Q h De We EEVdPEC EVBP#EPfe U h Q i qI C F CW a a HW YXQ WRI#dc t 7!VERGEBsYXQ C D Q x e F h I D C D H W Q U S W D CW C q q ! B`VEP#5BBU )t `VYRXBb E2e0Q BS Q b W UWWQ U h HW e U He #C ! t Q b W UWW Q U `VPP0b h Q b V#R7BXEfE54B7Q Q U Se I x D x H De C q D U H Q H We Q ce D Ce H U I H W D C I h H Q Q Red#dRVRPYVE7EVq )4D t g Q XPBnfrV7b U Q I q hc Q b H U g x
Y H X pX UX Y (fX )|V Mdl(X) u /X X Y fX (Z, O) 1 h Y gX pX G(X ) q = (qY )Y f -1 (p) V Flt(q) Y V U fX Mdlpqf (Y /X) U V Y VY fX Y ~ ~ /U k(pX ) = k(qY ) : V fX qY
Y (fX )-1 (g)
18 38 6 8 6 & $ " & ( 8 ( 5997%553 q 08
F
1 8 3 " " " @55XG
U
X
p = (pX )X g = (gX )XMdl(X) P(X) p {g} p=g G(X) X X pX G(X) gX = pX Y Y fX Mdlpqf (Y /X) gX Z(Y/X) / h (fX )-1 (gX ) ~ X ) = k(h) ~ k(g Y (fX )-1 (G(X)) G(X) HL Y1 Y H1 L1
h
X
u u
^ Y = (Y1 X X )es
^ Y X Z
(u, id) : Z
k
(Y, O)
(U/X) Ob(Etqc /X) fX U ^ Ui X U Colim Y
LimXMdl(X) (Et/X, Nis) LimXMdl(X) (Et/X, Nis) /X) (U/X) = ( U
LimXMdl(X) (Et/X, Nis)
/ Y X Z ^ pr1 (u, id)
ColimXMdl(X) Et/X
k homX (U , U)
Y Y fX Mdlpqf (Y /X)/fX Y q fX Y /X fX X ~ ~ h k(g) k(h) ~ X ) k(q) ~ k(p
Y/X
X
((Et/U )br , Nis) pr1 ColimXMdl(X) Et/X homX (U , U) Y G(X ) u (Y, O) ^ ^ homX (U X X , UX X ). / homX (U , U ) X (Vi )i ColimXMdl(X) Et/X / Et/X u: Z Z H Vi X Mdl(X) homX (U , U ) X H gX X Y fX (Et/X, Nis) H1 / Et/X H G(X ) N (Y /X ) L f|V : V Nis /U i /Y X U Flt(p) (U/X) Y q g (Et/X)br HL pX Et/X Z / Y X Y ^ /U N (Y/X) (Et/X)br Z(Y /X ) Y fX ^ Ui X U ^ Y X Z H f h
D g w z 8s z & z t z y ~~ z #i~ T#y x g s t t x r z t ~ s Xz w1g s 'z x s z r ~ U2&r z }{ !xv '}{ #aTt { xv z a(s y t z z z t v { z t av y r vs xwTt { t ~ xv `x #iz !av i y s z z us }'z wt 'z r t z s &r #y Xi z ~ s !}{ ( z z a 'T{ aa( !f xv y us ( ~ {z t r v z y r t x{ z y ~ { rs ~ v & S}~ s i !f'#pz s { v Tt X(x ~ r !z w! ' g }t y us z X}t #Tz X(W z u(uwxv 2x us v }xx 'XT~ s z ft 'c 'XT~ s i x r | ~ ~ y ts s i ~~ x z x { ~z y x y i x s x z x ~ z xz x r | ~ r y ts s z y s #p Tt y ws z X}t #T~ ! 6Wz (wuav x g { 'XT~ s i #'z (wuav 'z 'z z !(xv w s a(t z x z x { z y ~ ts s ~ x t ~ ts s r t r { ~ 'XT~ s i ! xz x { r { xv ~ r ~ ! #& xa z & x S2{ xv `~ xa x z 'z (wuav cz !(xv w s & ~ av { &xv z r z y{ x z r v { x t 'z !q z u(uw (z !(xv ax z & { s x t 'z et x!Xz t 'z w s aw z Q0g T{ q { z t av !t 'z xa z & ~ av { &xv r r ~ r t r ~ s r ~ ts s x t ~ x ~ g s z
aNis F ShvNis (ColimXMdl(X) (Et/X)) pY ~ ~ k(qi0 ) = k(pY ) y X F X // U (Et/X)br U VY, (p) fNis,y Colim Et/X X k Fy Y (Yi Y)iI (Yi0 /Y, qi0 ) VY (y) Fy = // (aNis (F ))y
(U/Y,qU )VY (y)
r ts r q !p!Si i r r y z r t y r a!!av }{ v z T~ r z ! z &r z i x ~ x h ~ t x t q ~ z s { v Tt X ~ xv }~ g ~ w z 1 'z ar }{ v 2'z s {
s i r q x w!&'pt z x xv t '#Xz !xv z ~z z r r 'z v Tt 'W!Wz ar }{ q z (wuav z !(&r i x z y t x t ts s r t
homVY,
r { z t xv x ~ 'z xv '!z ( { v z ws z ~z y i r t t x t q i x x x Xi z ar }{ v t 'z ~ z s { v Tt Xpv
VY (y) (U/Y, qU )
r v tz x a!g t z x Xe z { v x Xi T~ ti z
(U2 /Y, qU2 )
r !
t t ~ t z #xv ~ z t #i~ !xv y &r x ! z y i r r g #y z { v pz '| y W! 'z x z v r
z T~ v r z &p z t #i~ }'#y x z y ~~z t z #a! & ~ xw 0!x ~ r !y v r r v z x
pUY
tz iti { r t s v &r z x ~ r !#y s s { TXus x x y z z ! t rs &wxv 6y t z r ti r i ! 6y ( { v pz av !it 'z r ~ z x X6 z v 2x Xi }~ { i x r v z ~ z { ~ z ts s t ~ z r s v ~ z z s { v }t Xx xw T2T~ s ! t 2z (wuav Wx xv z#y~'z x Si xxx us ~ xx #y
Y X F ShvNis (Et/X, Nis) p Y U y = (pY )YMdl(Y ) (F )p = Colim
r r ~z{ t rs ~ t z }{ q { z t xv `z y av T'}T{ v &uav U! z x Si ~ z { v t rs ~ { x r t rs z y s ~xx 'i T~ ~z '#y { av ~ &wxv z xz i s y ws ~ &uav W#z u}t x XT~ xv Wz uw( Wz ! #y | }~ v { z r s x r z ~ BE V BP d B v v t x ~ v }t z g z #y z { v Az X| y (! v r { a11 z &2us }~ r QDP TX2XTP z wTt x '~ av t 'z r z #x6s ~ ws i~ awxv #z uw( s {z r ~ z t v r x t rs z y s x z t r v r z |s z { h ~ s { x z t r v r v z r z~ h t rs r ts r q y r x z & }{ &axag #y av 2x ~ r x #'&wxv !Si av y t z t a!Sz t z x ~ z & }{ &a'i y av T{ u( T!2x ~ r x 'z Tt tz #y r z }!x T~ ws y r z z z{ z s ~ t ~ z z x Xi #y s s G~ }Sws xx { iti { ~ r !y x ~ z y #~ r !i~ &uav #y x y t rs ~ z { ~ s xz x r ~ v xeax z !2 { v fx x6~ !!xv ~ &wxv #z w Tt x XT~ av z uw( z #y | T~ v s z r ~ 'S}~ s !t 'z { x i r r y t rs z y s { z r s x r z ~ v {z t r v s g z t z ~ v Tt z z a 'T{ aa( !y t ~ w& # T{ s x wxv { v z x z t x av s xx & z { x ~ {z XT~ v ! ~ # 0 r u6eeXpp!(j n mt q s r q o n m l k C B E V d I V F E B C B V E e C B Y P C e E e V I Da!H V $GSxY C B Y C Y CE H C B RB g V C B SuP V fS0x8$xY C e H H Y CE H B SXR R C SSSu XDaD`DXDP g B g g P B C H R B QE V DXS$w2d V 0#AaS$! B H F B YCC C E B Y P e E e }(B xxD(SDP V g I e#F B H E $ Qppx
fNis,y ((Et/X)br , Nis) X k y P(Y ) LimXMdl(X) (Et/X, Nis) X ShvNis (Et/X) Y (Et/X)br X p Y F Y X X Y
~ xv }'#y ~~z s xwTus ~ v v !x !z ! T!x ~ | !xc{ s x XT~ s 'z Ga!!xv }{ v z !2 T~ r z z &r z x w!&'ct z x v t z g #Sz !av r vs t z r t s i z r q { z y z{ { x x{z i r r y ~ ~ z r r ~ t r ~ ~ t r r i x ~ x z t x t q i x x 'z x ~ 'z av Az A&xv A~ t z #xv z A&xv y g z ws z z s { v Tt ' ~ av T~ t z &r { uv t 'z ~ z s { v Tt Xpv
FltNis (y) fNis,y (F ) =
(U,u)FltNis (y)
BC H XA P F #C CQ}Su(#DC P R C B P F B d C P d C H F E F Y H A#!Q(S#U#$e B F B XSE C !H V Y RB g V C B F R C B E H C CE B C B E H Y B P e Sue$Su$X2XXuaSuixcDaE V R V BE H $eH B E e B Y P e E e u V XaDXDP V g I B H Y E H H 0#F (!#F E B B Y P e E e SxxDXDP V g I 0#F B H r
k FltNis (y) k (y) // k (u) y P(Y ) ~ k(y) X (U, u) ~ k(u) ShvNis (Et/X) U
F V S V }0(#0$xH V E SSTP BR C eP H F e H H Y E C B F Y C e B Y C FCCB xcD}SxY I F V E XB h E H B d B g B P BE d F UX(XQE V STXR V }R I DC V S E H B S e1Qi6
u P(U ) k
r vs ts r q y r y v t x au}Si z ! { 0 x 6y us v t y t rs z y s t z ~ v } ~ &uav 2#z wTt x '~ av z wu( z ! }{ s X i `{ xv &r !y i i T{ s x uav { v z XT~ wu}# x r { z r s x r z { y { z { z ss t
U Vi
B E C B C F y r vs ts r q i h g f b B Y C H B C u0GB 0ay WxwTu!Sp#e5 d ca`SXWF V
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
(Etqc /X, Nis) LimUMdl(U ) (Et/U, Nis)
I eH V GDSB R EC
eGu1 ' # (z X| { 'i t z z !ws }~ v t z s z v ~ { x ~ z y v r !z '| y ! 6y t r s t s & z r ~ sU&r z T{ !av 'z}{ r! 6ycz &s { i!Xi AT{ q x {`z#y s s ~ iTXus {x z z x ~ { ti z#y s z r ~ &r z T{ !av '}{ #aTt z z t v tav }{ sxv(X( { av zT~ pt z rau}t '~}xxz}{ |w!r {'z#yApwu6eeXpp!(j s z i s ~ vs { z ~ s r {Xz r }#y s x X~ ! x{ z t z #y t z #av! x v n mt q s r q o n m l k h E H B g F e E H B d B F iXu V #XfXASXH P FC BP FER FPE #DUT#QSDQDC I 0#GDDB B H F EC
P U /U (Et/U )br ((Et/U )br , Nis)
ShvNis (Et/X, Nis)
ShvNis ((Et/X)br ) (Et/X)br y P(Y )
(Et/Y )br
VY, (p)
homVY (y) ((U1 /Y, qU1 ), (U2 /Y, qU2 )) =
t rs r z a z t z y ~~z #~ }'!y x &uav ! ~ t r ~ z ~ z xv z faav ~ z y #~ t z #av v
VY (y) pY
qi0 (Yi0 )
ColimXMdl(X) (Et/X)
(U/Y, qU ) ~ ~ k(pY ) k(qU )
YMdl(Y )/Y1 ,Y2
VY (y) = ColimYMdl(Y ) VY, (pY ).
Colim
(U,q)VY, (p)
(pY )
F (U ).
^ ^ (U1 Y1 Y, qU1 pY1 pY ), (U2 Y2 Y, qU2 pY2 pY ) .
Colim
p q U P(Y )
F (U ) = Colim F (U)
Y U Y (U1 /Y, qU1 )
F (U) = Colim F (U).
Y
pY UY
Y Mdl(Y ) p F
~ k(p)
(U, u)
q U U
(U, q) ~ k(q)
i0 I
aa
Y
U
aa
~ r !y #xxx u s #iz x Xi xxx us ~ xx #`z s { v Tt X i x x z y z z y ~ t ~ z r s v ~ z y g S r z x u!&i z t S s t z x 'z x x qs t z z Az uT&r !y s X~ z v v w! y ( z aft z i x z y z s r y z r r v z ~ 'z s { v Tt Xp!US r z x u!&i z ! 6UT~ v ~ s y a v & ~ xwu x u6eeXpp!(j n mt q s r q o n m l k C Y B H QR EC !H V (u V V}DDB H R C C F $QE V X'XC V 0B $u`X8$R E H B E H BE H B F B B XuSE BE H C Y B d d H H E C d R e Y B F H F B H X$A2SC XB 8D$!U$xX#0#F E d SBA$Y V H Q E HB d !UX(d V TX5e$R C SFC EB B B H H P F #$ DSxY I XW#XX0S`uP V a V $XC C F C C B R P F BE R H B B d H BE H $0H $ P F #$ C H E H EB H Q EC !0XcRDDB B F E HBE H d C H C XWH V S V !S$!aF P V #DxP$X2TQXH (XD0T0 V P V XDSXR V P V R P F C e I g H B BPE Y eE e P P e C P eE E SB RB }$g V B u0F}Ru`BX'XC V B RC B F R H F h F u0#21#X P F BE R H H F BE C B F #XX0S#pDx'6 G C H E H E $$eSB E DSR V U$R I DC V XSxxDpSSDP V g I xADQ$A#F C B E d F C B Y P C e E e B Y B d C P d H E $ 1Q fGCCB @ E D A A
(Vi , vi ) k x = f (y) FltNis (x) ZY ^ V = Y X U f ^ Y Y WY FltNis (yi ) f: Y Yi k v P(V ) // X FltNis (x) (Et/X)br VX (x) = ColimX VX, ((P(X) X )(x)) x X f -1 (x) = {y1 , . . . , yr } U Flt(x) r V = i=1 Vi Vi Y Y Y R // ColimXMdl(X) (Et/X) Y // FltNis (y) WY Y ZY Yi 0 (Et/X)br // (Et/X)br , y Y
r vs #y # xwTt ~ v v !x !`z ! }!x #p}{ s x wxv { v !it z ar x u! ~ t x ! ~ i `v r ~ t s t { z y z{ z y z { z eGu1 z i }'!'}{ au}t s x{ &av (x a v z { v 2x 'i }~ z r ~ s { z z r vs y x t r h y s t &r z }{ !xv '}{ !t 'z t s i {z t t z r s v r z z r ~ z aWar z pG z Xu &r z z X w z }t xx #z x Si ax ws ~ xa ! x v s { x z y g { x z t 't T{ # x h s t g ! au}s ~ v v ! r vs t z T{ v r z t 'z ~ z y s # z r ~ Uar z }{ xv T{ Ut z z z r ~ { ~ s y t i r t v !xTt { av x xx 1 { ~ t ~ z x Si xx Xi T~ z y ~ i r ~z ~ t r #z &ws { !'i 2'}{ q 'z 2&xv ~ ~ 'z z & z t z !us }~ ( ax &r z z t x ! ts v uxe t s r z r ts v ux s t z y s # z r ~ &r z T{ !av '}{ !Ut 'z z z r ~ z & z q z T~ r z ! t r vs y x t r { x y z { z i ts #y Tuw x us x !`T~ | v 'z x xwTt s a}{ &xv ixz z z ! }!x Xz w( 'z t z y z x { rs { r t r z r r s 'z & X!r ~ &r z }{ !xv '}{ t z s i s t z z r ~ z & ~ xv { &xv z { v px Xi }~ y t 'z !a(! z av i}{ | y s ~ xv a v & { v r t z r s v r ~ t t &r x r z &xv t r t 'z !aW z t ~ s Sz w!g z T~ v r z y #z x Si xxx us ~ xx!it 'z z a z t z { v r s z #y t z r s v r ~ v r ~ ti z x Xi xx us ~ ax!pt z z (xG6 z y i r t rs #z &s { !Xi &uav !pt z ~ r !y #y x z z xxx ws v z x S z { v x Xi }~ r y ! 6iz &ws { Xi i}{ q x t 'z i r z ~ ts v tz z & z t z #y z { v z '| y A ux v r z & ~ z t z t ts v ux z t ~ s Xz wspg ~ | txv x { ~ z y ~ i r t rs ~ z y ~ z z ~ z y s { ts s '#z &ws { Xi ~ &uav '#'z ! ~ r #Az wTt z v !cz (wu x &r x r z t !xx { x r vt v ~z s r v r t r i { z ~ z y z z s v r i { r ~ #y#y Arx!i( Xzu !t ' z & ~ xTxx '!y Axaav ~ ~ ( Xuz z & #}{ ~ x xi~ v ~ r { x z t &r z Xi }{ v x xv r z i ts #y Tuu x us $x s t z r ~ ts v r v i r r t t x x s z r ~ U&r z zT{ !av T{ pt 'z { z aaxx0g z &!xv y ar x 0 i {z r ~ z r ( Xu t 'zxxx usxv ~ g zT! v pz ~ s T{ W!t z xz (av 't 09z 7 }t 'z { ~ t z { { v r ~ g z 8 r 6 s ~ z ~ s !}{ ( z #g z & v rq ~ u z X| y (!t ~'z v r s i t v r s v v t z#y}{ s x &r z 'u av z z t z !r z r ~ z s wxv }t x { av &r z T{ !av '}{ !Ut 'z s z z r ~ t t v r r v v z & z 1s 45~ s xx 23t Xz !xv ~ v x`~ u (z { { { z !x6g !y z { v 2z '| y ! { a u6eeXpp!(j z v r n mt q s r q o n m l k R g B H C 0 B V P F E H B d B P g F R B P H F E C CP ( $u ) 'X0D1xTw$}E `#!X(XT#$T#DSB E XB B C EDx'B2 E F E P F B F BC F #$#$WX XxDC H C Y C B E C B d C P !H V Su V Se SA#!Q(d W E B E d F B Y e E e XB SR V U$R I DC V X xxDP SDP V g I 0#F B H E H B X$ % &1Q $#i! " " BX SE F C B B B aY u2d V ce#F B H r vs euw z i { &x(Xi y t 'z xwTus ~ v v 2v t r v ~ r vs t { x ~ t 'z !av x au}t s { t T{ { ~ z t z s ~ | !xz s { t z Tuw a i z t Xz &t z S}~ s !2t 'z { x x ~ i ts x x z x r ~ z & ~ ! ~ v x }{ u x r z & z t s x g v s s t z ww x # # z ( z z z t ~ s Xz u1 S}~ s !(t z s g x z x r ~ z (xG z r ~ U&r z }{ !xv '}{ !t z z z r ~ r v s rs ~ i r r t t { xx e { xv x aeg z wTt 'z $w`t z z &(!!av y &r x e tz r y it r z ~ z &p z t av z t ~ s Xz ~ r !y &r #y i T&r x ! #y | }~ v x t z ~ r !y #y x z v r t r ts v ux#6S S}~ s i !2!z ~ s T{ z &&r x r z z u}t z { ~ t z t t s z xx w s 6g { x6 z & ~ av { &xv x z x { z y v ~ r v g xz x r ~ v s s t s x g x x# 'S}~ s !it z z (xG0 z r ~ i&r z }{ !xv '}{ !t 'z z z r ~ z uw( 2x !& # r y tz z ( z z g { { x z & z t xv `z t ~ s Xz & ~ r !z uT&r !y &xv ~ x y ~ i qs t z s t r r vs z ~ ~ v t s v s rs ~ &r z ae Tt 'z $wpt 'z y v y r t r ~ xwTt 'TU'z 0g { x0 z a ~ z t z a { 0 x 6~ av { &xv z !ws }'z ut 'z z & X}{ &x(# f~ x!p# ~ { rs ~ { z t r v t x v r t z iti { ~ r !y x z x Si #y s s { }Sws xx r t rs awxv #aTt { av z wTt 'z !ct 'z t v s s ~ r ts v ux z ! ~ r !y x z &U z t S}~ s !z ~ s !T{ ( x z x z y v BE V B d F t ~ t rs z y s q i z us z { xv U~ &uav !`z u}t x XT~ xv z ww z U&r z { v { z r s x r t ~ 'z z &s s z s { ar A8DP ASa XB vt t rs ~ v }W y &uav r ! xv y t ' z G r r r i x ~ { z t av y xv T}{ v xx { z t av z g x r ~ z{ ~ z s { v Tt X s v v g rs t rs ~ z s y ~ r ~ t r ~ t r v r ~ z y z #S r z x w!&' i u y 1Qs ~ w1 ~ &uav 2'#y u!xv 6{ z t xv z 2aav x &xXs { v }{ z t xv 'z g ~ }'#y G z #2av ~ ~~z s t
FltNis (y) I0 I WY ^ (Yi Y WY )iI0 (Et/X)br R // ZY (Y - T ) (Y - ZY ) T = ZY ZY Y ^ (Yi Y (WY ) )iI0 Y ^ R Y (Y - T ) (WY ) Y Y Yi i I0 ^ ^ Y Y WY Yi 0 = Y Y Yi 0 ^ Y WY )iI0 {i0 } (Yi ^ R Y (WY ) Z = LimY ZY R = ( Yi i0 I Y ZY WY WY ZY Y i0 RY Yi 0 Y // Y )iI qi0 P(Yi0 ) P(Y ) // Y
t ~zU{ z t xv Xz & T{ &x(xg r z z t r v r v z x z av z z s v Tax ~~
(U, u)
i }'z g ~ w z & 'z s { v Tt X#S r z x uai z ! i x z y z s r {{ x h ~ s v tz z S r z x ua i { xv z aGe r r y z r s { z t xv ! 2T~ v ~ s y r z xv ! a
u = (qU pY pY )Y Mdl(Y )/Y VY (y) // FltNis (y)
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
(U, u)
(V, v) FltNis (y)
z Z i0 I qi0 P(Yi0 ) (Yi0 , qi0 ) FltNis (z) Y Yi0 zY z Y zY RY Z Y zY Y
Y T ZY
(Yi0 , qi0 ) FltNis (y)
ShvNis ((Et/X)br ) ShvNis (LimX Et/X) fNis,y ShvNis ((Et/X)br ) fNis,y ((Et/X)br , Nis) // G f : F fNis (f ) y P(Y ) X Y (Et/X)br f f a1 , a2 F (Y ) f (a1 ) = f (a2 ) y P(Y ) a1 a2 fNis,y (F ) (U (y), u) FltNis (y) (a1 )|U (y) = (a2 )|U (y) (U (y) Y )yP(Y ) F a1 = a 2 f b G(Y ) y P(Y ) b fNis,y (G) fNis,y (F ) (U (y), u) FltNis (y) by F (U (y)) f (by ) = b|U (y) f (by1 )|U (y1 )U (y2 ) = (by2 )|U (y1 )U (y2 ) (y1 , y2 ) P(Y )2 (U (y) Y )yP(Y ) F b F (Y ) b|U (y) = by f (b ) = b G U (y)
^ Y X U, vi ) Flt(yi )
VY (y)
k
(U/Y, qU ) ColimX (Et/X)
(U, u) FltNis (x)
u V (U, u) Flt(x)
ColimX (Et/X) VX, ((P(X) X )(x)) vi V // U
WY = W Y R Y
^ R Y (WY )
y P(Y )
U = U
(V =
z &s { Xi z s { S(x'i r f'z as Tax i~ t x ~ i { t ~ xwTx!r x r z ar #y av x ti v t s t z r z ~ ~~ z r vs t v r vs z ~ t z z y t x{ q i s t ~ ~ ~z{ '}T{ v xwTt 'T~ ~ av (z Tt S#U~ t x ! ~ i { t p~ xwTx!r z z & }'z s { a { z t 'z z g z x ws { Ut 'z z r vs t v ~ z r vs #y au}t x w x x z & }{ v v
I Q ` B 8 Q v u I s c 8 q Q i @ ` @F 8 a c 8 a ` Q X VF S Q I BF D B @ 8 8 6 Ch9Txw3tCArpRphgfedA'bY9WUTRPHGECA975
BE F B F d DC $Y XxTu(!}R z g z ( pv
F X k X
B Y F P & B Y H C H B d a#D2'aD!X(Y V P V iT$X V fDDB BeP d EC H C H B d $DSA$Y V SR V U$R I DC V X xxDP XDP V g I 0#F B H E $ B E d F B Y e E e $ D A %1Q GGCCA @
" ! ! b
#u ! 6 s v ti v t x r ~ z{ r ~ z t s ~ z i s ts { ax z as { Xi z s { XaXi r z &r !y xv T}{ v xv p#y &r x w! iT{ s x uav { v z far z z t x s y ((uf! y i y r x t
U i ~ v # Cg Wb #z !av i y t x u! ~ i z #g t 6 d z y { t s i t z r ~ t ~ v ~ { rs ~ &r z x uaW&r z z w }{ }t x t z z us z z (aGi z !ws }'z ut 'z
((Et/X)br , Nis)
XMdl(X)
eGu1 z &t z
t z s &r #y Xi v r z t&r #y Xi z! | x z & Xz !xv !#` rav y t ~ a z ~ s !}{ (xv ~ s s ~ax &!it ~z z s { { z y s s r z z ~ s !}{ ( z z &it&r z # ~ r z wXi! xf xv {x r z x'XT~ s fz#y`z s { v iTt xX(x { ~ t x!Sz t ~'z v ~ z z x ~ x { z t xv `z #g # 2T{ s x uav { v r vt v raus ~ w w s !qXi ~'z us pz#z ~ s !}{ v( z v r { x s r y t ~ y z xx a { ~ xTx # xwTtus ~ v v !2x !yfz us ~`#yz ~ s !}{ v(xv ~ s #y &r z &$v {! y ~ r yUz ~ s !}{ v(xv ~ s z 'us z#yzpv ut6qsereqXpp!(j r vs | { z ~ t z z t s x n m o n m l k E F E P F ##!
F pZ e : (Et/X, Nis) // Re F Hp (X, F ) Nis e // ((Et/X)br , Nis)
XMdl(X)
P F C C H B e #DpSX V S$TxcSF V X V 8X02#F CB F P HB BR C e P H
y B F H R B d C P d C H F C P BE C B S$0H V A(DQ$A$X#u V Da51 F E $ XR V U$R I DC V XxxD`SDP V g I e#F B E d F B Y P e E e B H E $ 1Q W#f "
X k
z &s { Xi `z s { S(x'i r z ti v x ti v s v v r| x s ~~z X 2#y s s z s { S(x'i r z r ~ U z s wx(av v x #y ! X(z z ' }{ !& ~ xv }'#y x x z { z s z x r s i x ~ z { x z x z y S}~ s #{ z t xv z }{ x 6y ar T~ v av r U~ r z u'! '0 xv x r S}~ s i !'#y { x y v t x z z x z x { ~z x !p{ z t av 6'Xy r z Sr ~ r z u'! '0 xv x! z r ~ S}~ s 6z { xv 6z x xw r ~z ti s i x ~ z { z y s s ~ v z y ~ x z x ~ t xv 2z '| z p r g T'}{ ~z t v r r vg r ! Q z { v ix Xi T~ g r T{ #y s s { }Sws xx z !t z iti { r ~ s z x!a'S x Xi T~ ~ z z
~ k X k X k X p p p p HNis (X, -) = R Nis (Et/X, -) HNis (X, -) = R Nis (Et/X, -) (X ) (X ) F (U ) = F (U ) = F (U ). p
s v v s z z r ~ z s wx(xv v Wx #y ! s z '| { Xi t z x r au}t }~ }# r vs x s t v i x ~'z s { v Tt ' z w z }! }{ w!r x x ts x z |s r vs z ~ r i y ~ xwTt 'T`{ z t xv y &ws { i { z eGu1
t 't x ~ i z x v h { y i x z s { v Tt X(x g
X Mdl (X)
t &r z z ws ~ us X i `~ x z & z t { v !y x it z #xv { xv t { { z r q ~ z
Colim F (V ).
~ v }{ u x t z xx y
r z t ~ s Sz ws ~ z & z { av ~ x }#y r }!#y t ~ us z{ z{ z
C Y CP FB g !H V `#Xu V h v { ( z
t s ~ ar x u! z ( z z ~ aTU 'z uts `!y ar #y &'i z ~ s ! r vt v ~ ~ z t z s iti { r y v r ts v ux t v r s r z z z #y s s { TSws x z 6Az X| y (! z x!r Q!q 0 t !y
X k pX X
euw }{ z t 'z i~ i T~ s x f&av 'i~ us x Xi 'pv r z wXi ~ r ! t 'z g t rs &uav ! 6i'z x Si 'ax ws ~ xa '!pz s { v }t 'Ax r y ~ t ~z r s v ~z y i x ~ x ~ t r { z ~ t y ~ z s z ~ t { xv }x { x v axg 'z r x z afs y &r z z t x s y ((ws z !av i #y av 'Tpv z ar #y &'&r z #y t x t i t z s i t s z x r y z r z~ x s { x z |s z { r v r ts y x t r ~ z Tt }{ w( }!Ax g ~ z { v xx Xi T'#p~ r U'z x Si xx us ~ ax #Uz Xz r 2z z y i x!' i Uu! x{ Wa v & }{ v v ~ ~ ~z y x y ~ t ~ z r s v z y ~ { t z
h OX,p = ColimpUX (U, O)
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
(U/X) Ob((Et/X)br ) [ColimXMdl(X) (X ) (X ) F ](U ) Mdl (X) Mdl(X) X U U
~ t x `'z X| y ( '!(z ( { v ~ v ~z y i
r v z z & ~ xw 0!x !
{ 'z X{ xe x v
a i ti z &s { !rXi T{ q #y z x X6 z { v x 'i }~ z z q t s ~ ts s z x rxv z s { v iTt x' ~ av ~Wx &r x w! (z (wuav tav nut6eeXpp!(j s m q s r qo n m l k B C P B g H E C uXDS$fDDB
P FC #DWF V X V 8X02#F BR C e P H
V E H B Y g e B d C P d Sa'XAA#!Q(B BE d R R B P e R B H 1 Sw0U$'(D$&}E 20#F E H B X$ E B B E d F B Y P e E e SXR V }R I DC V S2xxD`XDP V g I 0#F B H Qp W" #! "
X k
(Et/X, Nis)
~ z & }{ v v
Et/X
X
ColimXMdl(X) (X ) (X ) F
k
1) 1) 4320(
Colim Hp (X, (X ) F ) Nis U \(Et/X) (X ) (X ) F (U ) = Colim Hp (X, (X ) F ) Nis U
ColimXMdl(X) (X ) (X ) F
U V U \(Et/X)
// Rp Nis ((Et/X)br , F )
XMdl(X)
Colim Hp (X , (X ) F ) Nis
X : ((Et/X)br , Nis)
F
// F
F
LimXMdl(X) (Et/X, Nis) (Et/X)br
Hp (X, -) = Rp Nis (Et/X, -) Nis
(Et/X)br Et/X // (Et/X, Nis) U
i }x' !ft ~z r s # z#U}{ s x t&r z 'u av Uz ~ r !y}{ u x t ~z awxv Upv i t rs z i { z t rs Xu &uav {{ y z x ~ v z &e z t r ! z t ~ s Sz wgt x t z s #e r t z r v i s ~ z#y i( 'zu &ruav !`t ~z z & XT~ v ! ~ ! 0 xeg { t s {z 'z x ~ t z av ~ !w x s ~ z t ts r ts g t { i ~ }{ v #y xws ~ r z t Spz !far x u! pT~xx #y tar z zx r r! }{ s `z u!p z r v z r t z x z x z t rs #y &wxv ! uxv ~ ar z z &#v s i tz r v x}t z rws ~ r!it 'z ~ ts r v i r z t x~~ x r 6 ~{ r z &U! y i y xg z &ws { !'i `}{ q x &r }Ta 7 z !ws Xz us ti ~ ~ { rs ~ t s z x s i ~`T{ q }~ t zz x St 'z w s & ~ s Uz !ws Xz &wt z z x z ~ s T{ z z ai&r z 0# v z #y i( Xu &usav !t (zxx w s &rxv y r ~ z t {z t r ~ z { v p~xx Xi }~ z y #Az ~ s T{ !u! y i y r z r v r ts ~ r !y x t r r z &av xci( 'u &usav r v r r ts v ux#1 z r ~ sUAi {}x' r!2t 'z s { ~ r {z t r z & ~ xv ~ v ! u6eeXpp!(j n mt q s r q o n m l k B V P g C C V E C B B F P R e V P FC EC #DE V 0pDDB D}F DS2XT0 X'XP C V X!SC I a'X0D0 XDP V #GDDB de R B Y R g B H C e P R H F E C SXR V R I C V S2SxxDDXDP V g I C B E d F C B Y P C e E e
~ k
B Y R g B H C e P R H F E C a'X0D0 XDP V #GDDB
U
u
C B d P SAD$SC V X!XC I de R B Y E F d d R eP R H a 8E V #2XDP V #F E $ Q1Q x C B P VE H B d e e R g B H C ST SASwS}X0D0 CE H B d BP g F RB ~ 8XAST#$TP 'z ~ w 0!xx &xv z &r !y i i xwTus ~ v v ! x z r ~ U ~ i }X c~ ws v Tac~ &r z T{ !av '}cv i z t r ~ t z s { r vs t { { x s {{ x x i ~~ x t z z{ ~ z
// M(X) M(U ) q M(Y ) M(X ) (C) p = u(q) Y U = k(p) = k(q) k M(U ) k(p) = k(q) p ^ Y = Y X X Y
eGu1 s r r z a!! z t ~ s Xz wg s z y z #xxx u s ~ r 2a t ' z r x y ~ x ~ z t ~ s Xz u#g s B C P R g B H C E C uu V }X0D0 DSB
// M(X)
~ }{ i r r t t z &!xv y &r x z s r v h x xg ~ u z 1 s i r t rs z y z #`xxx u s ~ r !y t z ! z &s { !Xi &wxv x ~
p X
vu x r x & z t z & z t r
z &p z t ~ s Xz z &ws ts g z 8 uxv ~ 't 09z
zt ws r t rs s r r { iXi &uav z &w!! r 6 u6eeXpp!(j n mt q s r q o n m l k
(u : U
B Y R g B H C E H B d B P g F R B P H F E C x}X0D0 X(XT#$T#DSB e P R B XDP V uf # $QQ aY V cx($QE V $e2SuAaTP V H H E H e B Y C H E H C B B Y
// X )
B F w0$R pQQppx
kK
r v S r z &'i ~ av T{ # z !s xh ~ t z av i~ !u x T{ v #y r z t x r s v i ~ aus ~ r z t Sz z ~ z s t z ts ~ t s ~ z ~ z &r x u `ax #y &r z x r iz x t ~ t & z r ~ U ~ i }X#pTTax 2x z &&r !y i i pz ( z y u! y i y r z r z t z x { x ~ z s {{ x ~ z y z~~ {
f f f
eGu1 t 't x u! ~ i z x v ! ~ ( Xu~ &uav z { }t 'z { ~ t z z ~ s !}{ ( { h i { z t rs ~ ~ s ~ v vs t r x t v ~ { rs ~ z z &s ~}~ A}{ &x(uws rxwTus y av pv z t Xz !xrxws Xz (us z r!ar z Xi { v t 'z t x t 'z x z t r v ss { v v ~{ ~ g ~ z (xGp z !ws Xz ut 'z z t r i z t s z t r v ss ti ~ s { v ~ z & T{ aav(6s 'z ~ ( {Xuf~ &rwxv f~'z { `z wsTt 'z { ~ t 'z ~ ~ z & }{ &aGuws r v s t r x xwTus y av (pv z x Sit 'z z & ~ ! Uz t Xz !xit 'z x av x s y xws }'z (ws v r r v ~{ ~ x xv xX2z ~ r x!y swus s wus X}{ &x(Xi cz X( ~ x!s 'z ss { z t r v y r | x { v r z !ws Xz &wst 'z ~{ r ~ z ~ s T{ z !g #y &wx ~ s y t z t x av s xx & v z t rs v x ~ s t ~(z & 'T{ aa( z# rav y t ~ e {z t r v y z !ws }~ ( ax t Xz !x!`t ~z ~ s y { v r z z ~ ~ s y x { ~ z t #ypz us T~ ( x~ t Xz !xv #y &r z z w }{ }t x r z t r &xv t s t r vs s r v r ~ ~ z ~z '#y s s ~ TXus {x 'pv D S s Xz#Sz r z i T~ s x pt 'z x us { au}t x w x x #px'2z x ws { t x!Ut 'z { iti ~z z { ~ ~z z y ~ x z #y r vs xwTt x w x x z a xXT~ v ! ~ xuws r v ss wus xvwTt ' w w s { a1# T{ s x t 'z wus ss r s x s v z ss ss r v x s v n mt q s r q o n m l k wus aus}t ' w us wu6eeXpp!(j {z ~ B C P B g H C C B Y C e E DaDPSXeDP V g I xY(BA#!Q(u S B d C P d B wSDXGEDSB h BE H B g F e E C 2Xu V #X'iDDB $Q$e$R V UXB QAC$Q$0CSu`$ H E Y H E H E Y H R B C F #& $QxY V aY$QE V $eCSuBxTP V H H E H Qe B C H E H B Y Q1Qp $#i! " " t s ~ ar x u! z z z ~ aTtU r v v ~ 'z t x av s ~xx a #y s s { x ~ z s ~ z ~ t i i { z y v ~ z t z x { z ~ {{ ~ z~ ~ TtSws x #~'z ~ s !T{ ( 'z Ut&r x w! Txx #y &r z a r pxz ! x t ~ &r z T{ s x t zW s z r ~ s ~ i }xXU'z#yTT~ax Axpv E XB Q Q V DDB EC E$(#c2#$XGDSB H H F h BE F Y e P E C BP }SfQ V e d P B E H E F E P F ADSSe!e##! BE P B g F H C P B d d B H DDS$#$$DX(e#F Xu V SAADQ$A#DSB E B B E e B d C P d H F E C H B C eE R EC !XDDDP V SB eP R B XDP V 2u C BE C C E H C C BE g F C C H E Y H R C B s 2S V u8E V XSH V #D`!$Q$e$DwC z r ~ `i }' #2uX!a s { { x H F e B
f (C) x X -U u f -1 (x)
f
C B E d F C B Y P C e E e SXR V U$R I DC V XDaDSXDP V g I
B xY
8E V #$(XDP V H E F d d R eP R
QQp
p6
0
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
(C )
X - U
x y
k
U
k
p ~ kK K x = Spm(l) X
// X , f : X
f
Y
k
p M(X) q M(U ) Y Flt(p) f (q) = p
U
// X
X - U ~ k(p) red(x) = p (C ) X k
^ X X Spf(l )
X -U
(C )
X ^ Y = Y X X
(C)
f
U
U
U
x
// Spf(l ) f -1 (x)
u
k
u
u
// X - U
U = f
// X
// X.
// X
// X
x = Spm(l) X // X x : Spf(l )
p
f
f
// Y
x
k
{x }
k(x) = k(x )
Y X
X
k(x) = k(x )
(C)
f : X
X -U
X U
(C )
X -U
// X X X ^
k
X
a& y
Q SASwS#c)Q uX(XX!X#DSB E H B d e e H F h e B d B C H B H F E C C B E C C E H C C B E g F C C H E Y H R F B Y C B C E H B d B F B C E B C R g B H C 2S V u8E V XSH V #D!80$W SaDuDX(Xu#XSGX'X0D0 F V }X V #GDSB BRC H F EC CP $u ( P F C C B d B C H Y #DSuSAXSSB pWF V X V #F BRC H E $ XR V U$R I DC V XxxDUXDP V g I e#F B E d F B Y P e E e B H E $ Q1Q x % r z ! z X| y (! 6z x s i TT{ q x { ~ ar !y av T'}T{ v s v r y t x r ~z{ ~ z r s v { x r v y us z ax z &ws { 'i xX(z ~ r !~ xwTt }{ t ~ a(Xi y x ~ x x y r vs x r x av #2W z &s { 'i `z s { X(xXi r W z r ~ ti v s z y x x z s x x i Xx}~ xv t z xwTt ~ ~ xv 2z t X2#2z t '}v { x r ~ r vs z ~ t z z y ~ z { z t r v xi { &x(Xi y t pz X| { 'i t pv & z r ~ Ui TxX !t 'z s {{ x r ~ ~ z v z s
g0
z ax T{ u( T!x T{ w! ~ i { i i !G~ ax ~ a#Qs ~ xv s v z |s z { z s { r v x v s i x!' i Xz r z T`{ xv # A}{ s x wxv { v z p&r T~ uwT!r z t '}{ s r v r z { x{ z~ t x ss t z z iti { { ~z s t y ~ z ~ | !!y &r x w! ~ r `z t T{ s z xx x ~ }Sws ! v 'S{ !r z #av 6~ { z t s z z y s v { x v v v z s wxv }t x #z &ws Tax U~ TSws ! v !z &w z & #p'#y s s z u x r z y ~ ~~ iti { { ~ z y ~z { ~ eGu1
U
a
z r ~ i&r z }{ !xv '}{ !it 'z s t s z z r ~ z &`t z { ~ t x t 'z ~
X h-1 (V) // X 0
i }'v {{ x z t ti ~ ts r v s z & { s x &r z z a t 'z { `z x Sit 'z x i ~ ~ z &U! y i y x#g Tt 'z { ~ t 'z ~ v ts v ux z (xG #y s 5~ s x 1t 'z !a!t z x z 4 { 2 { v r ~ { x z y z #2xx w s
X0 (C0 ) h-1 (X ) X0 h X
h
t r ~ z { v 2x Xi T~ z y s t ! Tt x #2av z r x s { W! { w x &r z T#aa!g ~ v t st v r v { 'z r $(`z u!p x { t ts t z r ~ p&r z T{ !av '}{ t 'z s t ti r ! z t ~ s Sz w s & {xT~ s { }t x~ x z v x z z s z z r ~ z & z t z x X z { v (x Xi }~ t z r v g r v r { x r s s t {'~ v ! ~ # 0 aq ~ w z '| y ! x z {z 'Sax Tz TaQs ~ uU z { t ts g rs z y v #z X| y ( 6y z r ~ 2&r z T{ !av T{ z z r ! t &r z T{ !av T{ ! s t v z x z z r { x s & z r ~ U&r z }{ !xv '}z Xz r }{ ! 0 xg ' # z '| { Xi t z x z z{ { x t z r v { Du " b t x t z t Xz !xWxws Xz (ws z ! t x r z s ~ }t xX z x g g i { x { x z { v r v ~{ r
X C U X // X X a X V X h: V
g
C B E d F C B Y P C e E e DSR V U$R I DC V X2SxxD`SSDP V g I i }' !t 'z {{ x r ~
(C )
B aY
R g B H C e P R H F e H H Y C H C }X0D0 XDP V #0$x$XxF
(C)
&r x u z X| { 'i t z x r z r ~ t s ~ v s s R g B H C 0 e P V r t s ~ }X0D$XDP R !us y t 'z # xwTus ~ v v !x ~ r !z (xv r vs t i }' !g z x ~ r x h ~ { x y { { x r ts ~
t s z &&r z 0#
^ X X x x ^ X X Spf(l ) Spf(l )
eGu1 ~ { av G'z x s i TGT{ q XTTap!{ w x t ~ ~ ~ z x {z~~ x z y ~ v
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
f
U
U
u
u
// X
// X.
^ U0 = U X X0
e P R B XDP V u
BC H C B F B S$!$ TXpSE
B aY
X0
X
f0
U0
U
u0
u
// X 0
// X
f0
f
(C)
k
(C0 )
k
C V S}XC I de R
B aY
v r { t t z i }X y {{ x z '| y ! 'z !av { i# 0 r u6eeXpp!(j n mt q s r q o n m l k C B d P SAD$S R g B H C e P R H Y B F P e H e B P h B P d C E C B C F B Y F R g B H C 0 e P R H F B P R H B E C }SeD0 XDP V #F XDSeS&UTXGQ V (A!Q$A$XGS0ix#X'X0DRXDP V #T}eSDDB
(C) (C)
E d F E P B g F H F BE C B SR V U$R I DC V XfDX$#$#Da' F
" 6Q1Q#f
{z z xi X!T{ t x u! ~ i z { !r z #xv t s t { s ~ v x r s { w x xv z r ~
F () F
X
k
V
U0
U
g
U
a0
a
a
// X
// X
(( X.
// X
// X 0
// X
X
// X
g
g
U
(u, f ) X
X0 (U/X, X0 /X)
F
k
(C0 )
h
R
Etqc /X R
# y
a& y
x y
y &a y
R g B H C C e P R C B Y E Se0 DXDP V SxiSB CE P B g F Y BE C C P R BE F C B H F C P BE C B F 8DS$#$ pH V }TX#0#u V Da51 # B Y R g B H C 0 E H B d B P g F R B P H x'X0DBSAS}'#}T#F E B B e e C E B B E d F B Y P e E e XcXP V XfXcXR V U$R I DC V XcxxD(SDP V g I 0#F B H E H B X$ #QQp $#i! E " "
F (Xi+1 )
eGu1 r vs s au}t 'z ! t &r z XwX i ar z }#xWr xi i ~ t 'z i t s t v { x s r t r ~ x { x z y z~{ r z t #`Xz us xxx u s &r x r T r z z{ ~ z & ~ ! c a i xv y &xv i~ # x t &r z 'u'i z ' z { ~ i v t r tz t s v { x G~ i a&r z &xv ~ z &&r z 6 { ~ i a&r z t 'z !& ~ s z t ~ s Sz s z w6g T~ | v x e z u! { t ~ xv t { ts r r v ~ xv ~ v ! 1 { xv s { t z s 'z x ~ { ~ i a&r z t z !& ~ s {z t z s r q z t ~ s Xz w s aag { iX r }T{ i X}{ &x( x (r ~ }'#y s z (xv ~ z z{ { { x {z t r v ~~z z & XT~ v ! ~ # r 8!Si y t z z & ~ T#y x ar z }{ xv T{ ! y t tz ts v ae r t x ~ ~~ z z z r z y ~~z y #p~ }'! U~ xwTt 'z 'T{ i y s ~ av &r z z us x x r vs z ~ ~ { z r| x{ z s {z t r v z y x{ s g { x t x r t t r x z ' }U~ r }t 'z { ~ t 'z ~ z & X}{ &a #( ~ uai x i ~ t 'z ~ z afs T~ x v z &&r x r z &ws z t r v s rs ~ z a }{ &x(#s z a z u}t 'z ut 'z z g s { x x | ex xDus x~ s t x t r ~ z &fus }~ r { xv s }{ t z t x ~ i z z & ~ xv ~ v ! # { s x t 'z s S g x ~ { ~ z z & }{ v v z & s s { iS r T}{ i ~ av { &xv ~ z { v 1ax 'i }#A z r ~ ~ i }X!'z &s { !Xi '}{ q 2 a i z r ~ ~ i }X'#y ~ z z{ { { x r t r ~ ~ z y s {{ x z y ~ i r ~z x x s {{ x ~z ts y z { u!r { xv # Az ( z z ! T!(x z & ~ xa x z r v { s t s & z r ~ U&r z T{ !av '}z z & ~ z t z z{ z r }{ x
X R = (X0 , Y1 , Y2 , . . . Yn ) |Xi LF (Xi ) |Yi+1 F i=0 F (Xi+1 ) F (X0 , Y1 , Y2 , . . . Yn ) // F (Xi ) F (Yi+1 ) Xi |Yi+1 LF (X) // Xi+1 R i=0 (i )|Yi+1 i+1 (F ) : F R // LF i i (F ) (1 )|Xi+1 i (, i )
s { av z r ~ ~ i TxX#y t z s {{ x ~ z ts ~ r z ! z ! z t ~ s Sz w#g ~ xv T #y s x ! Az ( z z x # 'z & &r z r z & #y s ~ ~ z { ~ x i t r s z t v r r v t z {z t z z xt z z t x av s xx a x ~ z & XT~ v ! ~ # 0 r v r tz t s v { x ar z a6xi i ~ t z z #y t z #x! !y z ~
)
r t r s r vs t r ~ z s s ~ av y &av ~ us t z s ~ xwTus y xv 'v z wTt 'z !ft 'z
//// F (Y Y V )
~ t Xz xv 6z &r }T~ xv S { y t x~ s { tz z a ~ T#y x y ~~ z z t #y &r z }{ !xv '}z z z{ t z r t z r s r vs z z#y &r z zT{ !av '}{ { ~ ar #y s Qv a ~ au}t ~ y v z & { v0 x 6y}{ &x(ag z &#v ! s i cx { x z t r v r v { { '}{ s T}'X !r ~z x~~z i z y t s i z a }{ s 2 # x w!&'es z z t x!Sz t 'z x ~
F (Y ) F (Y )
~ v }{ u x t #i~ z y | ~z 'us { t ~ z t z T{ &r
//// V V
i i z s v wX2t Sz xar z
ts ~ t r v r x { z ss t z u! (x z &`i { &a6s ~ ws Ax r z & XT~ uwT# 6y t z r t s ~ r !y x z y z s r !2xxx us z &w! z x ar #y s Qv ~ r !y #'ax ws z !ws T~ v #'z z &&r z & x z y ~z ~ ~ z y ~ ~ v s vt i r v g z 8 r 6 x z x x x't eTz z a { w z &w}t x x}# x t 'i~ &r z XwXi ~z t i ts x r t r t i ~ | s s i ~ z t i ~ | z T{ t&r z Tuu a i z & ~ xv { &xv ~ &r z XwXi z 'us ~ xv { t z }{ &r z Tuw a i iz &r z x w!&'t '~ &r z XwXi z Xws #y ~ z t i ts x t z i v & i }{ x$aTaav ~ T#y s p~ t z #xx & #`'z ~ r !`~ ar z XwXi }'z ( T!z eQs ~ we 'z a &av ~ i ~ { s z { ~ g rs ~ x i t r x ~ v vt t r ~ ~~ z { y z y ~ x y t r vs { v t ~ au}t 'z xv ( #f'z g z t 'z !acxws }'z (ws z Gt 'z z y ~ { v r v ~{ r ~ z (aG! { ~ &r z ~ v s v z ! | x z & }{ u x { ~ z axr z z ! | pv x
(1) (a, b) F (V ) F (Y ) (pr2 (a), pr2 (b), pr2 (a), pr2 (b)) F (Y ) F (Y Y Y ) // F (V ) F (Y )
pr1 (a) = pr2 (b) b F (Z) F (Y ) F (Z) (2) (1) (pr1 (a), pr1 (a), pr1 (b), pr1 (b)) V
s x z y x z ts ~ r ~ v x s t z uw( ~'z ! ~ r z `z t a#Xz u! z ! }{ u (x r z r ~ i&r z }{ !xv '}{ !t 'z s z z r ~ g s i r t ti g { x z t h z s iti { x 6Dsus ~ r !(z av z r ~ i T{xX r!2r!xv y &r x St xT{ # x 6 #y }Sws xx x y s { x x #2z &r { !xv 2z !us S(z !`t '#y z y t x { r ~z z s y v y r S}~ s x i !Az z & ~ ! s r z ! x r & z r ~ U 'XT~ s !t 'z s s s xz x r ~ x z { z & }{ x 6~ xv ~ v ! u6eeXpp!(j n mt q s r q o n m l k B BE H C C B Y d F B V Y C d P V d e R C B Y R g B H C e P V R H Y Y V H V B Y B F P e H e B P V aY X8SBXDBuXx(H#XC (BxXBAD$SC X!Sx'X0D0 TDP #F G$F $I xXDSeS&UTXGQ hcB& V CC V UP V p8H V XTP0 x& y 'gX0D0 De}DP V R F V }$0XTQDDTXf#X!$D}SxY I XR $$e$R V H V ieUB E H B g R B H C C P B P Y H B P E C B P B C E F B H C F C C B H E Y H C Y C F C B d B C H Y H B C eE V R e P V R H F H B Y R g B H C 0 e P a'X0DRXDP V R DwSASXSB pXSDP (XDP #XB C B d B C H Y B d C P d C H F E F Y H SuSAXSSB pAD!Q(X #(#$0
H F B d P C P #(DSu!H V 8E
CB dPBE CBPE S(DXDTQ#F V pY H
C Y C B Y P C e E e !H V 2DaDSXDP V g I
&a y B Y R g B H C e P R E F x}X0D0 XDP V f#E Q
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
(g, v)
Et /X
qc
X
k
F
F (Y )
//// F (V Y V ) F (V Y Y ) F (Y Y V ) F (Y Y Y ) .
Etqc /X
g
V
// F (Y ) F (V
V
v
v
// Y
// Y
g
)
F (V )
F () =
(2) V Y V
~ 0 }'#y ~~z ~ x & 'z i ts x z }w &
X
i F (Xi ) F (Yi+1 ) Yi+1
i (1 )|Xi = (2 )|Xi (2 )|Yi+1
X 1 2 (X0 X1 Xn = X)
// F (Yi+1 ) F (Y ) F (Xi ) i+1
X
k
// F (V ) F (Y )
F U
(X0 X1 Xn = X) // Yi+1 Vi // F (Y ) F (V Yi+1 // Yi+1 F (V ) U/X
Xi
// Xi+1 Y Y Y = Y U =X 0in-1 Z R X Etqc /X Z Et/Y a F (Z) 1 , 2 F (U ) F Y 0in-1 Y (1 )|Yi+1 = = (2 )|Xi+1 R
~ !&av ~ ws v Ta z r ~ r t r ~ i ~~ x s s t x x~ r t r {z t r v z y ~ v t S}~ s { z z &f&r x}f~ !aav ~ x z x ~ 'XT~ s i Gz z & X}{ &x( #f}{ u x us xx '~ # xz x { ~ ~ x xX `z X( }{ x r| x t s n mt q s r q o n m l k z r v T~ xag z y vt v z t #Uz &s }(xv T{ q x &r x r T! 6 ai T'}aXpz 'z us { t !y y r z &r z r z Gt ~ u6eeXpp!(j z{ r ~z{ ~ x { t x z
0 (A, F [n]) G= F // 0 (A, G[n]) 0 (A, F [n]) X Z n F F // G
B C P B g H C C V E C CP $u ( uXDS$DF DDB E F E P F B C P B g H E C ##$uSDX2DSB B d C P d B B F B C F (D#!Q$AuSXxDC H ( B Y E d xcSR V U$R B E H B P E C B H B E C P I T xXD0}}0 V C B P H B E C P I Ti xXD0}}0 V E H B d B F E d E B H X(XSe}$A$SXx #F E $ H P B Y eE e P P H P B Y eE e P P QST XXB STQDD}P SDV SB E HB Y CC XxX}$! E XB B F B XpXE C Y B d C P !H (DQ$Ad #F H E $ Qp W#G "
M 0 (A, F )
v g az X| { Xi 6 b #yz w XG{ ~z a ' t ~'z ! }!`x &xv y &r x w! Uz X| { 'i t z x r auis v }xc & z r ~ U( 'XT~ s av ~ i x t s ~ v s xz x r z z z z{ t r ~~ x s tz 'XT~ s i !`z t v r v 'S}~ s i ! xz x { z !rx#g ~ r !f}{ z x t z g x y ~ { ~ xz x { r!!!rav y &r x i i r t t t v !xTt { av 'z &r { uq 'z u(uwxv 'z t}t 0 z #2(xv ~ t x t ~ ts s ~ s z x t r t vt v t t z #x! & ~ au x r z s ~ }t xX v &r z z &ws Tx(xv v r r v z i { x t vt v { z t av Az s ~ t a xv ar z z &s }(xv s y t 'z x ~ t 'z wGs ~ t&r z z T~ t z s ~ r z s ~ Tt X ar z z &s }v(xv t ~z i { x t vt ~ s z z #y i {TxX !pg Xz u Tt Sxx y au}Sp10005 bd { x r { s s { x r 6 y y r vs ts r q i h g f s T~ x t z "f" b { xva z ! x t z X| y ( z#z s { v Tt 'z !t z s i x r ts r v { qs ~ v ~ ~ v y ~ z & z ! ~ x ~ r x!}'~ v ! ~ ag Xz uu ws { x y x{z z &us zT{ t T{ xws Xz xTAz t&r z x ws!&'iit 'z r z t Xz i v f t r ~ r { ~ z r v ~{ ~ ~{ " # 6$# ti { { ~~z z 9 7 " z #y i}Sws ! v Ax g { ~~ | txv 'z us x t XSU~ xv ~ g z & z A$xi~ av }'!y s a # }{ s x wxv { v zv ~ z ~ r {z { V V V E V H B P E C B C B d P V d e C B Y R g B C C e P V R B Y C F P H e C B P e gB HC eP V X(DSC S}RXx}SeHD0 D}DP (xDBXDXe0XaD}SfQ F F }RX0DV 0 CD}DP V R F H XTQSTP B H C F C C B R C E Y R C B XC XBD}SxY I X!H80H}SuBGQ DXg Pe B CP FXSu B B F E SA$Y H r z ~z r { z ti { z t&ws }{ t '}{ r z t ~}'z i v f #y i}Sws ! v { (EX(dY { HB E VR E HB dB F XDCDeDP fSAXS C r z t Xz ~{ p6
I 0(!XDP V #S'X0D3XDP V #AD$Su!H V 8E E d e P R H F H B R g B H C 0 e P R H F B d P C P V V C Y CP FB g H V 2#Xu V Ah r { z iti { { EB d i v f #y TXus ! v ! fSA$Y
C E B B B R H B g F e B H F E C DfScwS V }eSw V #S(0#DDB P F #DC F V }X V TeA#F XH B R C e P H F E Y
F
r vs ts r q y t x au}!Si x s r 'z X| y ( #cz s { v Tt Xz ! i x r ~ v z y
eGu1 r vs t &x# xwTus ~ v v !(x #y t !z xv !z ( z { z z x y r z y i { x v s {{ x r {z t r v v i { x {{ x r i z T~ wu}#x6r z s ~ }t xX v i }X { ~ i xr z t ' z r ~ UAi }' ! a X}{ &x( { ar z s ~ Tt ' i }' ! { ~ r z s ~ Tt z ss t r v g {{ x r ~z s r y s v ~ s r ~ ts s r z { x {{ x r X i TxX ! xv z axr z 6 z !q 'z u(uw z #(av x t { z t xv pv u6eeXpp!(j n mt q s r q o n m l k REB C Y C B Y P C e E e H V 2DaDSXDP V g I x}X0D0 B Y R g B H C XG V 0H C e P R C B P F B P g E C B eE e P P B d DTDP V 2Du#$i V fDSAXD0T0ASH d V H B C eE VR R E B H B C eE VR e P VR H F E C YB SDSDP 'iSSDSDP AXDP I #DDB I aSAXa$uewcP V WSe0 XDP V R #F pa V B Y E H B d B H B R g B H C e P H Y B d G xxDXDP V g B H e#F E $ B Y P e E e Q1Q fGCCB @ D A A
eGu1 t x x~ z &is pz !xv i y r x &r z ( }A ~ { av y y x{ t i {z y z t ( Xu i}{ s x &r z Xw xv i z a0 z t
F
r z s { i t z !it 'z v r ~ z t x~ s { t z &Ut Uz &r }T~ xv S &r z z t s { t ~ t 'z ~ ~ z y ts ~ #z u! Ax t x z ~ t z s i t r ~ ~ { z z{ it i { { ~ z z &us pv z &r !y Xf&xv 'z i X!TG~ }Sws ! v Gv ~ v x z { #y t Xz !x v { xv }{ u (y r }! r z{ z ~ v r
F Xr = Xr-1 Z0 Xr Yr X Z0 // X
s t t r s V~ s x 2 xxx us Tt x #(av z r x s { !t z ~ r !y #y x z 4 { z r r ! { xv t | z r i r t rs r t s &r z z 'u T{ ft z xx awxv Uz ! #y z &s { !Xi &wxv !fs ~ av y t rs z i ts v r t r t &r z t }{ s x &r z Xw xv Uz a& { av ~ { t ~ av ts z ~ xv ~ v ! a { x i x x !r ${
X-1 = (X0 , Y1 , . . . , Yr ) Xi Xi Z0 p // Xi+1 p Z Yi i<r Z U
s v r rs r ~ xx! { w z t ~ s Xz ! S s y wws x t 'S ~ v x z {z r v z s r v r ~ aag #y x!ft z # ~ r !y s x z #y x T~ v r X r }T{ i fz !xv ~ s }{ r z z z{ { { x r r x t z t &r z T{ !av Tz z & ~ z t ~ z z{
Xr-1 X Z Q X Xr
i~ t Xz !xv 6y s z r ~ sU ~ i {}{x''!y t z z { v x Xi T~ ~z y { v r z t&r x}~}~ sav Xfz ! ~iz r! T{ ! { t ~ rxv !y{ w x t ~ z#y t x fz X| y (2 z x { ts z s z ~ v ~ z { v ax Xi }~ ~ z #y t z r t z r v v s z r ~ s&r z zT{ !av T{ r! 6yz &s { i!Xi izT{ q x t ~z z & {XzT~ v ! ~ ! 0 xxg ! z X| { Xi t z x u6eeXpp!(j { n mt q s r q o n m l k B 0H S V P R g B H C 0 E H B d B P g F R B P B B F C B 'X0D)X(XT#$TuXXE
k (X0 , Y1 , . . . , Yn ) // X )i X X R R
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
k Q = ( Ui (X0 Xn = X)
Z0
Etqc /X
k
// ShvNis (Et/X)
Et/X
X
SmRigqc /k VarRigqc /k
k
// ShvNis (Et/X)
Vi
// Yi+1
^ U X Xr X = Xn
Etqc /X F
77 U
M
M
z &r x u ~ t s ts v v t v t { a1 z &ws Tx(xv cz &r x s xx z !t z xx a ! xwTus ~ v v pv r r vs t y { x
homHo(M) (A[n], -) : M F Etqc /X
QQp
F (/X)
//
// Ens
F
// G
" # ! $
" # !
M
Nis (A, F [n]) 0 M
M A Ob(M)
PreShv(Etqc /X, M) A // Nis (A, G[n]) 0
n Z homHo(M) (A, F (-))
M
F
G
U X Z0 =
Nis (A, F ) 0
n Z
f
B P e g B F C P CE H B SB R B Y B P e R B H F B e B E C GQ DXuXB SX$w8XSR 'aDa}E V 0#AuX!a V DSB
C B H H C B d d DQ `SA2SC R}$g V Bu#Fx&H V D8EDa}E V R V TY0XaSB#FS B e P B P e BP HB H Ba8ESR V U$RXBASBFX0(!8SXx pY Y C d E H d E d CE B B d B C H H F BE C B uX(XX!XB #Da' F C E VPE E d E C Q E V C VE E H BxH$QE V XGBSuXBSuS V DeSu V #XSB Y C P Q C B E C B C B R H B g F e C DBH 8Q CSB$RSBHBC#FY'TPe}RP SBuS DC$XC Q Q B E E B H P SuS V DCSC!8E V XGQ RP V uX V EXXDXPe6DB}RF V 2(T0T0SB B HBCe h BP P EC CSH V #SSDe}TeGSu BE g F C C eE e P P C B 1Q i6 C B Y d B Y B P e R B DuXxA(xWD$&TE V AeH
M E Ho(M) M Z M M
xv ~ &r z s cXz v #`z s { v Tt X t z y i x x t z !(xv i x r ts v ux z y !z s { v Tt Xz
ey yxwvsTut!rXUp0005 d b 'z i'Tx r s qi h g f x {z{ x s t i vs s r y r z r! yWz u}t s { t ~zT{(~ w ar z z}{ | u rau}t!qSi z r! ~ avTt v xx(~ av 'z ar }{ q z u(uwxv y ~ t x t ~ ts s r v v t z #av ! & }{ s y r 'z X| y ( r x { z t xv z ps ~ t x av t&r z z as Tt(av t 'z x ~ z !y ~ v z y z ~ #xxx w s t 'z
eGu1 iti { { h TXus ! v !x a v { x
Xd 0 = |(Y-T ) Xd Y (Pd ) (Xd ) F Yd // W // V A // Xd+1 M (Pd+1 ) F
g { z r ~ !}{ v r z t 'z s z y z s ~
i }X(x #'t }{ ! x h {{ x g { x z t i ts v r v r v aus ~ r z ws y v #2}{ v r z t z z y z ~ ~ r !y x z y z t !T{ s x &r z Xw xv z z &`axa0g ~ r !y x v r v xws ~ r z ws y v #y t z t v r y s v z ~ z aG ~ r !y x z s r ~ xxx uxv { ~ &r z z &s !}{ (xv ~ s av `z axr z X~ r S r T{ i #xx av ~ tz i r vt v t z x{ x y z z y r z y z t #T{ s x &r z Xw xv z ~ x}0 z & z Ut z z a !y r x i t r t 'z !ac! }{ u pz t ~ s Xz iti { { TSws ! v !x ~ x!&u ! r v s ~ z z t T{ s x &r z Xw xv pz G&xv y #y z { v r ~ v x t &r z XwX! { w x u!r { xv i i r ~ v ts z &p z t
homHo(M) (A, F (Y )[n - 1]) d homHo(M) (A, F (Y )[n - 1]) (Pd ) |(Yd ) = 0 T S Xd+1 = W - S
z t a#Xz u! v D x z ts ~ x
g { x z t h tz z x{ r v { z w!t z 't }{ # x ! ~ r ~ z & #y r x~ u aus ~ r z ws y v #{ w x z y ~ v i( 'uzv av r vs z t 'z t i { z ~ tz { z z T~ v r au}t ~ ~ z & z ( 'uz Ut z r t T~ r z!y t Xz xr! ts v ux t r v z z z { v { z w!`t z ~ r ~ z a z t z z }{ z s { i ~ aus ~ r z ws y v (#y z s 5~ s {x 3t Xz xv! z t ~ s Sz wg S r T}{ i !T~ | v & xx 6 2 { r t t 'z i t r 4 s z z { { z y z { ~ z t }{ s x &r z 'u av Az `&av y z z ! | z ! ~ r ts z z { v z ! u! y i { t 'z &r z zT{ !av '}{Xz & XzT~ v ! ~ t! 0 x0g }s xx t 'z ~ t z r v st y y ~ z aGe ~ 'z { z !!r x '~ ~ r z & zt s t z r z t z v s z r ~ U&r z zT{ !av '}{ ! yX r z t ~ s S z XT~ v ! xv y # r g ' # Wz X| { Xi t z x {z ~ r { ~ i&r z ( }A ~ r q t x~ r v aus ~ r z ws y v `#y t z t xU}{ # x # g { x z t h t s ~z '| y v z { xv ~ { { z y r 'z !av U# xv y t ~ a z ~ z &i&r z &e t z ~ r !y r vt v tz r v xws ~ r z ws y v #y t 'z z & { s x t 'z ~ x z z #yiax u s ~ x}U z ~ t 'z !a(!&xv { t ~ av ~ v ! a ~ t x z '| y v !z ~ s !T{ ( r ts v x ux#aDDt x av z a z t { v r i ~ z y v r s ~xx &t ~'z r {z t z z & xv y t z z a '~ v ~ # 0 r ~ }'#y s Az (av ~~z z~~ }xx x t z { st Xz T&r z r z q r vs s { z r v z xwTt xX u w s X}{ t&a !y t ~ ws#g }{ w av { a1 z r v s 'XT~ s i !#y Tuw x &ws { t x z aw us x z x { ~ z i ts s iti TXus {! v !x g s z!y 2t Xz !x v s { s x ~ iti { { x TXus ! v !pv { { xv }{#y r T{! t# rx v z & ~ ! z s }{ (t z z z z t z x{ r v z a z it z z & #y r x~ u aus ~ r z ws y v i t r s 5~ s {x 2 t Xz !xv ! z t ~ s Sz u1g #y t x z X| y (2#aTt { x v t v v z !y t c}{ s x &r z Xw xv z &xv y ~ z z t 4 { r s z tz g t s v &r z x z &r x w! ~ t s iti TXus {! v ! x zT{ | y s ~ av r Xz T&r z ! r r ts v ux { { st r Tus !a xv y t 'z st y y x r { z t xv ~ x z s z !ws X r z x w!&' i !y t z z z v z ~ s !T{ (xv ~ !g v s r
nZ // X X d+1 |(X0 ) = 0 U (P0 ) d W = X d Z0 (Ui ) // X F () |(X0 ) = 0 Z d ^ X W] - F V = [U d+1 0 (A, F [n]) 0 (A, F [n]) U U |(U0 ) = 0 F (U2 /X) 0 (A, F [n])(U1 Z d+1 // F (/X) U0 Z0 Z X U2 ) V n Z U/X Ob(Etqc /X) U0 U 0 (A, F [n])(U1 ) 0 (A, F [n])(U2 ) U X Z = Z Xd d dim(X) R = ( Ui s : Z0 X-W // X )i Z // U
r z s ~ Tt X &r z z &ws Txav i }' y ! y i y r i { x t vt v ts {{ x
z y #i~ t z #xv !y z
tz
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
0 (A, F ) d 0 0 (A, F [n])(U )
(U1 /X)
(U2 /X)
^ Y = VW Xd
// homHo(M) (A, F (W )[n])
F (U1
Etqc /X
U2 /X)
() = |W Y Yd T W S X
z x{ z & !y r x~ w z x{ z & #y r }aG~ u i { z s { x x z ( Xuz Tt (pv ~ r v ~ u aus ~ r z ws y v `#y t z
V
s {{ x r ~ z x & z r ~ Ui TxX !t 'z s
t s v &r z x
T -T W
M homHo(M) (A, -)
|(Xd+1 )
Y-T
// V - T // F (U1 /X) // homHo(M) (A, F ((Xd ) )[n] F (V )[n]) (Pd ) (Pd ) (Pd+1 ) U =X X X0 X Z = X - X0 X0 = X - Z (X0 ) (X0 ) (Pd+1 ) d+1 d+1
Nis (A[n], -) 0 AE nZ 0 (A[n], -) K M M
t s ~ s t t &r x u z &!r 'z t x u! ~ i z ~ xTx r vt v {
LocNis (K) K(C) K nZ (A) Nis
e Gu1 x!av ! t ~'pzT{ s x uav { v pv 'XT~ s i {!z!`z !ws zS r z x uai z !t 'z i { s r ~ z z x z x y x s r ts r ts r g x t 'z t x z & }{ &raps 'z 6 z t v { z ~ s T{ vav ~ !Uu! y ws ~ z aUu! y i y xv!t&r { q ~ z &Uus `z Ut z g ! z '| { Xi t z xx e v tz r s rs g s { { av z r vs s x { #y xwTt xX u U~ | xz ~ s !}{ v(av ~ s!t y ww z 80g z ' u x v r r vs t g v r i ~~ z X r z x ws!&'i z !(t 'zWz ~ s !}{ v( z S$#y au}s ~ v v { pfe5 d b { sxa y z q ~ av !y xwTt T~ u X vw z ! us v }xx ~ z r vs x s x r s x s xvwTt }~ u xX u #yA{ z t rxv z (t z v z xv r v { v ~ z ~ s !T{ v( z }{ | y s ~ xv rx1g z a#v ! s i {x { x z Sa# t z w y y t av #y z r r z ! xv y ~ z ! | i ~~ x s s t x z T{ s x uav { v `z ~ us v }xW z r ~ S}~ s G'z { Uz ~ s T{ av ~ u! y us { x x z x ~ ~ v s r ts r z &us y r vs t xwTus ~ v v { !0e5 bd !`z !av i y s z 0 z !ws S r z x u!&i z !t 'z g f s r ~ {z t r v z y x z z & X}{ &x( z r q r z !r z } xv y t z r i x z s x{ r ~ z y z ~ #z X( #y t z r z z !ws S r z x w!&'i t 'z z t z r vs x ~ z y ~ x s { ~ z ~ s !}{ ( v v z z y t x #~ &r }{ q ~ t z #av }{ &r p~ xwTt { q #pz X Tt Xz
// K(U /V ) K(U/V ) // V /U V /U K(V ) Fib(a ) M // K() // Fib(K(U /V ) K()) Fib(a) // Fib(K(U/V ) K()) : K V V
a a
i TxX !fu! y i y r z r { { x r ts
z y ts s r #z (wu z ! { Gz T~ r z 6A 'S}~ s i !#pz (wuav `z ! z xar z ~ ~ z y x z x { z y ts s r s v
// U/V.
i { x r r v ~ z y 'z !2 T~ r z 6 'S}~ s i !`#y r z s ~ Tt ' v i }Xz ~ xv { i y s ~ xG xz x { z {{ x z !ws X r z x w!&'i z !it 'z x z s r ~ ts r ~z ~ '}{ q z { ~ u! y us z ~ s !}{ ( v {z t r v t x z t z r vs x ~ z t r tz z z a X}{ &a # xv y t ~ G ~ &r }{ q ~ t z #av }{ &r W~ au}t { q '!y &xv ~ z y r ~z '! | 'z g z !2 T~ r z 6y ~ ~ z t z z &f!!xv y &r x r z s ~ Tt ' &r z z &s }(xv t 'z i r r t t i { x t vt v r t r 'XT~ s i !f#z ~ s !}{ (xx( #y &av ~ xz x { z y ~ v v r v ~ z t r ~ z & ~ av { &xv
K(a) K(a ) Fib(K(a)) // Fib(K(a )) // U. a a
BE H BPE C BP H BE C P XTQSTcXXB CP ( u G
s {{ x r i r r ~ x 'z t a xv s ax & 'z x Xi ~ TXus x ~ ~ t iti { z #y s & z r ~ Ui TxX !!xv y r r vs ts r q y s x z s r ~ ~ av ~ v 0x ! au}!Si Ax #y }Sws ! v T{ u( }!Ax x v z i t i { { z |s z { h z & ~ ! z !ws S r z x w!&'i z !Ut 'z z s iti { t #y S}~ s i !z z r s ~ i y z ~ s T{ z $g #y TXus x v x z z s x z x { s r z t }'z i v f #y TSws ! v !Ax z us { ' w r { z iti { { q i x v t z &r { par z z wTt 'z $ws t x q t s r S}~ s i ! ! & av ~ av { &xv r y r t r ~{ x z x { r r vs x au}t { q r z t ~'z z | x 6g ~ z y z !2 T~ r z 6 'S}~ s i !#2z ~ s !T{ (x!a! xz x { z y v v r v r i r !rxv y t&r x Sig z '( z t h g 'z ! ~ r z y S}~ s i ! #aTt { av g Xz u Tt z t x q r ~ #y ar }{ (t z #av ct z ~ z x z x { t v { s s { x r t x q s r t z x v {z ~ r '~ v ! W xv y # 0 r 8 X w z t i ti { { }{ X y TXus ! v !x !Wz X( #y t z r z 1 'XT~ s z y z ~ xz x 6 ar }{ (tar z z wTt 'z $ws i !!'z !ws zS r z x w!&'i Wz a '~ i ! t z z &ws }{ t '}{ T'}{ r z t Xz i v f !y }Sws v !pv u6eeXpp!(j s { x i { z { t r z ~z ~z r { z iti { { x { z y ~ x ~ ~ ~{ n mt q s r q o n m l k B E H B P E C B P H C P B V C h B E H B V g F e E C B H B E C P I }i x(SDe}Te V GS#XQDP V AH H P B Y e E e P P P B F R E STQDD}X$g (Sw #SfDSXDXB I H P B Y eE e P P Ti x`SDe}Te V 8STXDXB I }i xXD0T0 V UGQ DX!eDSAXu#SXpSD# V u I C BP H BECP H P B Y eE e P P B P e g C E H B d B F B C E B C R EC DDB B Y E B H aSXx#F E B CE H B SB R B Y B P e R B H Sp8XSR '`xDa}E V pe#F E $ 11Q fCGB @ D A A
(M, E) K Nis (A[n], K) 0 PreShv(Etqc /X, M)
" # !
" #
t s ~ z &r x w! T{ s x wxv { v z x r tz ~{ r { xv z !ws Xz us t 'z s ~ &r z z ! T~ t z s cz X u t z s r ~ ~ ~ x v z S r z x w!&'i z !t 'z z ~ s !T{ ( r v z ' w i x z y z i x ~ r v g r vs ts r q i x v ~ v 'z X| y ( #z s { v Tt X#W}{ }t { t z ! 6y z y ~ r ~ z s { v }t X(z { WT~ v T~ s y x1Qx y y xwTu!XUh ~ z g f d { s p0005 b x z r ~ z s uav Tt x #z !0g t z g s v v g g g i x ~ z y r ts v z y s r ~ 'z s { v Tt 'U#2z ux CE H B B R B Y B P e VR B H 8XSR S (x(D$&E 20#F EC DDB B F C P V P Sc$w V $Y H C V Y CE VE H B C eP BP C B Y E H BEE B d V E XB CE B C B XXx(Su U !H AH XSSTeTDapX8X(Y E F E P F B F E B CE P R E H ##$USS8H V XGQ U$XC C Y CE B C B C F E B F !H V AXXua(Su(X V XXxDH U ##$US0#)Q V SXC P B C F C BE F E P F C B H F e PB B d B C H uX(XX!X B
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
S E Etqc /X Et/X VarRig/k SmRig/k PreShv(S, M) // L Nis AE A[+1] (A) A[-1] E E Nis // Nis (A[n], L) 0 (M, E)
AE
Nis
" #
homM (-, K) M
(C)
/X
K X aNis () X
aNis (/X)
Nis F homM (F, K) // F1 F0
K(U )
V
// K(U/V )
// U
// U
Nis homM (/X, K) = K(/X)
// LocNis K
homM (F1 , K)
LocNis
// homM (F0 , K)
// homM (, K) =
Nis
i { x ~ r !y r z s ~ }t xX x r z { z t xv pv
K
t '&r z z u}t T'}{ s z ~ z tz g it v r t t r {z~ ~ ~ Tx!G&r z z ws ~ &xv XTG~ a #a &r #y r 0 i pz r '#y s s ~ }Sws xx ft ~ z { v xx Xi T~ g ax 'i }GS g z y ~ x t z z y ~z { iti { z ~ ~ ~ ~ z z & }{ v v g r q r 6 #7 z x s i ~ z y i r z #`z &ws { Xi `}{ q x Xz r s ~ i y x{ r vs t i z x }{ q }~ r z z y #z u z { v xwTXu av x ~ v x t v }{ u z r r r ts r q `u!Xi y r tz gs r z z y 8!q 0 t !x Xi }~ r ! t s v &r z a 3 w& 0#' x Xi }~ ~ au}t x x!r 'z a z au! y v { &ur r vs t v ~ ts t rs tz ~ z ~ s !}{ ( z &r x w! 'Txx #~ &r z x r #T{ s #i xv y t z v ~ t s ~ ~ z ~ z y t z x ~ z y z x z y r tz s v z y x t t x i ts x i r ~xx 'i T~ #z (xv ~ z y { ax #}{ t z X0 s z e {z z &iar T~ v ~ r z Tuu }{ !Xi x xp xv y ~ x r t r z ~z{ r &ws }{ t '}z Wr z s i x{ x x ~ s i x{ x z z v `!!}{ (xv ~ s z 't z w!r z &wTt ~ s { Tt a}xX a i 6y t 'z g 'z w! }t x C z w!r z &Tt ~ s { }t a}xX#y { s s { x r 6 t 'z z y s ~ i p}{ v (z z &t z S| S}~ s y t 'z z t { x{z r v g r vs z ~ t z z y ts ~ ~ { ~ ~ z r vs #y au}t x w x x z & }'~ v ~ xxwTt 'T~ ~ xv (z t S#z Wx ~ r x h
p
eGu1 t ~'z t s x z s z ai&r z 6& z us UX r z x w!&'i z v ts r r vs ts r q y z t#x}t { xv z &`! y i y av ! xwTu!Xi cx #y z S r s }~ 2 xv y a y }{ z t xv U'!y xwTt ' w ~ x r ~ ~ z r vs x s ts v ~ x ~ z ux z us 'X r z x w!&'i z z t z Xi y s ~ t
R(X, ColimI K). K K // L // L
'XT~ s i !`#z !ws S r z x uai z ! s r x z x { z y x z r!t z ~ g iti { z ~ s z x }~ uit Uz t a xv s ax & #y s s { }Sws xx ~ z ss r vs t r wus xwTus y xv cx x ! X u t r aav ~ tz {z t z { x v z a XT~ v ! ~ # r x { x !xU~ | !xz ~ s !}{ (xv ~ !p&r T~ ! y ws 'XT~ s i #z ~ s !}{ (! v s r t x s r xz x { z y v r x y { z t av #Uz uw( `x z & Xz s { z t T{ ug T{ u xv { x r ~ z y s x { q i ~ z s z r v ~
HoColimI R(X, K) K L Nis X K(X) // L(X)
s ~ x ~ z 'z !ws f'X r z x w!&'i v r ~ z ~ x v s y t x t ts s r { x z t h ~ ~ v z S r z x us!&i z !t 'z X w ~z 'X r z x w!&'i 6z &r { uq z u(uwxv (z !t xAT{ # x r z t Xz ~{ z { w xx r z xX w ri v f z!y i}Sws ! v {x z s { ti { q i t x r v r vs t t ar z Sax T{ ! { z &(us }~ xa ! au}s ~ v v !x xx { { z t x #y ar }{ q z z r ts t av6 z &r x}{ q z s { v Tt xX(z r! z x t i r r i r r y r { z t xv !2!!av p~ xv ~ v ! r z t ~}'z ri v {f #y iTXus {! v !x car z us { &isa X ar z z t x!Sz &xv ~ X w ti t q i { z { z t x t r x v S}~ s i !A'z && ! }{ s x wxv { v z x z x { ~ g z t&r z zXxx }{ r! t s i t x ~ g r vs ts r q y { xwTu!Xi cx { xx z #y ' u x v z 'z x &r z x w!&'c&r z z us t z ~ | z !ws X r z x w!&'i z ! s r { x z i r r t t {z t r v z s g x t vt v z &U!!av y &r x z & 'T{ &x( #y t ~ ua't x xv &r z z as T(av t 'z ~ ~ t x t ~ ts s 'z &r }{ q 'z (wuav A Uz #(av x t
hom(A,-)
r z s r ts ts s ~ z r ts r s i z v s ~ !T{ av ~ t 'z x y { z t xv v ~ !q ~ u! y v ! v #z &r { uq z u(uwxv (x t '`z &av z & y v ! v !#g { ~ r z u'! x { z y t x t { r s r q ts r r z g z 8 r ~ z { z s r q ts '0 av a2#y Uu! y v ! v !r z !p y v ! v ! z { v ~ r { t { z t xv 2S 't 09z 6 'z ar }{ q z uwwxv (z #2av { { x ~ t x t ~ ts s x t
homHoNis (PreShv(SmRigqc /k,M)) (X Acst , -) : PreShv(SmRigqc /k, M) // Ab
r {z t r v z t ~ y v y r t r { z t xv z z & X}{ &x( #y &r z z { !'i i~ u t ~ # ~ t a xv &av ~ x i r x t r ~ t z !xv z z & }{ x 6~ av { &xv 'z !#ws 'z (xv pz x z u r x s { t z s { v }t X(x { xv z ( z W y i xX i Xz !xv ! i i x r v r { { ~ s r q r ~ ~ ~ x v tz s z z y r #p xv y t ~ & 'z ' u ~z '}{ }t { t i'#iz ! ~ ~z y ~ s r z t }{ &r z r z uw!p #y d { x s r r v g y r vs t g f v g z S r z x uai z ! x x0Q$xy au}s ~ v v { 6pe5 d b x y z X| { Xi D(i ~ v # eg W b { v z y z s r ts r t z s i r v r ~ v Tt #cX r z x w!&'i z !2! y us z &r #y Xxws ~ u us v u6eeXpp!(j n mt q s r q o n m l k E B B C C E B B E d F B Y P e E e XXGS(SR V U$R I DC V X(xxDXDP V g I 0#F B H RB g V B d P B Y CE B C B AD$S V x28SXxSuP V SQ C B H H CDA$X}$g V u#H V DQD$&TE V SH V S'X0XaSiXASSR V U$iDDB B d d C RB B e F PE B P e R B F E E H B B e P Y H B H B E H B d B E d R E C B e F P u#x&H V D8E B P e D$&TE V R V 6 !& 8xY V aSXF V (SXR S 2aDa}E V e#F CE H B B R B Y B P e R B H E H E H Qe B Y C H B C Qp px
(M, E) HoNis (PreShv(SmRig/k, M)) X Acst SmRig /k SmRig/k
zS r z (xv r r vs ts r q i h g f xwTu!XUp0005 ~ z z T~ r z 6y 'XT~ s 'z 1t p x z x ~ z
}{ x t r t v r r ~ { z x #y { ~ &r x t ~ av 'S}~ s i !z z xz x { z !xv { xv (z &1s }~ db s v 'S}~ s i ! ! 6(~ v T~ s y x6g ~ r !`}{ z x r v { xa ~ r !`}{ z x x y ~ xz x { r y z x y ~ i r r t t y r vs g f xz x { r 'S}~ s i ! ! ~ !xv y ar x Qxy xwTt 'z ep0e5 bd Xz u! ~ xv A}}{ av p{ z t 'z z g { t r x { 'XT~ s i !'z &r x r XX xv ~ xwTr z { x z Xu av v t Gz Sw xv z s { v Xz t Xfz ! v t| xz x { ~ t { z r r vs t v ~ z t| x r
I gAeCApR'I c cF ` V i @ 8 Q I
` IQ
Q QF I Q ` g gA` Ch0c a
M
BQ c CRi eI
6u
eGu1 ~ r !i'z &r { uq 'z (wuav i'z !q 'z u(uw #y au}t x !(xv Ax x y ~ t x t ~ ts s x ~ s r ~ ts s ~ z r vs t t r t x t i t t x z!y t z'z t&r { uq 'z (wuav xi'z ! x t ~ &xv ~ ~ x t ~ ts s { x ~ z y r vs x ~ #i~ xwTt { q z z as y ar z z t x s y ((us z !xv i y t x u! ~ i pv ~ &r { q { z t &r z z u}t 'z xv 'S}~ s i !c!'z (wuav 2'z X}{ i y s ~ xv #c av y t ~ # xwTu!Xi x #y wus xwTus y av x #y s r z y r r vs ts r q y { x z x { z y ~ ts s ~ { z z ss r vs t r z t &r z z t x s y (ws z !xv i y s z 'z ar }{ q 'z (wuav 'z ! x t ~ aav ~ i t r s ~ z ~ t x t ~ ts s { x ~ z y ~ x ~ z #2'z !ws S r z x u!&i v u6eeXpp!(j n mt q s r q o n m l k C B E V P E C B E d R V B V E C E C B H B E C P I H P B Y e E e P P V E V P e C !H V Y DH QQ S}P iuX GDDSDSB T aAXD0T0 GH Q DS$g P FC # F V X V 8X01SxX}C V XR V ! 8XSR 'x`D$&TE V e#F BR C e P C B Y B C E $ 1Q px CE H B SB R B Y B P e R B H
"
Etqc /X
CP FB g #Sw V h
" # !
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
ShvNis (SmRigqc /k) PreShv(SmRig/k, M)
i r !Xi t x u! ~ i { t r { x x r v z `~ au !xx 6Duy y 'S}~ s i ! !t z xz x { r ~ r }{ z x { ~ x y ~ t Gz us ~ ux ts v z t r
B Ri 'b` CQ Q 8 6 a I
Q8 i IQ AeeCdI
S
M
S
homHo(M) (A, (X, -)) : PreShv(SmRigqc /k, M)
A
F
R(X, K)
C
C
(M, E)
C A Ob(C)
X Acst
k
PreShv(SmRigqc /k, M)
qc
M
X
Acst
k
(X, Kf )
EF
R(X,-)
M
Kf
R(X, -)
Nis HoNis (PreShv(SmRigqc /k, M))
// M
S
C
// Ab .
Nis
K: I K
Nis
// PreShv(SmRigqc /k, M)
I
Shv Nis (SmRig/k)
AE
A
K
Qsch (X, f ) p
~ k X = (X k k )[V ]/V - f. ~ Qrig (X, f ) p p=1 Qsch (X) Qf or (X) Qrig (X)
a y
) 1) 320(
k
Nis
k ~ k = k(())
ColimI (X, Kf ) = (X, ColimI Kf )
Qf or (X, f ) p Qf or (X, f ) p Nis
f (X, OX )
Spec(k) Spec(k ) // Spec(k) ~ Spec(k) Spec(k )
k
ColimI Kf
p N - {0}
f
Nis
Qsch (X, f ) p ~ k(())
k
~ k // Spec(k) ~ K ~ k
// Kf
E
i { x {{ x ~ z { v Gxx Xi }~ ~ z !y r z s ~ Tt ' i }Xz z { v r g r vs au}t { t ~ xv z a ~ xTx r { x r vt v z y ~ z x ~ z r vs z ~ { t r z y ~ ~ z #'! r }{ '#y xwTt 'T}'z &u s #i~ T#y x #}{ q x t 'z z y z ~
e0 e : StU a k
-1
~ z { v Gax 'i }~ ~
t r r y v z y #z x Xi z ~ s !}{ (! z !xv i~ x!r v r
Spf(k {U, U , S1 , . . . , Sr }) Q
z y ~z x ~ z #f'! r { #y
t s ~ z ~ &r x w! ax #y &r z x r ppv z t z x z $& ! U~ au}t x !r 'z { xa r vs t v ~ s v t s r &r x &ww !xx 6 t t z y ~~z #U~ }'!y x z y z t !T{ q x &r x r }! r z r z #xv z x Si z ~ s !}{ ( z z{ v
f or
t st v r z x tSi z ~ s !}{ v( z far z }#xxv raus}t Xz aw s v z~{ t r r vt v ~ xTx
e0 : C o[U, U
-1
{z t r v z y r g S r }T{ i `#~ | xv t z & X}{ &a # xv y t ~ ! ~ r !y t 'z x ~ z z{ { z y z ts v ux#& { r vs z ~ { t r xx axwTt 'T}'z &ws { z ~ s z ~ z { r vs 'z u}t '0T'}~ au}t x &'i 6y z y ~z x ~ !'! r { z r vt v r v z ~ aTx# z #y z { v }{ t '0`z ft z z z ~ ~ z & ~ xw 0!x t r ws t x v z z x t 'S ! { xv {z r z x Si z ~ s !}{ (! z t ~ s Xz u s & ! 2z ( z z c 'z !x!r #y v r r vs t t { s x t ts 'z &ww !xz u!& au}s ~ v v !x xXT~ v ! ~ # 0 x#g { { x {z z r v z z y #y t '}{ s i{ z t x !it 'z x r ~ s z v t rs r { x t x ~ ~ x z y ~z x #A'! r { #y x us $x }{ a!r z { X r }T{ i cxz !xv ~ s { r z ! x t ~ ( ~ z z v ~ z z{ { { x r r x z (xG&uav ! x &r }Ta t&r v r ! r vs t z z x au z { v 2x Xi }~ ~ x aXz Xz s }{ t `z X( ~ xg xwTus ~ v v !(x t z & # z ( z { t x r | x { z rv { 'z x t x ss t r 6 r t r t 2TT~ wt z t x xv s xx & !y s s { }Sws xx z ! iti { r t s v &r z x z `&r }~ wu}## z & ~ xv { &av z z ~ s z x ~ z eGu1 t &r z }{ !xv '}{ X r }}{ i { z z x z z{ s z r v i x ~ x #y z ~ x aXz 'z &ww !xx y t ~ w#g }{ w av i{ x! { s { x { z z z i #y r x r Uz s { v }t ' ~ xv (x ~ r !y x t&xv ~ r tz t x r v z z{ s z ~ z tz z &Uus }~ x#g X r }}{ i ixx '&r z z u}t T'}{ { { t { x t z z{ z &r }{ !xv '}~ tz r vt v r ts v ux ~ x z t a xv s ax & #y s s ~ }Sws xx 'pv ~ ~ z { iti { ~z ~ x}!t z tz g { x { z z i !y r a r z i i x ~ s z t r v g i rs y z w! r x s { t z s { v Tt ' ~ av x ~ r !y t 'z x ~ z & r z ! }{ &aDxx !y y i y wTt ~ s cz r x s { t v z ts z i ~~ x s us v }x z r ~ S}~ s !T{ (xv ~ t ! &#! | x #}{ xv cx ws v Ta z r ~ 'XT~ s z 0't 09z 6 s x z x x z v s ~ z z x z y z i ~~ x s s x z x g z 8 r z y z q
X Acst U1 Acst U2 Acst (U1 U2 ) Acst i0 I I = I - i0 k (Uj )jI (Uj Ui0 )jI {Uj A, = J I} X = U i0 V X I = {1, 2} k X (X - C) Acst (U1 U2 ) Acst
U Sta
ax v s { z x z y r vs x q z y ts ~ r r ~ s { Tt z p Xa#~ au}t { v #z u! z ! x r z a ~ xv ~ v ! X}{ &x( r z s A 0wxaTt z us ~ !ft z z a { v v #y x ws Xz { S r T}{ i Gxiz !av ~ s { r u6eeXpp!(j r r ~ ~ r y{ x {z t r v { x r s g r v z ~ z z{ { { x r r x n mt q s r q o n m l k P V 'X0XaSB BeP Y HB H B e F PE B P e u#x&H V D8Da}E V R I XC V USQDP V x V C F h E H BE E B SXx Tf P F #$ BE H H 0$f P F #$ CE V d I V F C B C C d V CE P B g F C B Y V B xY H E HB dBP g F RBP H F e H H Y C H C Q X(XT#$TA#U0x!S$!aF Qp W" #! XR U$R DC XpDuXD2Y DX#iSxUP " 'S}~ s i !`z !g t `}~ wt z t x av s xx & y v y r vt v ~ r !y t 'z x ~ xz x { z z ~ s z x ~ z ( z z }{ x 6i~ xTx0 iti { r ~ s z & X}{ &x( !y t ~ u # xwTus ~ v v !x x X$x z { v x #~ t z #xv z s y r vs t ~ { x z#y s s { }Sws xx z !t 'z ~ {z t r v z { { z y i x ~ t v r z i #y r x r z t ~ t '}{ s }{ z t x(z ! x t ~ { z z z y ~ x { x z y i #(z w! r x s { t z s { v }t X ~ xv x z r V de R E XB B C C B DX0H S V X!SC I #F H RB $g V
p N - {0} X =J I UJ = jJ Uj {Uj Acst , = J I} T ~ k AE HoNis (PreShv(SmRig/k, M)) A Ob(M) f (X, O ) X Acst [Qrig (X, f )] Acst p // U1 Acst (Ui )iI X X Acst U2 Acst AE I Ob(T) I // X Acst . X k
xa B d P B Y CE B C B AD$S V x28SXxSuP V SQ C B H H CDAd$XCR}B$g V Bu#H V DQEDP$&TE V BSEH V HSB'eXY0XBaSBiHXAdSSR V U$iDDB B d e F P B e R F E B P H H E B BE d R EC B e F P u#x&H V D8E B DP$&TeE V R V !& 8EexY V caYAHSBXCF V HSXR S RaYBDa}E V 0#F B C CE B B B P e R B H E H H Q Qp #f E " t s ~ ~ z z x { x { x z z x ar x u! z t 'p!!x}aX`!y s ws !`z '| { Xi t v r { v z
r v g r vs t v ~ x x `~ au}t x x!r 'z 'z p @ 9 7 54 3 ) " % % " A68'6!210$('&$#!
a = (a1 , . . . , an ) D1 , . . . , D n
X Acst
(M, E) HoNis (PreShv(SmRig/k, M))
Qsch (X, f ) = Stf. ~ k ,p , p X
, S1 , . . . , S n ]
U Sta
// Spec(k[U, U -1 , S1 , . . . , Sr ]) ~
-1
X Acst = Cof((X - C) Acst X Acst ) Ob(T). (X - C) Acst
Q
k
f or
(X{R, R-1 }) Acst
(e0 ) : Q
// Spf(k {U, U -1})
-1
.
Qf or (X, f ) = Stf. ~ k / p X /(),p
f or
(C)
-1 StU {S1 , . . . , Sr } a -1 //
a a k {U, U -1 , T1 , . . . , Tn }/(T1 1 . . . Tn n - U -1 ) T1 = 0, . . . , Tn = 0 C = D 1 Dn T
k
AE
f: X
X = X
et Qrig (X, f ) = Stf. ~ k / p (X /()) ,p .
X Acst
// StU -1 {S1 , . . . , Sr } a
V = jI Uj V Ui0 = jI Uj Ui0 V A (V Ui0 ) A card(I) = 2
X Acst Ob(T) X k X
k
StU a
-1
k {U,U -1 } Qf or (C)
^ k {U,U -1 } Qf or (C)
Y1
e
e
// StU -1 {S1 , . . . Sr }. a // StU -1 {S1 , . . . Sr } . a // X StU a
k -1
f
{S1 , . . . Sr } f
y & t z s &r #y 'i z ~ s !T{ ( z v
n
xv
r v xp { x i r r y r 2z !!xv 2 av y t z xwTt 'T}'z &ws z t z ~ r vs z ~ { t r t r r vs z ~ { t r xwTt 'T}'z &us v z#y s Qv r ! x xg z ' z i r v h 'z x
i-1 . . . Ti-1 Ti i
t !y !r $`T{ '| x ~ z z v x z z !xwTt }{ t 'z !a v z x r r vs x { z ~ s !T{ (xv ~ s v r r y z ~ s !}{ (av ~ s ! xv x r v
b (T1 1
iti { z T{ X| x x #U}{ t '0pz t 'z z y z z ~ ~ z #y s s { }Sw s xx pv z t i~ t '}{ s U}{ z t xixz ! x t z u! r x s { t z s { v Tt X ~ av x ~ r !y &xv ~ i x ~ z z y ~ x { x ~ i x t r
k D(Ti |Tj ) // Stv Acst X,b // OX
tz
X}{ &a'i r z s !w$g {z t r v y { x r s
.
v s {z x z ~ s { Tt z p Xa#y i! wTt ~ s yGz r x s { t z {xx rs tz t r r t z z!y y}x!y r x t ~ ar z }{ xv TGz X}{ i y s ~ xv xv y # r Q x i y s z av { x z z{ {z t&r }{#y r zx r z tz s z ~ ~ ~ { rs ~ v t z t z V ! r z ~ rxvwTt 'T'z g z !ws }'z uGt z z (xG B t ar z sa& z }{ i!Xi rxws ~T~ T~ s yz r! {xpzX r z (xv r v v x r ~ x aX Xz r z }T~ # z cr c E x { x{ z t '0x Xi #y r `#y | v ! r r y z ~z t z z z { av ~ v ~ r r v { z au x { S r }T{ i fz !av ~ s { r u6eeXpp!(j ~ z z{ { { x r r x n mt q s r q o n m l k ~ z & { v v t z y
-1 -1 X{U1, U1 , . . . , Ur , Ur , R} ^ n-r Acst ; p N - {0}, 0 r n et (mi )r Zr k Bk mr m1 i=1 (Rp - U1 . . . Ur v)
g rs { av Qs ~ we ts r u! y ws
B d B C H uXASX!XB P V iX$0X&&XfX8X}}$#SXR V P V B e P Y H B H B E B CE R B P Y C P F BE B E uS V DC B xY(Bu#H V DQD$&TE V R I $XC V H V DSB e F PE B P e C F C Y EC E B CP SXx TG$u ( E H B X$ H HB Q XE #FHH0HUCPXQXB YESueF I H#F SB C B E H B E B C C B Y P e E e SSaDXDP V g I e#F B H CQp W$#i! " " &r x w! z ( z z 'T{ &x( #y t ~ u!a ! (z '| { Xi W z ! }! (x Xz wr( 'z t t s ~ s g v t y z{ {z t r v z { s { iti iti { z#y s s { TXus {x z #y s s { TXus x x pv { x x z z }&r #y s '~ x qs t z r v z ~z '}{ X| x !z ~ s !T{ ( z z y z #xx w s ~ xw 0! z y v { x
e0 k f (C, O ) [X A] Ob(T) [Y A] Ob(T) Stv X,a StU a X
-1
s y x r v z z{ t rs ~ z S r z x w!&'i 6z r (x x a&g S r T}{ i #~ | xv ~ av ~ s ~ w { z y z
[X A] [(X - C) A] [Y A] [(Y - C) A] ^ (StU -1 - o[U, U -1 ]) k{U,U -1 } Qrig (C) A a .
~ z ts '#y ! y i y r z r
// X.
s ~ x v 'z ' w ~z 'X r z x w!&'i i { x t r ~ { { x ~ z z y ~ i r ~ z ~ g z ~ r !i~ r z s ~ Tt ' v &xv ~ i TxXS#iz &ws { Xi '}{ q z !a# T{ s x uav { v z x x y {
(StU a Nis
-1
s r t st v t Xz !x v a ts v ~ z { v Gxx 'i T~ ~ z y s # z r ~ ~ i }XW #y s ~ ws x &r z T!xWr {{ x z { r vs t r y i {z z { v ~ r r v ~{ ti ~ z (xv ~ r z au}s ~ v xv i z xv `x r z Xu t Gz t 'z !at z ~ r !y y r x z y #z x av x s y xws Xz (w s $g z x St z
P0 Y Y1 - P 0 C C
e0
r z s ~ Tt ' i TxXz iar z }#xa#g i { x {{ x t st v r v
t s ~ z ~ &r x w! `Txx !y &r z x r x z t z x {
) " " % 3 3 ) 4 10$8$X&% &0) a11800'
A Ob(M)
e0 P =C
^ - o[U, U -1 ])k {U,U -1 } Qf or (C)
Y-C
o[U, U -1 , S1 , . . . , Sr ] StU a
^
Acst
k{U,U -1 } Qrig (C)
k
U -1
P
// o[U, U -1 , S1 , . . . , Sr ].
P
// StU -1 k {U,U -1 } Qf or (C) ^ a
(StU a
v (X, O )
-1
(StU a
// C
n
{S1 , . . . Sr }
-1
~ k
e0
-1
) k{U,U -1 } Qrig (C) A Ob(T).
.
n
xa
t rs v z &ua ~ s y s z & ~ !
n
(D(Ti |Tj ) D(Tj |Ti )) Acst
OX {T1 , . . . , Tn }
1i<jn
D(Tj |Ti )
D(Ti |Tj ) Acst D(Tj |Ti ) Acst {Stv Acst , (D(Ti |Tj ) D(Tj |Ti )) Acst } X,b
b . . . Tnn
n2
{x D(Ti |Tj ); |Tj /Ti (x)| = 1}
- v)
Stv j X,b
{Sj }/(Ti .Sj - Tj ) =
b i-1 (T1 1 . . . Ti-1 Ti i
D(Ti |Tj )
i
-1 OX {Sj , Sj }{T1 , . . . , Ti , . . . , Tj-1 , Tj+1 , . . . , Tn }
bj = (b1 , . . . , bj-1 , bi + bj , bj+1 , . . . , bn ) i
a = (p, . . . , p) v
b
Stv i X,b
b +bj
j
// D(Ti |Tj ) Acst D(Tj |Ti ) Acst
j-1 j+1 b i+1 Ti+1 . . . Tj-1 Tj+1 . . . Tnn - Sj
b (T1 1
b
bi = (b1 , . . . , bi-1 , bi + bj , bi+1 , . . . , bn ) j
OX {T1 , . . . , Ti , . . . , Tj-1 , Sj , Tj+1 , . . . , Tn }
b
b
T i Tj D(Ti |Tj )
b +bj
b
^ ) k{U,U -1 } Qrig (C) A
// Y
j-1 j+1 b i+1 Ti+1 . . . Tj-1 Sj j Tj+1 . . . Tnn - v)
Qrig (C)
b
a
~ k[U, U -1 , S1 , . . . , Sr ]
b = (b1 , . . . , bn ) b b OX {T1 , . . . , Tn }/(T1 1 . . . Tnn - v) v D(Tj |Ti ) StX,b
a
b
n HoNis (PreShv(SmRig/k, M))
-bj
ShvNis (SmRig/k)
v)
b
n=1
.
b
D(Ti |Tj ) D(Tj |Ti ) k X-C Y-C Stf. (C),a Qrig // Y .
D(Ti |Tj ){1/Sj } // (C, O)
z v #y !r $x { x { z z i !y r a r z ts r z &pu! y i y xv
r vs x xwTt T~ s { vTt xxz &ws! z ! av z t ~ s Sz 1 #y awxv { ~ r r r y r s v z t rs z i { z t rs #y ( Xu &uav Wz z axr z t z &r !y sXi z ~ s T{ z 't }{ ! x U au}s ~ v v ! x r z wXi ~ r ! t 'z v g { x z t h y r vs t s 'z r x v ts v ux { x z ~ x r { tz r vt v ~ z v ~ aTU# #y !r $x # ~ z y { x z y ~ s y { v t #z !ws T~ ( xx t Xz xar z }{ xv T{ ! z z r t v usx t &r z T{ !av Tz z r ${ !& ~ s y z z{ x s z #y z !us }~ ( ax t 'z !a(&r z T{ !av T{ ! # q r { v t z z r z r r t { av !xv t z t x u! ~ i `z z &Ut z ~ { z & t z s i t ti { xv &r #y '&r x Ses 'z z y ~ s y { v t #z !ws }~ xx t 'z !aG&r z }{ !xv '}{ xpz r }`T~ z z r { x x{ z s t { (S r }T{ i `~ xv { &x('Up ~ z z{ { { x r t r v i h z r }{ x s z r ~ &r z T{ !av '}Az z & ~ z t z z{ r v x{z XT~ v ! ~ a&g z &
p P(Zt ) // Xr Spec(OQt ,p ) OQt ,p (X0 = Y0 , Y1 , . . . , Yn ) r-1 (Pj )jJ an Xr Spec(OQt ,p ) Xi // Xi+1
an Xr-1 k
~ z #y t z
s V~ s xx 2 ~ t Xz !xv 6`z &r }}~ av X &r z z t s { t z u! z ! Xz !xv { i! 0 r 4 { { y t x~ s { t ~ ts ~ r { t t z
A // X
an
s s z r ~ ~ i TxX {{ x n mt q s r q o n m l k u6eeXpp!(j
R A
an
B Y B C C d E P B g F E H B d B P g F R B P H F aAuX$Y V DX#fXAST#$T#P V i0H S V XC B P B B ud V V i B R g B H C 0 E H B d B P g F R B H aY 'X0DRX(XT#$TP #F E B H B IE B XQ }xY V X!SC I de R H #F E H B X$ B Y x0H S V TX'& I 0#20$xAX$Xcu V DQ0SB BP V B H F B H H Y B C H E B B g PE H H E C B Y H F a$QE V V g V XX$!aH B F BC FC C1Q W#f $ "
= (uan i X : Uian )i R = (ui : Ui k // X )i X
an
&r x w! i~ s i !i~ w t x t! ~ i { y ts z r ~ i&r z }{ !xv '}{ ! }{ v r z t ' t s ~ s t { x y s z z r z ~z s s t r r ts r r v g r vs z ~ z r ~ far z }{ xv T{ ! 6y xwTt X uTt x r x v au}t X }t x r x 6f{ z t xv !Gu! { t ~ xv x aa# au}t 'T~ ~ av 2x ~ r x h z z r r vs x qs r vs x qs y I Hh%V A3 R06 teheI c Y9WRRPHGD CHF g c dc cF ` 6 F ` 6 I i 8 Q ` V ` Q X VF S Q I BF B Q c a
eGu1
// D(Tn-1 |Tn ) Acst D(Tn |Tn-1 ) Acst // Stv X{Tn },p
n-1
t r v xi { &a'i y t '`z z pv }{ u xv U{ av ~z z z r
(D(Tn-1 |Tn ) D(Tn |Tn-1 )) Acst Acst
v s { z x z i rs y z z z{ t ~ s { Tt z p Xa#y ! u}t ~ s `z r x s { t z t S r T}{ i `#T~ | v { w T~ uwT#r { z y z ~ v x z ss t
g g
n-1
aXcz Xz us { t ( av Az t '}~ a!68s ~ xv r z ! s y ~ x { t x r y ~z{ v r s 'z x r z &r x t ~ av t z t r r y s { x ! `{ Tt t z t v s t v t z r v #x}t { !r z #a`! 0 xeg { r v r r r r g rs r v aTt { av Wz ( z y i a!X i t x !av a v Gs ~ 8s ~ w i x ~ z i #y r x r z t ~ t '}{ s { z t xxiz ! x t iz w! r x s { t z s { v Tt ' ~ av `x ~ { z z z y ~ x { x ~ i
{Stv Acst } X,b (i, j) n a n
x r tz ~ t Xz xv z { ~ { x iti { t r t z #y s s { TXus x x #y({!y r x t ~ &r z }{ !xv 'z}{Uz {Xz}{ i y s ~ xv rav y ! z0 r Q x i y s z (av z y x z B V rau}t 'T~cv vs z x iti { r r v z}#y r za r z { z #y s s { TXus x x ~ av { i y s ~ aG E T{ w xv { xv t z z r r vs xwTt { t ~ av x ~ av (#y &r z ( }A ~ &r x &u !x r s z t x~ t s r r r v { xv s T~ x t x !xv Az x6g t z ~ z tz r ~ t rs v y r v z us z 2&wx ax xf z a x x z { x z ! ~ r z r y &av ~ tz t s x t r z a&r z !
ax
n (p, . . . , p)
n-1
x ~ r !y t z x ~
g
i x {x`z i #y r a r z t ~ t '}{ s i{ z t afz ! x t `z w! r x s { t z s { v Tt ' ~ av ~ x { z z z y ~ x { x ~ i t x r v r v { xv z &(us }~ ag ~ t z { z au x { X r }T{ i x x ~ z z{ { { ~ z #y r t z z x { qs t z Xz uT&r #y s xv y # r
D(Ti |Tj ) D(Tj |Ti ) n n
U v StX{U,U -1 },c
-bj
@ 9 7 54 3 ) " % % " A68'6!210$('&$#!
(Qt )tT
(Y0an , . . . , Yran ) r>0
p p OX {T1 , . . . , Tn }/(T1 . . . Tn-1 - v) Stv n },p D(Tn-1 |Tn ) X{T
n
D(Tn-1 |Tn ) = Spm
OX {T1 , . . . , Tn } OX {T1 , . . . , Tn-1 , Sn } {Sn }/(Sn Tn-1 - Tn ) = Spm p p p p T1 . . . Tn-1 - v T1 . . . Tn-1 - v OX {T1 , . . . , Tn }{Sn-1 } OX {T1 , . . . , Tn-2 , Sn-1 , Tn } D(Tn |Tn-1 ) = Spm = Spm p p p p p p (T1 . . . Tn-1 - v, Sn-1 Tn - Tn-1 ) T1 . . . Tn-2 Sn-1 Tn - v -1 OX {T1 , . . . , Tn-1 , Sn , Sn } Stv D(Tn-1 |Tn ) D(Tn |Tn-1 ) = Spm -1 p p X{Sn ,Sn },pn-1 T1 . . . Tn-1 - v
b
Stv j Acst X,b
-1 -1 X{U, U -1}{U1 , U1 , . . . , Ur , Ur , R} ^ k k Bn-1-r Acst ; p N - {0}, 0 r n - 1 et (mi )i Zr . p - U m1 . . . U mr U -m v) (R r 1
i
k
Stv i Acst X,b
j
bj i
Stv n },p X{T
bi j
D(Tn |Tn-1 )
Spec(OQt ,p ) b
n-1
Vi
bj i
p p = X{T1, . . . , Tn }/(T1 . . . Tn-1 - v)
// Yi+1
c = (b1 , . . . , bi-1 , bi + bj , bi+1 , . . . , bj-1 , bj+1 , . . . , bn )
// Xr . R an Xr 66 Yr
U v m N - {0} StX{U,U -1 },c Acst
Z
an (Y0an , . . . , Yr-1 ) Z = Xr - Xr-1 Zt = Qt Z an
= X0 Xn = X
-m
0r n r=0 k Stv n },p X{S Tn-1 X Stv X,p
n
) 1) 320(
//
Q q bfQ S fQ e y S r aV X Q a Q y Q S fQ e q 1B#1sy vc a Q r Va `S ` e g H H H H H r a dc X pdY@a y ` V c 8g ` Wc Wx hQ i y X Bi qQ )g y H CbH hQ H qdc X ` WYCqQ y c c ` Q e r H e H X rS exa y V cV r Tvc sYg WqQ H i r hQ Ti y CWSg Qi q V e ` H y e XTi b x H` x r H vc hv` x H r x c H Wx gYH g sBWvbWH Dy Cr H i H t YH c g Q He q H 1BpYksyg r V a ` S 1 e x i Q ax g b 'dc S x w bS H e q )T#1S i c H q r r H YV Wx ` ga g g H c r q a q c t V e ` qH H H i i g H Wx Ti TWg TWx g Hi ii w w i $ H H H H i H q x H
q c ` X X ` e a aS a a c r XV Q a c ` } } ~} Fr3| g vpX 1sy ldc X C` Cc dYe Q Csq ` H H H Q a fQ g ` Q c ` e qQ V `S rS a vWmc Ba 1YQ aq ` H i e ry p$ B1 lQ sp` H y b1W#'y H V H r S V e Ha ` a Q y ai ` #vc g YH Bi H Bi x G H H i H H Wx x i H x x @G i H Hi TWx X Qq WU B@c @c pg Q sYS r X r Qq ` aq Xqy r ` a YCq X X rS Q c ` X q psq aS c r bv H g Ti x H x H x x H 1 H iH i x W1H c q vsq a e c Q BQ e WU YV aq X Qq r X qQ V `S rS a Q c a c vCr dr g c Ba YYQ x x i g bHV G e q Q V ` S a H la q ` Wx $ i q c t aq Xq ` e c rq a ` H sBpTa H kCbTi e y WDIc H B1Ta i YQ H X H X r S ` gy rS Q V a r g Q WQ sy g 1TpWc H H Wx H x H H x H i H H Wx H IG a c dr Q a ` c We Cr e Qa Tdc X WCy Q c r i YH H H lH H H g I87FFY8fW H G # B b $#F d ) pRRbr # # v # u'$1Y)@lb b b )p)Y pW W # d T # vY$CR 'YY$ 'bR v b TF#ufvY dp p qC @F#$ r # RbF#b83v $ p b YR bCfC$p bY$ Cf f W#R#DvYb bf W'Y u'$1Y)h u Y8p8 W'Y #7 38 ospp a c hdp` q Q y Bl#'y 1V fdc X CYBr h Xq Q a ` a Q y a y ` V Q c ra x H g x qH i H H X q a c a ` U ` a `a Q `y ` a r g xg g bppyyx kX y y r C` q Q q v'S l#'y V S e H i H xHi q i S gr H c r a ` U q v'S S lvsqBQ eQ y e Q c r r adpT QQ dH z yg H psi r $bS y x @WWCdp'x a x qH H CWYTWg r YH TQ i 1Bii c `Q a x g x qH ` y y r V Bc i c r g e a c ` g w H i v H e CBi X g 1WBa H y H sTa #Ty Q1H V Wx gg Q qQ e x XH WU kBi p$c H Q Wx rV U Q q y q a `H a a Qq r w XH c a q ` i a H cq a c r sBdYS vC v vw )v hi x )g a S c 2 2 r E 6 4 ! ( H H Ti H H x x qH i H H H g Ti x H H x i x WYH H G FI)87#d g g c rq aq c t s# z z sBWvbV w F u # # vl ) vbY b bb b Y$ T bR'fC ) # Rf5Y $bY' C$p Y$ #$ 5# I RbF#b ) k Y s bl lC Rlsvk #3b W'b Y okW 85Y $Yb' s bl Y l R'l8v $p) 1d ) pF ss oWvppp Vd#r c aQ a S y y Bc aq Q QQ a S RP aQ e c rq a r c ` uCTpWky r i g g c rq a c Cygvz vz sBqWvtYV w )) ut Ti WxYH H H wH H H i x i fWx BWx Ti Hi x H i Wx ) o bR'@ RbF#b@8f$v@ 5m W'b bC$#) 1d ) vY$#F p38 orWWpp Q c qTadp` q Q pCr rbS Q q `yy e fQ yldc X #ksldc BCc a a `q e V X Xq a q a e r x Ti H i WxYH H H H H a `a Q g #'y V c#` q e WsBBx `i y eq Q Qq X e Hg i yx X uH YW'TQ Q e rH ` X V U SQP r x a qH r H i 87FFY8fW } } ~} Fr3| y jr vg c p` a WsTkWQ oQ y eq Q Qq X e ` ry ` a H H H adc X r YS c H Q i H a aq r i g Hi H c H TWr x Ti x H a e q Q ` pq Fka r 0c x i Wx g H H i iH YCWx x
U D X Z = Z (V1 , . . . , Vn ) U D Z
a dc X r bS g H ai H j Q spmc g BWx ThIG r y ry ` ` aq r H H i iH YCe Wx y c r i y a e r X i vx g H Wx H H H
V ^ U X V
x
CY$'' vY '' C b ) o u Y sb$' # # ' f7 v Y#' b 'T Fv#CY# q bRY$ ) b$p) vY$#F 338u7l rQ } } ~} Fr3| g Q BTQ e P { g x qbS Cz yg X X y e y bbS sg rS V e q v e WsBB@dc X y eq Q Qq X e ` a r YS c Q a yyq X ` ` ssk$usy x dX a c rH w H i ` H iH ey S uH r ly 3g x vBBdc X H H YS ` a H Q x n H y e q Q Q q X eH ` a r c Q H s$x a i S x H r a y y q TX W a T q a Q q y cq sX H a r sg Wx YTi i H H H Wx g i x H H TWx Ti x uH Hi H H H H Wx i uH Ti YH H t H w i i i 1H Wx c #` Q a r Q c rq rvCBQ ksq 7e sp$qy X X j yyq X ` ` pbpQ spoQ y g q `y S ry ` a Q sq 1Tbsq ` eq Q V a V e n H XqWU i mc dc BCmr H vCTWx Tsq TC$lQ kTYsvc y ` )g vx i i Hq a i i H H i i H Q r X a q a e r c Q c YrC #` Q r a S ` 1H e V a q S q H j Q c ra h vTpr Fg i H g Q Ti sy a x H w i 'dCr H fWvt x DWdc Q a cq e q c e Xi r c aQ sq sq C` bRC` Q y aS X r S Pq Q a V c c r e a c ` WWCdTa H Wx H g uH H Bi x H bH w g x H H Wx
Op // Qt // X an .
r
H
c a vhH x H
H
H
H
x
q a ` T#Ta Ydpvvc X X rS a c ` e r Q fa a ry ` c r WvusBpTdc 1V q c t c rq a ` a Q e a c `a fdTbV x ` XV USQ H YW'TRP Ti iH 1Wx i i xH Y)g w H H WhH i g H IG
Xr Yr
FE
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
k
Op Q t
p
U 'S
e
e r c CW#`
` cq Qq pBsCr
c
P(Zt ) (Op1 Z) (Ops Z) D t = O p1 O ps an // Y an Xr Dt r (Y0 , . . . , Yr ) ^ (Pj X an Qt )j Dt
" ! $#
vp
vp
^ an Qt Xr Yran 88
// Y an r
p1 , . . . , ps P(Zt )
Zt =
H
(C an )
X
Et/X
A
h
A
Rig h
// Et/X an
X
V
k
U
h(C an ) h(C an )
Rig : ShvNis (Et/X) h Rig
P
X
X D
X -Z
B1
X
6 B"A@"
A
X
k (Et/X, Nis) (Et/X, Nis)
k
A
(SmRig/A, Nis)
k
X D = D D X - Z X = D V (V1 , . . . , Vn , D)
Rig
k
Z X DX
Et/X
// ShvNis (Et/X an ) h(C)
(Et/X an , Nis)
(C)
// Shv Nis (Et/X) Et/X an h Rig : (Et/X an, Nis)
// (Sm/A, Nis)
ShvNis (Et/X ) Sch/k Sm/A A
(C)
Z
k
8 9#
6 5754"1)(' 3 2 0 &
SpectS 1 (op Ens) (M, E)
$ %#
X
A // (Et/X, Nis)
(Sm/A, Nis) P (SmRig/A, Nis) P Rig : Sm/A
Qt
an
Spect1 (op Ens) S P X k (Pj )jJ M
"!
@
Shv Nis (Et/X)
// ShvNis (Et/X an )
// (Et/X, Nis)
(Dt )tT
// SmRig/A
(C an ) Rig :
t @H
// homHo (PreShv(SmRig/k,M)) (B1 E, K) Nis X
tq a e j Q c rq a ` Sq Tdc q 1CT#1ssy ` a S CTvCof` ` Wg TS U WlQ a c rq a S c r U e qa q e a i H g Wg H H w i H C H H
homHoNis (PreShv(SmRig/k,M)) (X E, K)
i 1H
g g vWx
c rq a ` Sq CT#1ssy ` y j H ` Ts$bV a Ba S Qq ` r r a c r dX Iy g e a Q n Q ` 1S y e )p` 7e p` c g 'S r e j aq a U q kX ` y Q c g Vg e q Q c W rvr WBvr x v w x H i Wg x Wx YTi i g w iH H x a a c e t a c vt TibS a H i e Hr q a U cq a g b X a H X i dp` H vpYir S g g q H dp` vYdc BvgX BS i sQ H v1H ThH 1H 'q i r bw Q H y fWi WCdpTHa H H H Wx Q a c Cr Q Wsp$8Q vc p` q 1Q 3dc y eq ` S y V y a Q 1V Q c i We Q t bS q i Q a e ` a e rS V c c r e a c ` H xH Y)g w H H H x H H i i TYH Wg H Ti x Wg i H Wx H 87FFY8fW
E B1 (o, f 1/p ) E X // (B1 (o, f 1/p ) - o) E X f =1 M
Q dr
x v wsb$ ') p F Y 1 C#Yf5v @b b8 v TY $ pF C # blv# R qyYR b C$p Y$ F b$pF a 8 ospp r z g g g Q c rq a ` a r c vsBpTWfQ y H r W x g c w w w i y X y r Q TQ RbCTBCr r j q r Q e Pq S c rq aq Q `y Q c ` e y eq ` S y fpW@Q v$Q vc ` q P V e hQ y BhQ y X Xq Q Q sy H g bH i x H H g Wg i H H H x H H g H g YH Wx Wx )Wx i Ti H H H H g H B k q ')Y o R#) b u D 5Y'@ # I c CfYY ' C$W#' q CY # b q 5Y'Tq# RvY' '@ $v ) ) BS q pr X qa u a W D C$W#' q c bY B $ Y$ ) )tl #v sb H sH l $#F# s C1 $ dT k) b b hs $p)p b Rv bY c q c drYYdCm 8WpY p8ofo 8g a a h a j q BQ aV X y ` e r c r sq YkBQ WCWX g Q 5y H Q Tsq y e `aQ a WTloQ q r Csq 7lCc q r 5y H fQ y x X i e q CYBHV pYk` e a c j e a c j e rW r a `S a dc X CYfQ y &g vpTQ dr y ` V a y p y e `a VQ Q VaV hTbD` H Wx x g WqH H H fTi x i H H Wx H H H i G H H H qH H H g g vWx B v T 1$ CR @R5Y' p)l p B yY$ Rb'#fb' b C3 WY$Y1 b f b vY 'bq$ dBR b B Y 8W R#)# 8'b $ b rb b' bq$ CT 3 1R#F b $ p Wr$ CT W R p) T Y 8 F Y c #5Y'Yi b qY Y p p8ofo 8ug a h e Cc q q a q v y eq ` S y YS )g v$ vc p` q 1V S Qq ` e Wsp$ c p` q V c Q y eq ` S y a } } ~} Fr3| g ` Bp$YV x H i g WhH H H H x H x H H Wx Bi t H 1Bi H H H H H H x r vBBQ 1RP y eq Q S S ` c oQ a qQ a BCc X y a @Q BbS U W@dBhvsBpYy @Q S Q qQ q a q e a c r Q Q c rq a ` Sq ` a i g Wx H H H x H H H YH H x H H g vg H w H a c `a Q dp'Cc V c a C` X e qy a vp` dc BQ Q e c a X X ` r g BbS U WQ qa q e a i H vYTi x H i gH H Wx H i H x i Wx H H H e c ` vpX e a dc q ` y S sBbvofC` ` fg TS U We Q c rq a S c r U e qa q a c rq a ` Sq sBpYy ` 5y j i H W H H g Wg H Wx qx i a e YH Cc q H pt 1P H YTi oQ H w q I i yvz CsBBCr r H i y ` S ry H a e a dc X ` vc jy e c g g g c rq aq Q H Ti t H 1BIw Hi H H Wx Ti H w w g i w F mH H x i Hi gH TW1Bi 87FFY8fW 'b3vY bb 1s b #v p ) $ $ e 4 'Y 5 kY$ rTFbs h) b ssvY YpYs7 B ) b ' sf C r T vY DpYTYbBc YYdC# pFpYY$''' c $ kbW'1 b C@ b @Y' b Y p)upC) IvY ) p #7 Rp p8opr3p a rV qQ TQ ` r q c WW'TQ q r U US e c rq a ` Qqy ` S r CT#Tssp1sy e g g xg yvz X 1WU j ! ` W X YWTa ra C` q sBdr r rV U e a c Q c rq aq Q ` sy e w y Y1H e Cir T#Tssp1w y F S e x vc i sQ q )#g $" Y i r S V c T q a ` Q q y S s a a ` r S a a i XHsy sCq Tbsq ` HP QQ Bq e q V a V H Bi x H H x g vg i H H x H H H H H bH G H lH H i i S ` Q 'S U Q e Q Q `y S ` a c BCsYsy dp` q Q y X e Q e r a S oQ a S Ysc a a Qqy ` S ry c H C H v H H x H H Bvx Bi vH c H H p$uc k Q Hi x e H x e aq ` a g H Ti BWg Hi H x ysq svc Q CYa CBvc Q Q ` Q c rq Q Q rS c rq Q q TpWU ra r X r y e `aQ a Wp'olQ V c c r e a c ` @vWldp'a g a rqH H t c H BQ q ` Q wq BkCWU 7e r a r Xr g j x ` H Bp$lQ gy x lQ g BWH BQ dpH Tq vsq H ` vx p$YV S Qq ` H Q i X Q WU g r g X r Q q qi e ah ` Q e c S e Q WU fQ H fdBQ WWspg $Q vc p` q 1V X Qq r X y a c r Q i yx e q ` S y Wx H H vx H Q YP i g H TS Vr x Q y x kH X e WDH S oHQ c 7Teg i Cr 0r y H vCHr y H ` x r i gb a Q H DCCe y ga c r y ` TSi r y qa U e a j Q QH q e c j Q c H i Wx H H C H Wg H i H TWx Bi H x H g Hi x H Wx H g vg T
2 A R F # ! 0 1SQ&$"16QPIH9B&# ! F ( G ! 2 % 0 F 2 DEDB9@9&81"5 C 7 A 7 ! 7 # 7 0 % % 5 "643 # 2 0 # ( ' % # ! 1)"&$"
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
M
E E
k
Nis (A[n], -) 0
|h(x)|1/r |f (x)|1/p
0 (A, K(X))
(WB1 , Cof , FibB1 )
B1 (o, f 1/p ) E X
B1 (o, h1/r ) E X
^ ((A1 )an k X) E k
B1 (o, f 1/p ) E X
K
RigSHeff (k) M HoNis (PreShv(SmRig/k, M))
AE
0 (X A, K) // 0 (A, K(B1 ))
nZ AE n=0
A Ob(M)
B
M
nZ
// X E
Q
// Cr1 (o, g 1/q , f 1/p ) E X
// ((Gmk )an k X) E ^
B1
// X E
X
K
X
K
// 0 (B1 A, K) X
B1
K
k
x
// hom(B1 , K) k B
B1
// hom(B1 , K) k
Nis HoB1 (PreShv(SmRig/k, M)) RigSHeff (k) RigDAeff (k)
X
E
M
f, g, h (X, O ) E
Nis
B
X
Nis
M
B B PreShv(SmRig/k, M)
X
B1 Acst X
p, q, r N - {0} SmRig/k
k
B1 Meff (X)
M RigDAeff (k) Q
PreShv(SmRig/k, M)
K
K(X)
A[n]
PreShv(SmRig/k, M) RigSHeff (k) M M
B
// X Acst
X X
// K(B1 ) X
B
M (X, )
cst eff
|g(x)|1/q M
K
X
B1
y ` S ry YP
y
K
homHoNis (PreShv(SmRig/k,M)) (X E, K)
// homHo (PreShv(SmRig/k,M)) (X E, hom(B1 , K)) . Nis k X B
cst
Q BbS U r Q YP qa y ` S ry Q a S oQ a H a H dz ygg H ssr YS Bvx ` i Bi ix x ` a Cc X Bpsy qa ` c sq yy e c rq a S c r U e dsBofC` c Ie Cr e r p vz vz X YWU j Q Qq c { g g rV q ` y y r Hy i x gW Wx H H H Bi H fH x H x H g w H Bi ` w )) f H w H mTi i H i a `a Q y } } ~} Fr3| g #'y V jr i e S S V e Q Bpr U y ` Ua j Q c ` e y eq pWrQ WBQ cq a c r Q y ` Sq srdBpQ YBa Q 'S U q X x hH rQ g vc H BTWx fH H Hii i g i g H iH Y H iH Y H vfTY TWg 1H IG Hi Hi x H
Nis an Wk E // U an E V an E k k HoB1 (PreShv(SmRig/k, M)) // (An - o)an E k // .
Q r q Ba qP ` e Q c q 1V WsqBaQsesQ CWpq a e qWU rXc qBvsp`pdc qBeCkc y c ` XQ Q cq a a r g i YH lH x i H H gTi x adrWe H ci a H QTa rQ H y ` q rYSa c vg H i T YHCWQx r H g i g Q Svc i iV Wx ` xYH)g Bp` c c qa Q rsBqdr r ` y e 'S U X q a#` Dsy ` r a cr S cWc r Q q H Wx i G g ivg H H lH i x H CWx i HBi w i Wx H BiTWx iri gH H i i Y$ ')Y
[(An - o)/ /()] E k n n=1 n-1 U = (AZ - o) Z A1 Z // (An - o)an E k
n-1 V = AZ Z (A1 - o) Z
a c rq a oQ CTS Q Csq 5y a c j H YH z W g iW g Q c rQ vCH dr g g g x vW w 87FFY8fW )
B1
} } ~} Fr3| y eq Q Qq X e ` a WBTsChdc PH e t r #` vbS c oQ aH i H x a Q H y eq ` S y g Q Wsp$rQ vc ` H H H
q 1P V x
p ) 'Y 5 W'1 b I v D pF 8 orWWpp a `a yq Q S VQ c r a r c pvc g e TQ y sBa ` 8q vc bS c i x hH x 1H x H vx g i H H hH x g X rbS y a c ` Qqyq @dpTssTa c q c ` WpX Yy e p` a Q 'S X X ` aq U X q vsb` g sq q 5P cq S y ` a q a U cq c rq BbS ssBa HBQiWx `q q HBH Q BbS x sq TCr T#` vH bS e WvH vp` CHr TBCr vYS H c x j a G p1H sP i r y p i P a q a U c H vc Y q a e t r QHi q c t Q c H a i c q a q QH X y ` S ry S H y vc p` q H 1V c H H i H H i H H i g H x x H mH H H x H x aq ` p$ y sBa Cr Qqyq c U 'S X q p` 0y a ` r i H Wx H H x w CH vmH i x i Wx Q Q spldBfQ TQ bqV 1S fQ 'S e ry ` a c r Q Q ePq S ` e U Q Q a r g Q Wsp$rQ vc p` q 1P y y eq ` S y V Q e a S ` H i x H i Ti x H v H H x ` H H H x H H H
(1) (2)
1
H
a S sy dc X T#y aQ ` a qa ` c B1Ta Ybldc X sq BkCWV 1c Q spdc Tuc qQV `S rS a ra r X r U `S ry ` a q a e r )g x c c Ti Q Wr t YfQ y C ` T H Hi i H e Wx Wsg TkWdc Ti X H y eq Q Qq X e ` a i x bS i y sBa r p` W11S y r SH H r Q qH y q c y V cV Q ` r i H i Wx H i g H H H BWx BqH H Hi Hi x w i H Wx i
(-)-1 : Cr1 (o, g 1/q , f 1/p ) X // Cr1 (o, f -1/p , g -1/q ) X Cr1 (o, g 1/q , f 1/p ) X et (-)-1 : B1 (o, f 1/p ) X
Q WU CfQ ksq dc bV CYS fq e Q X Qq r X r Qq y ` a e V S Q ` e a e V a Q ` 1S a g dc g Ti c oQ H r e e H p1sWa g x 'S x y H w $ Q ` y b1Wu#H 'y H c rq Q S Qq e ` sTQ By ` y eq ` S y v$ c p` q P H V a aq y ` S iT q U ` aq ` e H ry r S V Ie G a ` a x i H g H H x H x H uH i H H iH 1H v H Wx x i H x
h=f r=p Cr1 (o, g 1/q , f 1/p ) E X // B1 (o, f 1/p ) E. X B1
y ` S ry a S oQ a p1sP H Hi Bvx Bi e x ` y dc X Bpsy a qa ` c Ba YYdc X q BpCW@V YS qWc qQ V `S rS a ra r X r U ` y aq r a ` H Hi H a vx g H y e qt Q Q q X e ` a WsTkH Wf Cc XB$ H bS i g r i ri e j ue B` IHe Cr Wx Q c g 'S Q i X r kfCvc dX g BvYuc H q Q q aH ` y a c ` a c q ` i eq Q c r S U H H Tdvx BhG Hi Hi H H H Wx i H WWx g g H vH i H Hi TY qaV Ua r X r BYWBkCWU ` CT` lCBsTa c c rq a S jy a c ` Qqyq q c ` WX 1Dy e WCp` Cr 'S X X ` c c r Qq c U g H i g x B1 H Hi H H H i w H v X q e ` sy r Wg SS r 1YV Q WIq 'S r c U ` sy e r e a aq U ` 'S C Q c q c oQ a i H YH x i Wx H g )Wx i Wg x x H v H H i t qH x H t iH hH 1 x H q Q cq ` a c a c r lBvspkd` CCX ra r X r q TpvU q YT#1kvpWkQ pdYdc vc q Q 'S r V a ` S `y Q c ` e y ` V a c e U e Q S a e V e q c c H Wx i H Wx g H i H H H 1H H v01H x H Wx H x H y c WU X y ` q vsq BsWU kCvc ka X Qq r a e c X Qq r X r e jy e V aq BBa g x H g Ti H i g x H g Bi H H H Pc q 5y dc q BpCWU c Wc e j a ra r X r aq r U 'S ` g Q e H H Ti hH g H H x i Wx H G i Wx H Q TbWTpWU ` sBb` c 7e Cr sc Fg p` W1fYS fvWc qaV Ua r X r c rq a S j Q Qq e y V c V Q ` ` Q c r c rq aq Q ` e ry y Q sp` y b1W#'y V rSV e a `a Q g v g sBdr r g H i g H x H g i x H H 1fT w w g vg i H i H x x kIG i H
X E B1 (o, f 1/p ) X Nis B1 (o, f 1/p ) E X // B1 (o, f 1/p ) E X B1 (o, g 1/q ) X // B1 (o, f 1/p ) X Cr1 (o, g 1/q , f 1/p ) X B1 k B1 (o, f 1/p ) X ^ m : B1 k (A1 )an k k h=g q=r mE B1 (o, f 1/p ) X // oX // (A1 )an k
7E
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
^ X m : B1 k B1 (o, f 1/p ) k
B1
B1 (o, g 1/q ) E X
k
// B1 (o, g 1/q ) E X
// B1 (o, f -1/p ) . X
Cr1 (o, g 1/q , h1/r ) X Nis
a
H
Hi TWx
" ! $#
Cr1 (o, h1/r , f 1/p ) X
y ` S ry YsP
B B1 (o, f 1/p ) - o = Colimn Cr1 (o, ( n f )1/p , f 1/p ) X X
/()] E [Wk /
n2 An - o Z n-1 1 W = (AZ - o) Z (AZ - o)
Nis
B1
(Rig , Rig ) : PreShv(Sm/k, M) Cr1 (o, h1/r , f 1/p ) E X E B1 (o, h1/r ) E X
(2) (1)
// [Uk / /()] E /()] E [Vk /
ColimnN Cr1 (o, ||n , ||-n ) X
SmRig/k
// Cr1 (o, g 1/q , f 1/p ) E X
// Cr1 (o, g 1/q , h1/r ) E X
// PreShv(SmRig/k, M) . // [(An - o)/ /()] E k M E B1 // // B1 (o, f 1/p ) X oX E : B1 // ((Gmk )an k X) E ^
e VS b1V H V Wg i Q x IG i H
c r a CQ sy y t H H H g c Dpg )y Wg y H H x qBaS q a r BX Hq s r V a ` e )tH CYBpYS H i H
C yg g x w ` c g y mv8y c c r c a ` #vc Bdc c qa a ` XV USQ 1W'TP c a q dc r g c i Bq x BC` YS BWx kQ YH H ka x a H TsoQ q a Q V a S ` H i e a i S Qq e a e w c rq a y CT#` p` y x dr 0Cr ` QH Q c H vx i i H H H Wx H g H Ti b H H e U e r H S c r e agQ g lx a &Cc vx vHWfvWx q w y c rq aq c t V usBWvbWe ` oQ a e Q qy ` HX V U S Q BHQ DYW'TRP x c WvH lx dp'a V c c r e a c ` y H c oQ a b u a y V a q a q y ` e rH c r Q i pgvCWX H S '$ r 5pBWc H aw &CH uc Q ` U e r y X `x e a H x H H Hi TWx Ti x i Wx g x H x w )H g bi H H g H q
! 2 ' 0 0 % R 9"'S47 & A %F ! D7 4BQQ49"' # 7 0 % F 3 ! 2 0 A 292 ' 0 12 ( ! 2 ( )9"'0
QvCT#TWQ y a y CbS c rq a ` a r c X r X Qq WU H Bguc rH X r Q q c rq a sBS H Q Ba Tkv` x a g Si Q c He sq H TH bsq ` y e q VaV ` e rH g y ` e ri c r X i H pvCWQ y X &CHmc BsCW'uc U e r aq Qq r U S H H x
& 0 '%
Q V aq a e jy a aq ` } } ~} Fr3| g Q U e r Ta &Q y @BBdc sq loQ p$ H H Wx i H H Ww c TS U dc X y Tdc BQ oQ q a Q a y qa Q a aS c r bv y aq ` p$ ka yg e g x H $ H Wx i H H H H H H i x vYH H Wx H x H w X 1WBa Q sp` y bpTy 1V rV U e ry rS V e a `a Q a c r dX q q W g e a Q y g q `yy r ssr YS e c p g bvCWYV 5y e VS c r c j g i rT V a ` S P r Q ` H w w H Ti i Br h W H Q H U Te r 'a &iCpQ y x ` i V vc H Wc Q WQ DBS ` y Wsq a q YCi YT#YRQ H B@y x e x q c t c q x i X G r Q XH V c ` Q c r ax H x H H i fH i H H H g H H H x i H i x 87FFY8fW Y' W '# b pW b vl ' kb m5bYms c 1 )YCbDb Y B # 'D b$v1 Y'# d u$ dBR h $ y x Wp$5 #FubmT#ff1s u $ Rb u T ' v#Y ! a 8u7 p Qq S b1V Q y pTy V kuCr a TsQ sBp` ` TQ a `a Q y ` c S Qq e a c rq a y ` a y c y q a Q V a S ` y ` j e a sToq Bb` 1S pdYV 7Q TQ r i Wg x g x hH i w H Ti H x H Wx hH x Wx H H i i H H Wx i G H c rq aq c t V e ` qq c rq aq e c r } } ~} Fr3| g Wx g CTvvby e sq sBvCbS g ` e ry y hQ sp` y Y1pTy V rSV e a `a Q y ` S ry a bS Q sy a ` r Tdp` vhTa hQ Cc w Q BsWU k@Q y H bS e sy S Qa c e t Q U e r a Xw Q q r X V eq Q c r ` g p1P i a S c r g a r H i Fy pg x Q p1sQ vx c G 1P a Q nH y ` S r y S y i ` H q V H Q y STWbWe Ti d g Q vx gvCpQ i e sp$qy Bi H H q V Bi Q H sq YTi X a H r i H a H i a x V x # yyqH X ` ` H t y i H cq i Wx g x H H H H w i Vx U U a `a Q y ig x a c a y r V a ` aHB i q r t SY q ` i i l y e q ` e g W'S 'S H #H 'y H 1V Cojrw WWYH dp` vHt q H CYT#YS vax YH a i Ba hrH Ti ` TsHQ $bV YH Wx Q Wsp$ i 1H Ti H x qH i H i y H #c sBpX )vH ` x 0vx g x HCTBCr WqH sy H ` Q vc ` q 1uQ uq vsq S y V e a e c y a i ` c r q a `H H q c a ` i y yv g c r q a q Q i g r rgQ ` g i H H x H x H Bvx Hi i i w w g Wg i i
Qrig (X, f ) Acst p k PreShv(SmRigqc /k, M) T X RigSHeff (k) M T = RigSHeff (k) M // RigSHeff (k) HoNis (PreShv(SmRig/k, M)) ~ k ~ ~ k k = k(()) RigSHeff (k) M f (X, O ) p N - {0}
LocB1
VQ dr Ysy YS q V 1a X rS ` X X r S jQ aS c r bv g Q C` Q sYS a c ay t aq Xqy r ` a kCbS X X r v H g H H Ti i xH g i x 1lvYH H H i H x H x pd g w o aS c r bv i x W1H y a c r dCkX sp` W1Q y oQ y Fg Tb` YqdCQ e y V cV a X rS a c r Q U e r Ta &Q y me )p` 7e dCkc r e j a c r X n QaS H vx YTi H iH H i x g H g H H Wx H i H Ti 87FFY8fW W v # pW Ykb'rbRb ' b$ Y$ F ' # k c b 5bYb v 1 )YCbqvY Y ' $p1k Y'# d $ dBR 8 oWvppp a X r S Pq Q eq Q V a V r CBQ c rq ` TQ e p$uc aq ` c rq aq Q ` e a sy Cc X psW1q a `q e V X X } } ~ Fr}3| g aS Q ` CbRB` Q sCq 1Tbsq ` Q y g v g CTBCr r H Q w y #d Wx g g WpCr H bS y H r i qg ` y y r y Y1eqbS vfb sHq q g q V fx dc H X y i ua i BH YWxy P g Q Y rSV q t yq q Q a Q q Q y ` S i r a iH H Cr qw a qH y `S r p1y g q nw g x g x H p`x Q c i r q a vCr H S H q w e Ti hQ H i H i H PH x a hdc H X vhvsBbqQ Ti X q e tc a S H Qsy H r i dp` vdc H X H TSH Vr H oQ a ` a c e t a q aY U a Hi BWx Ti x i Wx g i H C H Wg H H i H Wx H i Hi TY TWu1H Hi g x H G 87FFY8fW #v B #v T c 5bYbkYT Y r $ $ e Y b pbR p' Y b v q5v 1Y$ W v # u TFbs F b l Y'bRb ' m$ Y$ ) ) p F y y x #$v$C p0YR' CkRY$ # TY $ B v v p m Cv rFbY c Y'rbY ' kC$p Rb$ v XRv # F hv#$#)p'@ br 8pvY YpYrhs 1s F B c #5Y'Yi Y rb C 'b $ 8 p8ospp a a c Q a `a Q y y V yy a c r dBQ Q yYsP ` S ry Q a bS afQ fQ Wc X Q e q a g d` q l#Ty 1V k` c vg Q adc p` q 1Pc sq x x hH pbfoQ H y C g p1syP H f S H )qvsp`fdc TWx Cr Ti Wg Wpp` H x Yvc Q W g CTBCr r i x a q `y SHa a w Q ` y Q c q a H qBi e x c y H e ` y a r Sw a c rq aq Q ` sHy n yx ` S r g Hvx i H H HBWx Ti i x H H H Ti H lw w H g Wg i e ry ` c j Cvp` y a c p g a Bra b` XCbRB` sq Tbsq ` P Q Qqy aS r S Pq Q eq V a V S ` c H Wx H w Fs H H H lH H fH g x CH w i i H Q 'S Q e U Q p1rP H x S oksy ssp1FBBQ la y `S y a aQ ` Qqy ` S ry q Q ` c QQ yS ` T`bksy dc X B#y a qa ` H v H i Wx H H t H H H Ti HBvx Bi i x i YH x xH Y)g c rq aq Q ` ` syky S r Qqy sqBa c g Q Wsp$Q vc ` q 1P y eq ` S y V Q e } ~} F}r3| g g vz v g sBBCr r w g Wg i H Wx gqH i x H H H x H Q X sy fQ U'S r ` e Q y r c q adCrkX a c e c 8y n a Q i Wx H i H H v H H H H Wx YTi iH H H s )tH H x 87FFY8fW Y @ b k b 1s Wf Y1Y 1 v p b ub Cu#$bF#Yi F 'b c 8 p8ospp a a c d` q uCTBCr r Q c rq aq Q `y D` c x g Wg i H
B1 Acst X
1
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
B
PreShv(SmRig/k, M)
B1
A
k
k
// X Acst
1
k
X an K
~ k(())
X Acst
homRigSHeff (k) (X Acst , -) : PreShv(SmRigqc /k, M) M
M
(Rig , Rig ) PreShv(SmRig/k, M)
X
B1
X
k
Nis k
X
~ k
K
k
B1
X RigSHeff (k) M k f (X, O) Rig K X Acst X K(X) X B1
PreShv(SmRigqc /k, M)
PreShv(SmRigqc /k, M)
PreShv(SmRigqc /k, M)
// PreShv(SmRigqc /k, M)
ColimI LocB1 I
K
B
B1
1
p
RigSHeff (k) M
B
K(B1 ) X
// LocB1 ColimI
1
~ k
M
B1
X
0 (A,(X,-))
A Ob(M)
AE
A1 Acst X
// Ab
M
// K(X)
AE
PreShv(SmRigqc /k, M)
// Ens .
M
! % $ 0 #
#
"
C g
w o
PreShv(Sm/k, M)
B qc
// X Acst
Nis
)E
H
H
w
q 1P V c x H a aH H pyg qBQ Wx c aw H H X w tH BQ c )s jr g Ti r g x H i i Wx
X
c B1Ta YqV 1c qQV `S `S aQ oq s jq dr y Q Q S c r he i x g g a Q ppyg Q Cr TQ e X H WCr i q q ` y e y Ti dc 1bV bH 'bV 5y 1H ` i vWx T a e SH ` a x j i V x H w i H H i g w WH CH H i g H v)g i v y Q 'a r e fQ y ` e a fdc X r YS ` dc bV a e VS c rq Q S Qq sBBQ Be iH 1Wx ` Yy H X i g a H V 1S H ` x H TWx Ta x H H Hi Hi i W` g bCsy x X VS ` g Bi H Q Cr Qi a g c H e y `Hcq ` Cg r Ti pe 1S y Q vc H V ` 7g i g S S i ip9 g q W S x c H y g YH g yvW$x vY)p` q w 1P H ii e g` V c i YP Wyx HoQ Tpyx g qiQ y g X H X i s a H T W i r y x w r sy H yv`$ H vc H p` q x 1P H c H oQ eq S y V a Q c r Q pyg H H H vCdr pw i g i eWvtsQq FDdc x X HYS x c H w H a q r a g gX x V c V kWx BQ dc Wv ` WYC@Q y i qH c q H ` H a r y H y e q ` S H i ` y T qQ ` a q Q cq ` a c a c r g i Wsp$ H vc TWx q Vi c x TQ oQ pyw g H H IBivspldp` x dX H H H x H x x H w H vx i H
jJ Xj Qan (X12 , f ) E p Qan (U12 , f ) E p jJ (Xj Xi0 ) (Xi )iI X B1 B1 U I
H
H
Q c ` e Q cq a Qq e y c ` vWf1V WsBsWfQ Wsq a e x H i H H a Svc ` q P H y V c Q w g a vCdr Q c rQ x H x H w H oWx H
B1 U1 = X 1 U U 2 = X 2 U
X Qq WU c dc BWc sq Ta P ` r X a Q Qq r Q r q g x H i Wx i 1` H hg H a w i H g Q Ti g B$ g Hi w YS i c g e q Q ` a fq Fdc H X r g g x Wv H i H TWx Ti x Hi H g
X 1 X2 // Qan (X, f ) E
p
` y ` i t y ` c ` c r Ig VvqBadvp0CQ i e V cV q WYC e t ` vy g H Q Wxr ` i pWmc Ti H y a r c H 8g i H
Qf or (X, f ) p Qf or (X, f ) p
c Q a Q c ` Q c aQq e y c ` vWe1V WqTopQ CWpq a e WU X c 7e dr sWe jr Bq X Qq r j Q Qq c Q yS c r bvCbS r y g W$ c p` q P a Q ` v y eq ` S y V H x H x H i H H g Ti x H g x x g Hi WlTWx i Wx g x H H H H x C g w o a e dc q V BWU X y X Qq r a c r X Q c ryy ` CCrslQ r Wg Q c ra vTpr h h H H g Ti H vx 1Bi H iH x 87FFY8fW b$ TFbs ) 'b Y$ ')Y 1 F b u v XbRvY Cpb')Y 1 fv #YR v hv#$ Y' mYbW# 7Y Tf b Y 0qRY$ b W'Y Y pF 7Rp u7l a `QQ V X y ` S vc kCbS X X r g pTTsq We e kX c H WYQ H H H Cr i hH H c Q x Q ga c Fg e c rq msBQvc sqe y Q Svc X ` S 1V ` c c r q WBQp` c F vTS V @vCe c c r S a g y eq a e q S c r H H iWx H HBTiWx i ghH i i i H x Ti i bH H H H Wx YH WWx 1g i g g xH Y$ # TFbs F 7 W l h 'Y ) fs mp)Fb $vYu Y bYT W'Y b$pF Rp u7 a c rq a QP r a usBbS Q Q a S Ip` Yvq #'y V 7b e y qS c a `a Q y q Sq r H x H H H g i Wg x hH i a a dc X y sk` WCr Xq Q a r c c TQ r g H w w H Cr q a q Q a w CBQ H q H c g 7e Cr 1H r w C g g Wx TBi Cr G r r c rq X X j Q i Qq e c { c g iH i YH H x H g 1Ww 'TP w w kWb0c Wg ` XV USQ y aq c t V eg i aprvccra H 1Wx H e sy X H sBYy CbS y r c rq a V X r a H ysqY1V Q ` S e t `Q vTuc H g ` g X V TU S Q YWi 'TP c H WvYc H H H i aq c t V e $s) )$1 " x
Qsch (X, f ) = (X k k )[V ]/V p - f. ~ p Qrig (X, f ) = Qf or (X, f ) p = Qsch (X, f )[1/] Qan (X, f ) = (Qgeo (X, f ))an p p p rig Qp (X, f ) Qan (X, f ) p Qsch (X) Qf or (X) Qgeo (X) Qrig (X) Qan (X) k
` y
H H
E
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
" ! $#
RigSHeff (k) M
Qrig (X - Y, f ) E p
Qrig (X, f ) E p
B1
E
p=1
k
SmRig/k
X = X 1 X2 X U
Qrig (U, f ) E p
Qan (U, f ) E p
B1
[Qrig (X, f ) E]/[Qrig (U, f ) E] p p
Qsch (X, f ) p Qgeo (X, f ) p
X
U =X -Y
// Qan (X, f ) E
Y X
p
// Qan (X - Y, f ) E p
X
// Qrig (X, f ) E p
M
~ k
k
X = X 1 X2
y eq ` Wsp$
X12
B1
f (X, O ) p Qrig (X, f ) E p
// [Qrig (X, f ) E]/[Qrig (U, f ) E]
// [Qan (X, f ) E]/[Qan (U, f ) E] p p
B1
// [Qan (X, f ) E]/[Qan (U, f ) E] p p
p
// Qan (X, f ) E
p
p
Y
// .
//
U12 = U1 U2 RigSHeff (k) M
g b
Qrig (X12 , f ) E p
Qrig (U12 , f ) E p
rig rig // Qp (X1 , f ) E Qp (X2 , f ) E Qrig (U1 , f ) E Qrig (U2 , f ) E p p
an an // Qp (X1 , f ) E Qp (X2 , f ) E Qan (U1 , f ) E Qan (U2 , f ) E p p
=J I X U card(I) > 2 B1 jJ (Uj Ui0 ) jJ Xj B1
rig // Qp (X, f ) E Qrig (U, f ) E p
an // Qp (X, f ) E Qan (U, f ) E p
XJ = jJ Xj XJ UJ
a x
a c Wg c pYsQ Wsp$ vc p` y ` S ry a y eq ` S y xH 1g H H qH H H g ` y q y ` 7qQ k py Ig j e ry ` a Q n iH YWx r vg $ i vc p` q x 1P g ve q ` S y y V c oQ a x H c r q i T g CTWx a ` H c BsCW'x r aq a qH Q q r H U S c a Ba i r g H x a r ` w a S ` y i Wx g H H a r c g c pi WCr H
X
Q
evcCqT#` evpYS fQ q Q r a t r e a i SC`sy X H jr Q cH g HB$ i H ag q H Cr Bi fv x e c e r S H qH H )g `a TYV x 1H
UJ = U X J =J I I = J {i0 }
n-d Ak ~ Y
// An-d ~ Ad ~ ~ k k k jJ Uj Xi 0 // X B1 // // X X
Q lQ x H TQ IGy r x T i c i Wg
Q lsy a q a U BS Vr X Qq WU c c r X tq a S BC` Yfa X rS c r E ! H Wx H H WqH i g g Ti x H 1H i g xH Y)g w )#$" Y ` ` c #pvCr q e Q V a ` cq Q Bq sy WsfQ sBa B1V sy ` Wb WsBS 1V qdr e c rq y r Q ` Q VaV hTbk` Q Tq a ` S c r e a psy lQ Q a c ` dpd#r c i g y eq a e q V Q i XH W i x H x i i H x Bi i g g vWx H vx x g W g H Wx i H X Qq BWU X r a c `aQ c rSP c r c a ` Cr Q Q c a c ` a cq ` kCr dpvc dX e Ba ` q g dTvYCWsoQ H g Ti H IG H H Wx YH WWx H i g g w H H i i g Q ` S Q c ` a c r XV CCYe c q vC I7y g BsvU Bsq S c r e ` j c n X Qq r X rQ g CYS fvpWe iH ` c Q a a r c rq sBQ sq y g g CTBCr r 1Bi sy ` g t i 1S y p` a H c pBi Q Wc X X j { c r qH a q Q eH Q ` aq X g` r i H 1 iH i i y x sBQ vc X qH S S 1V c g eWx Q ` 1S y 1H p` a w sq x v g g WX y i y ` pb Q H TvCe 1H Q vi TS H V q i S x ` aq ew a Qw y Xg g i g Q Qqy S cHr i y eq a t e wCH g H TBvx lH Hii i H a Q c vCr q S QH spH c 1H V 1Q i dp'ovCbksBhH ` g H X i W'dc H X sCr S H q kc Ti r y ` e iY j a aH c ` a Q c x r S c w r q a w S c r w f `H r c Q a Qq Q q xe i g i #pvCr q H H s1Tq y W1HWBQ DsBa CV H Q Qqy q a U Ba BS Vr S ` i sBpYBb` H YS c H c r q at ` S i t q a S X r r a a c r G x ` ` c e Q Vi a t i x c q Q e x c r q y r Q ` H `y ` H a H c Wg g dpT vCbfoQ i p a c `CQ c r S a Ha r Q ` y Q cg ` CYS rvpve W g 3YCWx i YvBa 1y g vc w WF" BpfoQ sq i x W WH g e j ! Xi kX y x dpTsBa c x p i y a c `H Q q y q g H x i H X r V U x H q a E r 7e $ V Q qY ` a y g i w )tH g H s H H H w w kH Ti w H i H H w w kH w H H x q `yy r pCr bS a vc e S S 1V e Q Br U y ldpvc dspX y bl#Ty 1V U a j e a c ` a cq ` rS V e a `a Q } } ~} Fr3| g g yW g w H Ti i x fH BTWx hH H H Hii i g H H H x x hIG i H t a dc H
X -X Di T f Qrig (Y, f|Y ) [Bd E/(Bd - o) E] p k k f X X f // Qan (Y, f|Y ) [Ad E/(Ad - o) E] . p k k f: X f // P1 ~ k f: X f XX
p
H
r y eq ` S y Wsp$ vc p` q 1P V i Wx g H H H x
a S j q 1V YQ p g U 'S ` Q ` S Q c sy 1S lv` Wg c Q g i a a w c r H e S c rw e a CoCX v vCDH y 7g Q n H x H w H uWx 1Bi iH H x
Q? (X, f ) E p Q? (U, f ) E p Q? (X , f a ) E p Q? (U , f a ) E p
Q vc ` q 1qQ dc BCCr V Q 1S qpWe z yg CTBCr r S y V e a qa e r c Q ` y Q c ` z g x c rq aq Q `y a c ` Qqyq @CBsTa c H H x H H w g Ti H H w g Wg i x
Q? ((Ad - o) k Y, 1 f|Y ) ~ p ~ k // Q? ((Ad ) ~ Y, 1 f|Y ) p ~ k k
H
H
Q H H g Ti
H
e US 'q Bq h 1V YS cQ Q ` e a ldc H Q BYH U BsBi Q qYH 1e a X Q W r X r Q i e Wx e V q q H g Ti H x cq S sb` H
H
x YH
e y a qa e r c { Q dc BCCr p g S dc X p1sTa ` a y ` Sq a c `Q CBdr Cbc X rS H H w w H C H H iH 1 g
// X
q a S V c g H Ti i xH 1g
a H H
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
n = dim(X)
YH
{ CC g w o cq h Q1V YS Q ` e a qa e r c dc Tur q g i eT a x q e vi c dc Yc H Wh` H c g Q TYQ BsWU kQ CC Q pYBa y `w a V X Qq r X e a c r Q y ` Sq Q 'S Q r U y H H H H H g Bi H H iH YC H v H x
~ k
Y An-d Y = Y
F
X = X An-d Y - F
~ k
Q ` XV USQ CYW'TP
d = codimX (Y )
e US 'sq
U
U
~ k
// X
// X
a
(Ad - o) k Y ~ ~ k
U
// (Ad ) ~ Y. ~ k k
// X
b
f a gb Y [T ]/T p - 1 =
H
` V c c r e a WvoQ i gH H cq c rq a S c r CTvC` g G iH 1
t
H
H C
Hi TH
U'q cBQq h V 1c vc )g S `S S c r e ` YH `TbVu`XYW'TP i y a Ti V Ux S Q Q e e t vu` g H i Wx x Ti H G Q WBQ y eq cq c rq a S c r ` ssBbvy i vx H iH Y
X Ad Y ~ k
g = 1 f|Y
Y
y eq WsBQ
a dc
H X
qa ` T#y
a c `aQ c r dpTvCbS
1 Y (X , O ) Y Y [T ]/(T p-1 + + T + 1)
X [T ]/(T p - h)
c
X = X [T ]/(T p - h) - Y [T ]/(T p-1 + + T + 1)
p &
g
f a (X , O ) gb
H
g Ti
Q sp` c g ry ` x i
i YH
H Wx
H
X P
c a W` X e i qH x H
h =
U
U
// X
// X
a
U
U
(Ad - o) k Y ~ ~ k
U
// X
p
// (Ad ) ~ Y. ~ k k
Y X
// X
b
l
X Qq WU
r X r Qq Buc
`
a r c c j Wr 5y
Qsch (X , f a ) = (X k k )[V1 ]/(V1p - (f a )) ~ p
V1 l.V2 ? {rig, geo}
// Qsch (X , f b ) = (X ~ k )[V2 ]/(V p - (g b )) p 2 k
Q? (X , f a) p k
Q? (X , g b) Q? (U , f a) p p k
QC`X1VWU'STQP a
H
e g Q Q sq Bbsq ` P eq Q V a V g B$ H Hi i i
Hc
qBaeCc Wg r c g
S
H
`
i YH
r tq a ic e BdWx sq g aQq Q c r S oBvY)q H lH t x c rq a S c r ` sBbvDy Q y w x hH g
, g b)
US 'q
YH
g w o w cBq h Q1V YS Q ` i S Bi ` H
a ` x
Q c ra Br h
Q? (U p
? {rig, geo}
p 8
a g dc
X
qa BS
Q yp`1SyP r S H
Nis
? {rig, an} Y
Q? (U , f a) p
Q? (U, f ) p
// Q? (X , f a) p
// Q? (X, f ) p
Q? (U , f a) p
Q? ((Ad - o) k Y, 1 f|Y ) E ~ p ~ k
Q? ((Ad ) k Y, 1 f|Y ) E ~ p ~ k
B1
X X X -X = D1 Dn X - n Di i=2 Qrig (X, f ) Qan (X, f ) // Q? (X , f a) p X = Ad Y . h= f
// Gm~ = Spec(k[T, T -1 ]) ~ k
a c `aQ c r dp'ovCbS
a
p
X
H
qBpsy a ` H r Bi i Wx g
H
`
)E
z d g H Q H e a c lC` )
B1
XBQsWU rkX y q S1Vfdc XXrYSfQvr y q a c q `y Q c ` spWe vsy S c ` H g Bi H HBi i H x gYHwHCWxYHTiWg i 1H Y$ ')Y
h-1 (B1 ) E k // h-1 (B1 - o) E k
$#) s p ' # YY' VS c r c j bW1V y t
f -1 (o)
T#
W'Y
CC g w o # ) p W 'Y 5 R f v#R#D p$5 5bC$ 1 b C$#F Wp $p@ # $#F sv# $v#p )5vY Y# Y1 Q pCCQ e a c r H H
(1)
c ` Q WbS dg Q W$8Q c p` q P X r c q v y eq ` S y V H i x H H H H x
// [h/ /()]-1 (Gmk )an E.
H
a bS ky aQ ` r x Ti Q Tdxc Cr g i V a Wq H g
c QVBa `1SYdc X sq rBarXrWUV `1S c qcfq e1ec )g q rS a Q q ` a V c dc S Q dsq 1q a a a c jQ i H` i Ti xa VIQ H y re e Q H y Wx g r y x adc X j e r YS H FvH r H YBvr v x y Q c V iq H Q c e H x x YH H i g i YHCWx H H TWx TpH Hi Hi i Y$ ')Y )
B1 B1 k [h/ /()]-1 (B1 ) E k (Gmk )an (P1 )an k
a Q
H
iH Yvx H vx a a
H
H
iH Yvx iH Yvx
H H e
c r Q X rQ a a i c w c H vH c YV Bi Q eY a H 1 j H x H H
x d g w o s W # ) 'Y 5 f bC$1Y b vTY p v Y Fus v#R#Df )m5vY Y# Y# bYT '# W'b #7 a a PR8 ospp q a U BbS Vr H Wg i Q rC` ` Cr Q Q )y rvpWFp` vYrQ y pTy 1V c c r y Q c ` e y V cV a `a Q e ry ` rQ @` y bW'S 'S #'y V rS V e V U U a `a Q H Wx g YH WWx H i g g H Wx H i x g x x i i i 1H x i YH Ti H x IG i H g jq ` Q c ` X j q psq y tq a e j Bdc q YQ eq V a V sq BYq ` P ` q i lH YC H g iH t H H H i i G a c @dp` q p$ q 1c c g Q aq ` y a e V e a H Q Qq a $BbS U r a c r CCQ V c c r e a c ` WWCudTa XBsqWU rkCQ e g g sBBCr x r Q X r Qq c rq aq Q ` y H ` x ` r H c a { i Wx H Wg i H Wx g Bi H w w w g Wg i i g w H g ` Q c ` X j sq q 5y tq a e j Bdc q YQ eq V a V sq Tbsq ` P ` sq q i hH Y H g iH t H H H i i G g ` Q c ` X j sq psq y tq a e j Tdc q 1Q eq V a V sCq Tbsq ` P ` q i H YC H g iH t H H H i i G qTadp` q QlTp$ Q c Q aq ` e y Q c lQ vCr X ` a e X Qq r Q c ra Br h g Cc q V BWU X y x x WYH H x i H fT H H g Ti H elVBsqsp`X Wc y Q y r S ` q a U BbS Vr ` XV USQ Yv'BRP y Q c V eq Q c svr Bvr v H i H H C H Wg i H i a U XQ BqWU Xc oQ r a Q Cr Q fa g qTS Vr g gq x H XkX y y ` w H w Q WsQ YH usBbv1g g gc ` 7ec j e`1S y H ` a WgqH ak 7Tqi Q a q i e Q g ys y e q i c q WWx r q a S x cHr p &g x gW g i g x WsqQ i Wx H iH S V Yq C fa c y e c rq S r H sq cCTavcC@` ` iX r c q q WH g H YH vri adc H X Qr x S w w Qq wec H H S i ` H iH YC G x i H H i tfW1H i x x H C H Ti i xH Y)g e pTdc 1y CbS y Wx q `a XV X r c Q Qq Q q q Wec aQ Q c ` vWe g `X rWTadc X r S g x i H H i tlW1H i x x H H Ti H H g H Q Qqy Tssla TS Vr q a U ` XV USQ 1W'BRP c Q a g H H Wg i x H w w H
M (Gmk )an (P1 )an k /()] B1 = [Gmk / k X (Gmk )an [P1 / k /()] h: W (Y k k ) P1 C ~ k k k X = f -1 (Gmk ) - E ~ f -1 (Gmk ) ~ E = f: X ~ k k // P1 k
k
E
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
X
X
~ W = [ X k k ] P1 C
// Gm~ k
E
k
C
// P1
f -1 (Gmk ) ~
Gmk [V ]/(V p - T.)
g
Qrig (X, f ) p
k
(h/ /()) : [W/ /()]
k
// [P1 / k /()]
r 5y j
Qan (X, f ) p
(h[1/])an : (W [1/])an
// (P1 )an k
r 5y j
e
W P1 k an 1 an [W/ /()] = (W [1/]) [P1 / k /()] = (Pk )
k
Q
q c r c ` WCWp1S Q
Qan (X, f ) p
(h/ /()) : [W/ /()]
// [P1 / k /()]
a
r 5y j
e
g : Y
// P1 ~ k
~ k
Y
g: Y
f
g -1 (o
// P1 ~ k
)
s
C
// P1 k
P1 k
Gmk [V ]/(V p - T.) W
k
h: W
// P1 k
E
SmRig/k
// [h/ /()]-1 (Gmk )an E
B1 (o, 1)-o B1 (, 1)- k k Nis
H
y ` S ry YP
a
Hi TWx
[h/ /()]-1 (B1 (o, 1) - o) E k
[h/ /()]-1 (B1 ) E k
(2)
(2)
(1)
// [h/ /()]-1 (B1 (, 1) - ) E k
B1
S c r e
a
f : X
// A1 ~ k
~ k
X
pF a )R ospp a c @dp` q Q Q 'S Q y U a c r XV dYWe ` vxy n e S c r e Q H v H Wx YTi H iH H i x
" ! $#
Qrig (X)
D
// B1 k
// Spm(k{T }) = B1 k
E
Qrig (f )
SmRig/k
C an (P1 )an B1 (?, 1) k k k M Q(A1 ) = B1 ~ k k
D = Spm(k{T, V }/(V p - a.T )) a k - {0} ^ k R = Qrig (X)B1 D k
// B1 (?, 1) k h: R // B1 k
w o
g
Hi BWx
yS c r bbS
US 'sq q h V YS e sq 5lQ cQ ` ` X jy a Q c ` e V eq Q c r S vpvV BvYV YuvYS Q `S e c r x c pgH 'sq q Hh V 1kc Q V YS X i Hr r Cyyg q `yy r pCr bS y q ` S c r e i sy 1H vCpa Bi cQ e `S a ` q Q i qi Q V ` S r S H a c I xg x US g vc BBa 1YsdCQG i Wx g w H lTi i H Yvx Wg iH g xH Y)g YH H Ti i x H Ti 1H TWIG i i Hi g H H i
(hU )-1 (B1 ) k ShvNis (SmRig/k) // h-1 (B1 ) k (hU )-1 (B1 - o) k // h-1 (B1 - o) k
Q 1V YfQ e `S y ` Q psQ y ` Xq a c r dX sy S e ` r Iy Bi p1H P a Q n i y ` S r y g H i g H H Wx 1Bi H iH H H Wx g i x H
(hU )-1 (B1 - o) E k // h-1 (B1 - o) E k
H
t H
Q Qa e V 'dc q Q Q 1V `YS elQ y aCcCrX k` y g e Q n i Bi x H Wx i1Bi H H H i x iH Y a WYH Q cHr a h vCBr 1g H H w a ` e a fCc X rH bS iq g 1r H ` WYCvxYS Hi Q a S V c y VH c V T d ` i w i Hii BTWx i
// h-1 (B1 - o) h-1 (B1 - o) 2 1 k k I = J {i0 } X X i0 jJ Xj ^ ^ R = Qrig (X )B1 D RU = Qrig (U )B1 D k k 1 Bk
e XqWU c Q ` 0c 8g Q r X ry ` H H Q c a r c vTTirg pW@a x i H
)D p$5# d#$ Y)#' ' f
Q r q lq Ba fP ` e Q cq a Qq e y c ` f1V WsBsWfQ Wsq a i H i YH @H x H i r a a r c pWc g QTa r e ivx Cc X ` ag r bS c 7qCYS H ue )p` 7qvr WsBvr v H j e Q ` y r e j e Q c V eq Q c i YHCWx YH i g x H HTidvx Bi x H H i i 87FFY8fW T Y b DbvFD bb $ r'bC x x x #$ p$' bF#' Y# b T $ lb
X // X.
tx
# v ) T$Y)h )#v#Y T Y t x x p$5 Cp$' bF#' v# b p s 7b v#R#)rb # vY '' #Ff Rp uu7f a `QQ V X y r e j g BTq e e X c e )p` 7e a c r XV e e c r c a d1Dc Cg sq WQ q fsvpve Cr a Q `y Q c ` Q Q Q ` S Q c ` kc g 1S svpWe a c r XV CCYe H c q v Iy #g x i S c r eH ` H j c n H H Ti H H i t H a Hc ldWx q H y vg x ` H q BWWx $ Q Q r g bg ` aS H a e V e 1H q uc Bi c Fg ` i q Bb$ Q Q raS ` c rq a S c r ` CTvCDsy n H H Csr c i H dpTBC` H Q 'vp` e x 8pq H X g ` ai c ` Q Q H y lQ c a e X r CWc S TQ r g Wx e a cq U S Wc x e y ` X ` e X ri gy CWc H i Q vc H V ` Q sp` S S ry w x H H hTi H vx i G H H g li g x g H i H H Ti H Wx H BTWx i g i i Hii i c c r Qq WCp` c g c vpTRX uCTS V y e ` a Q Pq Q c rq a e a Q Q c ra r fvCBWc z W g sBBCr r c rq aq Q ` y a c r XV CCYe r i g H i H @H H x t H i H i iH 1Bi i U Cxr qvWfQ w BwV sspX Wc )g g Wg r XWQ q S c r e a Qqy `w r S ` Wx g v t ` Q cqH g vpve Ti e V Qqy ` fBsspX H Wc ` i r Q H YS dp g r e c r z X Qq H r WU C X y w oH g Ti H w e pvsr ` cq Qq a dc X y H sBQ x BsspX g WBTi Cg X q Q S c r ei a V Q q y ` r Hc i v g r c x CW#` e spRP y fspX Wc Dvr Q QH c ` H vpWe e y ` ` e V Qqy ` r ` e c r z YS CH g H g WU X H y X Qq r e i vsHc r ` q Qq Q ePq S V S c r c j g Q BQ blYvvYV 5y H H Bi H i x g BTi Hi w o@H g Ti H w H H x x H Q c ` vWe bS X X r X Qq BsWU kc 7e X sqQ BQ r X j r `y r a v g c rq aq Q CTBCr r `y Q c ` qvpWe YS X X r S ` g H H g Bi x H i x H w w g Wg i H H C
C Gmk ~ (Xi )i o ? {o, } A1 k k k{T } k{T }[V ]/V p - .T k{T, V }/(V p - .T ) k{T } k{T }[T -1][V ]/(V p - .T -1 ) k{T }[T -1][V ]/(V k{T, V }/(V p - p-1 .T ) // A1 f : X o ~ k h XJ X B1 k XJ = jJ Xj f -1 (o) D =J I // B1 k
p
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
k{T }[T -1][V ]/V p - .T p-1 .T ) V = .V -1
f h-1 (B1 -o) k
X1 X 2
^ RJ = Qrig (XJ )B1 D k
X
h-1 (B1 - o) 12 k
h-1 (B1 ) 12 k
// h-1 (B1 ) h-1 (B1 ) 1 2 k k
hJ : R J
U
U
// B1 k
// X
U
X = X 1 X2 = J {1, 2}
// h-1 (B1 - o) k
U
// h-1 (B1 ) k
h-1 (B1 ) = R = h-1 (B1 ) = k k f -1 (o)
//
//
jJ Xj
a
S vc
S V
` y
H Wx
q ssy
g Wg
g ` c 8g
H
X Qq BsvU Q y r X g X kX y Bi e Ba H ` q H lx dc H 1bi V pH ` a eVS g ByH ux r Fg H v'S v'S i i V U i Ug H Wx i i 1H
H Ti
a qQ TQ ` X Hq ew ` fy Qx Q T` x t H Q`YS Wla y YH a `H a Q #Ty H 1V hWc y aq x H i
h hU
jJ (Xj Xi0 )
^ RU = Qrig (U )B1 D k
g
(hU )-1 (B1 ) E k
(hU )-1 (B1 ) E k
hU
// h -1 (B1 ) E k
// h-1 (B1 ) E k
Nis // ShvNis (SmRig/k)
Hi Bvx
a S
x Bi
aQ ` oy
` c We r e a ldc X Wsy c r r a r H X rHU a c r r vc B1Ta Ybfdc H X qq H TpvdCQg Q qQV `S rS a i g Wx H i H Wx g H
SmRig/k
(hU )-1 (B1 ) k
// h -1 (B1 ) k
RU RU
(hU )-1 (B1 - o) k
(hU )-1 (B1 - o) E k
// R // h -1 (B1 - o) k // h -1 (B1 - o) E k X12 = -
// R
a r c pWc H c g i vx
r q x
v
Q q rBa fP ` q e Q cq a q e f1V WsBQsWfQ yWcsq a e XWU c 7e Qdr e c ` Qq r X j Qq a a S hbW1V vpve e VS c r c jy Q c ` c rq aq Q Wg sBdr r i H i YH @H x H i H H g Ti x H g g WhH H H i g a ` dcpvc c c eCr QWU X y a X q r Q ra h vcCBpr )g r z v g sBdr r c rq aq Q `y a ` c c r S c fvWCbuCr vc S S V e g Hi Bvg x H H H g Ti H iWx g w w g vg i w CH BTWx kH Hii i Q Tpr dU y ` #g Q 'v` e Ua j US c y e aQ q Wp`'oRPkX QqYvU'BRP ` XV SQ c oQ j Cg Ta a r eQ S kCTS Q dsq y vTpr pbp W W e c rq a a c j Q c r a h g { g g i v Q H g i H i tfH H x H i YHCWx YH H H x H YH i XX qr a ` Bsq F adc X q Q ` e r YS c a ` X e V a `y S BpsY1 jV H H G dw F i H i g i H H TiWx Ti x H H H G a Q WWYS q c r c ` g H g g Q BWU BsqkQ e q 1Ve c c rg X Qq r X rQ a e ekQ BQ e ` Q S ` sYvcbS eq S r H Cc X psW1q q V c Wx rz W g a a `q e V X X t H g Ti H x H H x H x H H H Wx H H H i w w c rq q Q sBadr r ` sy e YSvvYV 5yqvWe V c r c j Q c ` a c ` dpc c c Wer WU X y a X Qq r Q c ra h Br g ` XV USQ 1W'BRP g ivg H H TiWg x H H H g Ti H c Wc r q a e r Q c v`We ypq y a#bc 3g ` V e j `y S V QCr Qa c x hi H i vx g H HWx YH WgWx i g xH Y)g 87FFY8fW p pW l #v ' dd TF@R v bb Y R B YYb s '# $ rvY T bksb' fI B ) R $ p x x #$' # hC#$ Y)pT q hv#Y 7Rbs #7 R8 u7f a a c @dp` q Q X y X x H H r X y gTi H a qCr r v g c X X y e g S c S 1w V w sH TsssBa H c x p ` y wdp` Q q y q a c Hii TBWx i x y ` kuc )g r Q ` y Q c ` CYS vpWe z W g sBdr r c rq aq Q `y Q c ` e a a Q c r vWlq vCbS WU X Qq g H H w w g vg i x Ti H ` XV USQ 1W'TP e Q ` CYS y p` a l jq aq e a Q y a Q `y Q c ` e q spW` Wc g a r c H Ba x ` i q X q e1H ` y a c e t ` vy e Csr c a cq ` dpTBCx ` Q 'p` 1H e ei H DWc x y x Q a c `QQ U S Hc i e X r H i H YH g x H H H uTi H g i g x g H i H Bi H H Wx i
// StU -1 T ~ a k[U,U -1 ] C A1 ~ k StU a
-1
H
q Q cq ` a q a e r BvspfCc Tc g H
H
e psWDy e Tdc Ysy YX fQ y ` c r `q e ` q `a XV X rS V y a H H Ti H H g i YH H H
StU a
-1
Q pTYQ WU X uc y `aV X Qq r e a qa e dc BWr H c Fg H fQ Ti 'v` H Q e g US c e H H H i
. X
H
qQ V `S Ba 1V YS Q ` X c )g ` y ry r y `a p'bV WU c X Qq r X H misBbS Q Bi dq y i vCBpr i g e c r q a i a c jH Q c r H a ) Wx g X 1WBa e rV U ` y r X X rH S r y H CblQ sp` e i r ST c x h g g H 1H i w H Ti x W1Bvg H xHi i g Wx DH i H Wg i
e: X e0 : C
-1 // StU1 T -1 [S1 , . . . , Sm ] = Spec A [U,U ],a ~ k
Q ` XV USQ YW'TRP e TYV WU c 7e vc sq y dr y `a X Qq r X j S aQ j Q Q fa c g g c rq aq Q Cr #C v v sBdr r i q r e V y q 1sp` ry ` c c r spp` BQ ` ` W ` i a Q a r Wx y i e q ` Hv y g c HQ aH dc H YBb1H V Q y Ti a Cx g g H sHr BdYH r 1H 3Wvx dc Y)V X V Qq S c q a q Q r i ` y g a g a xHbS g e V X kX y y ` 1g '` i a w F Q T3d@B3Q W$7Q vxc ` q 1P i Q gy y e S V { i Q a Sg H rp$ dBx sgy Ba c w w r dc g X vY$ H q 1W@sa rH rH H i i i q g S ` i i B3i V g dYWroQ z H W g sHBBCH r r x vq Hr i H a cH r X V e a c rq aq Q `y a q ` y Wa g c ` Q q q H a y e H V e ` y Wg c g H Wx H x H i vx g H H H Ti x xH Y)g i g vx fH i H w w g Wg i eq V a V sq Tbsq ` P y Ip` bS Q BsvU X y ` e y a a X Qq r } } ~} Fs8| g H i i H i Wx Ti H x g Bi H ` aq r e VQ BCr by dc X y sQ dc U oQ Q X rS ` a Xq Q a a S ` P r aS c r vC y dWe a c r i H i mH g t g H H g H H x i x WYH H Q Q e aq X Qq WU YV r X y a c mdp` q Q Q c sq sq 5uQ ` X jy a aS c r bv a oQ dc U Q BQ a S ` P r aS c r bv g H H H g Ti i g W0H x H Y iH H H i x vYH H IG H H x i x W1H y a c r CCe Q Q e aq X Qq BsWU kX dp` q Q Q r y a c c ` X jy a q q 5Q aS c r vC H H H g Bi H x H YC H iH H i x WYH H IG H
(Et/h B1
-1
q q Sq sy Ba ` c a U ` 'S C TdCoffQ fCCIq fQ Q a cq r U e ` e a c rQ aS c r e e BCr Cbksy ` VQ X rS ` x 11H x i g H H x wi H t H x W1H i x x 1H H H g i g
I
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
ShvNis (Et/R)
a q ` a dc a kCbS X X r Q qXsyrYS c ` e V x e q Q c r x pWqV H vCbS Q i H
" ! $#
~ A1 = Spec(k[T ]) ~ k
(B1 k
(1)
- o), Nis)
Y = Y1 - P
r
US 'q
YH
cBsq h 1V YS Q Q ` e i x a q c Bt i V e c Wr Y WbufH a H
X =F
A1 ~ k
ha : R a
U X = Stb
b
h-1 (B1 ) k b
E
E
// B1 k
-1
-1 (StU T a
T
F
b = (b1 , . . . , bn ) k[U,U -1 ] C ~
B1
h-1 (B1 - o) k F hE : RE ShvNis (Et/R) h-1 (B1 ) k F h-1 (B1 ) k E bj P Y -C k
- o[U, U -1 ]) k[U,U -1 ] C ~
^ h-1 (B1 - o)R - k // (SmRig/k, Nis)
// o[U, U -1 , S1 , . . . , Sm ]
bi + b j
ha h-1 (B1 - o) k b
// B1 k
(1)
// h-1 (B1 - o) h-1 (B1 - o) j k k bi b
T
i<j
// h-1 (B1 ) h-1 (B1 ) k k bi bj
k[U,U -1 ] C ~
// ShvNis (Et/h-1 (B1 - o)) k
i
Y1
k
h-1 (B1 - o) k
i
X k Gmk ~ ~
n
hE h-1 (B1 - o) k E
StUj b
F
i
-1
(2)
j j
T
k[U,U -1 ] C ~
// StU -1 T [S1 , . . . , Sm ]
A1 ~ k
~ k[T, U, U -1, T1 , . . . , Tn , S1 , . . . , Sm ] a a (U.T1 1 . . . Tn n - T )
// Y
hF : R F
a
(Ti , Tj )
U X = Stbj
(Et/h-1 (B1 - o), Nis) k
C
// X
(2)
// ShvNis (SmRig/k) .
// h-1 (B1 - o) k b B1 k
i
StUi b
// h-1 (B1 ) k b
-1
T
X =E
C o[U,U -1 ,S1 ,...,Sm ] C
T
-o
j
k[U,U -1 ] C ~
-1
// B1 k
1i=jn
k[U,U -1 ] C ~
U Stb
T
k[U,U -1 ] C ~
U X = Sta
-1
X -C
Y -C
n-1
T
k[U,U -1 ] C ~
X
-1
o
//
//
T
// (Et/R, Nis)
k[U,U -1 ] C ~
bj i
// X.
// Y
n
H Ti
xH 1g
H Ti
` y a ` a ` j g w i Ti x g H
i
xH Y)g
H Wx
a lCc q y g Q a sCr Q lQ q 1e ` Qq a e a eV n H Ti x i sB#y H x y a H e V e cH fq Wx 1r c r qH a ` ` x w H Ti H
W d - aU = 0 A =
e ` y vp` Cc kBQ Q y j e c a X X ` H Ti w i H x
r q g CW#` e P qQ c e r c y ` i g vx Cw H H Ti
=
dp` q sBpTdc 1V a c Q c rq a ` a Q `y sla ` X e c rS bvc Ha Q pS x Q WvH oQ Wg j q c t a i XH X r v H Bi x H H H w Wx H i H H
k{U, U -1 , T1 , V } q (V p - aU T1 ) A
k[U,U -1 ,T1 ,V ] q (V p -aU T ) 1
k{U, U -1 , W } -t/p {a S} (W d - aU )
Y
ks B'b
a
H
` Wve Q V c c r cq sc a ` pC g X e X Qq BsWU kX y r i g H 1 iH x H w DH g Bi H w
C$#F a c qd` q Q cq c s jr H 1 iH
C @C1Y ) Q kX y vr dCr g X y Q c a c x i BQ H r g H H a i G H x
S p/d W -t k
` WWCQ V c c r e a i g H
H
H
X Qq BsWU kX ur r y y ` V a C1loQ t hH H V Qqy ` X Wc y H Ddc Bi X sbC$ q V c sg a r r ` a g yq SH` t e c qa i H Wx H H H i x H H srB#`y TH c x Hi
k{U,U -1 }
H ` g
a c r dX Y$y p1uc g yq S ` Sy ` S H Ti H x } } ~} Fr3| X X y e Ba ` q X q e ` g W g g w H w H x H i H 1H g x G U Bsq a ` a r X rQ jy g t qH Bi g H Ti x H X ` Cg e Q pQ TQ g g # W W X i w w H H G Cw F li H x c rq aq Q `y ry ` a ` c c r S sQ vWCbc Ig z W g CTBCr r g i `y a ` c c r S c a cq c rq aq Q z w W g w CTBCr g r Wg ss i vWYr Csr c g w w g Wg i w g ri x vTY
A1 ~ k B1 - o k
U Stq
n -1
Q WQ BTQ RYS X y r c r Q e Pq X a c `aQ c rS a g CTvYoQ x x H H H IG H H } ~} }Fr3| c `q e V sW1X 'S U ` sy a eV e lq 1uc c g H v H Wx x H H
e r c ` CW#RP y ` WWCloQ V c c r e a Q ` CYS y i g sBvCbS H e H oQ Q v8y n Wx V eq Q c r S c r a r cH H H 1Bi iH t H Bi x H yS c r bbS Q spa X e ry ` g Hi BWx H i H 1g iH U BsfQ Ta r X r Qq a Q r e fQ S sBS Q d jq e c rq a a c t H Bi g H YCWx WYH iH x H H H YH i G U kCQ c lCCfTa r X r Qq a jy a c r e Q r e ` dc X a r bS c a ` X e t hH Bi g H x iH YCWx 1H i g H BWx Bi x H x Hi H ` BsqvU Bsc Q XQ r X r Qq a Q Tsq r x H gBi x H H Wx x Wg i Wx g e V a `y S j kBpsY1V 5y aq r g r y ` e y pVsq juc i Wx g a c Q dp`TBQC` Q 'vc` e H e DWc y Q vc US H e X r S S ` 1V y ` i g H i H Bi H H Wx H TBvx i i Hii i 87FFY8fW B )f $ 3 tx x p$5# f #$ Y)pT Rp u7f a y ` V a CYlQ e c ry `y r Q ` WssuCYS k` c ` ` X g t V e q Q c rH i i rq1S y qH BvYS H k YWpsq W g Q ` e a vx H Q x y X X x y y 1H ` i i g H YTi iH H x w w w H H H i a X X y y eq cq a lQ g V r Q g WsQ H Q IG S H H' i H y YC Q rH r X r Qq e a c r Q CQ fdBQ Y Wx a Wg Q phBa X i y H YqH `S U e H H i bH H iH YC H vYH H x Q c ` e pWrQ H X Qq WU g Ti
E
U X = Stq
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
n-1
-1
^ Q(E)B1 D k
T
k[U,U -1 ] C[Tn ] ~
// Q(X) 1 D ^B k
e
E
k
d = p.g.c.d.(p, q) sp + tq = d q k[U, U -1 , T1 , V ]/(V p - aU T1 )
` y
e
aq r Q Ca
H
d g w o e spRP y ` H Q c ra Bi Br h
W
k
V = S q/d W s
V
p/d -q/d T1
Spm
V
R
k{T, U, U -1, T1 , V } q (U T1 - T, V p - aT )
S q/d W s T1 s S T1 V t
T1 = S p/d W -t
q k[U, U -1 , T1 , V ]/(V p - aU T1 )
a
E
q ~ X = Spec(k[T, U, U -1 , T1 ]/(U T1 - T )) k[U,U -1 ] C ~
1
Qrig (C) = Spm
StU q
k[U, U -1 , W, S]/(W q - aU )
X
n-1
-1
T
a = (q, . . . , q) = q n
-1 k[U,U -1 ] C[Tn , Tn ] ~
q k{U, U -1 , T1 , V }/(V p - aU T1 )
// k[U, U -1 , W, S]/(W q - aU )
T1
(Tn-1 , Tn )
RigSHeff (k) M
k{U, U -1 , T1 , V } q (V p - aU T1 )
X
k{U, U -1, T1 , V } q (V p - aU T1 )
H
e spRP y ` H Bi
H
A=
k{U,U -1 }
T
Qrig (C).
A
k[U,U -1 ] C ~
U Stq
n
-1
T
k[U,U -1 ] C ~
A
V c c r vWe
`y a qQ
H
e H
V Qqy ` X Wusy dc b1V r c ` a e VS c rq Q S Qq e ` sTQ Bsy ` i i w H H i Wg x i
A
k[U, U -1 , W, S] (W q - aU )
(V p -
q aU T1 , W d
k
k{U, U -1, T1 , V, r1 W, r2 S} . - aU, S p/d - T1 W t , S q/d - V W -s )
k{U, U -1 , T1 , V } [W, S]/(W d -aU, S p/d -T1 W t , S q/d -V W -s ). q V p - aU T1
r1 = 0
r 1 , r2 N
q V p - aU T1 = 0
a Ya
c rq a ` sB#y
q Qq r U S BW'fa
c j Cr y
A =
=
=
k{U, U -1 , W } r2 k{U, U -1 , W } r2 { S} atq |S pq a-sp |S pq = { S} atq |S pq atq-1 |S pq (W d - aU ) (W d - aU ) k{U, U -1, W, T1 , V, r2 S} (W d - aU, S p/d - T1 W t , S q/d - V W -s ) k{U, U -1 , W } r2 k{U, U -1, W } r2 { S} W dt |S p W -sd |S q = { S} at |S p a-s |S q (W d - aU ) (W d - aU ) k{U, U -1 , W } r2 { S} W t |S p/d W -s |S q/d (W d - aU ) n2 A
=
k{U, U -1 , W } r2 k{U, U -1 , W } -t/p { S} atq |S pq = {a S}. d - aU ) (W (W d - aU )
` i g 87FFY8fW '#
a Rp uu7f
C g w o y ` e r c r X pvCWla sy vpq a V c ` a S c r vCc CBq q a c `Q e c H x i WYH i x x x p g w o c sq yy e c rq a S c r U e sBbvo` c 7e Cr r p W vz X j Q Qq e c z g g X a W g sBdr r c rq aq Q ` sy g H Cr f H sq YdBQ pWHe Wx CX H sy g Wsq a w q 1BpYS c vQw )F H H G w q YT#1@` vg ` # W vz X YWU rV q x aV X y ` r c r V c ` r V a `H a r Va `Sg i i g g g i H Bi ` w )F li H $Wx i H H x i H i H x H H i G v R#)# yY$ Rb'#h' bp vb$Y Q b sCq p BYq ` P Q b Wsp'oQ 'p)0kC' q $ YY T#1S k b Y' e q Q V a V e y e ` a s q Br pf W U r V a ` ' r a X r k ' C$ dBR l Cm$ dT qb Y 8r W u5vY$b 1' H i i H H H Wx g H i h h Cu $ p W$ dBR R#Fd ! Rp olo 8g a h a Q c rS y vYD` r a S y y Q vpTfQ y X fQ y e `aQ e r e a S oQ a g g c rq a p z CTS H iBa x U BvbS VBrx i Q CTS vYuka Q S Q Q q CYBpYQ w sBF vYsy x r sg TWx fWYH H H w H e c rq a a Q c r S `y a e r V a ` S H c r Wc V hY`H e q aq t e i Q H p1P H y ` S r y q i ii g i x H x H i Wx H H a bS y e q 1HWe Ti Wp'og Q H BbS H U i r a Q T` i a e V H y e ` a qa a bS H oTQ e Wc Wx ` Q aQ ` q Xi m Q q bH WQg a V X i Hi a x Q H Q H e x q 1BH pY` H WuTWx Bi H i g Hi x H H fdp` vbdc X BbS Vr Cc X Csy X i Q TWx S Ti P r Va `S e a x c e Yt H r S a qa U a S ` c oQWx i aH g H H Bi )g H H i G q q H i H H Wg H i H H g Bi x H H H Wx Q BQ e Q WU YV aq X Qq r X a c r dX Ba W` dc X Wvc bS Q TQ RYdc X ` j g V ` e a y e ` c r Q e Pq S a H H W0c g i Q Ti e dr w Yuy H sbBBC` CQ VQ Xg r S ` V q SH r Q Q H c `H X w q x psq a SH c r bvx i y G ` Q c ` vpWe e psq yH H ` X j a r c Ti Wg H H g t H YC H iH i x W1H H i g H Q aH c e t r S a qa U g 'C` vpYfdc H X BbS Vr ic ` Bp$YV S Qq ` a Cc Q Wsp$ Q vc p` q 1V Q y y eq ` S y Q V aq ` y a c ` Qqyq up$ dpTsBa a Q TQ RbmV YS i c Q e Pq S ` a c H H sq d` sy H ` Wg ieg j H H i H a Wx dc q H y 3g i Bi x Cr Q x spH Cc Tr 1k#'Ha y V r y ` a q a WuQBi a ` Q H e t H r S C`By 1H a i e x vbuDQ e Ht r Sx `yH a Q Q x Q spWg a H r y g ` g H H e x vbskoQ vg n i V H H xH c i g Q WsHe p$Q vc ` x q IGQ e Q p` H Ba Ti Q 'S y q ` S y y q H U Q a yH q Hb U BWSx Vr H sBp` pbS Ti c a c rq a e Ht r H a c e t r S Wax i q g bgS U d` H pbdc YXH BWWx Vr i a S 1`g H H H x H H Y H v H H H Wg iH H i i H x w i H H Wg H i H
GmZ // A1 Z
q q
Q c ` e q a ` vpv3B#'a CbqV 1c Cr g C` q Q X X rS `S aq a c q c ` X ` vpksy e e r S c Cr Wsp$ y eq ` x i Ti x H x B1 Hi H H Wg i w H S y c p` q V c Q a t q V r pg i H vx 1H r i Wx C` vpYS dc X BbS Vr i d c ma a c e t r a Q H H TC r x uQ Csxr r Ba Q q mWc Q BQ e q WU YqV q Q Q q H y rH e Qq X q aq X Qq r X y S ` Q r i qa U Hi TY i H g H x H H H g Ti i g WH H H g H i H H Wg H i S ` Q c ` vpWe y eq ` S y Wsp$ vc ` q 1V c t c r Q X Q a ` 7c p a j e q q H C H H H x H u x H w H YH Bi x i H Q Q Qqy hQ Bha Q #hCYW'ThQ e sq y q CYBpYRoQ Buy c ` Q ` XV USQ c r Va `SP rQ ` S ` Q c ` pWe g i Wx H g c ji e a e H H dp` vbfdc X H y H X H DH 7Cc q r x a c e t rS a S ` Y@` Cr X X Q Q HC r Q WW'S vBp` Q e c q c U a Q c r Qq r i H Wx H i q H H H g Ti x H H Wg i Q c ` vWe i e YH C` WWx pYCw c vX x BbS UH r dc i X Csy Wx X a c g g vt r S a H q a e a SH ` i gq c t 0c g b` CbCc X q q TpWa dp` vpYq e a X rS a r a r X rq U a c e t rS S c r Qq c ` e vpW` H dr Q Q c di jr uTvc a &C0WBQ H ` H Wc Bi V a q H H U e H r i rg c a r cg H aS x H g g q Ha p` q a V X y ` e H r c r g X a q g bQ H WWx WCkoQ Q ai q 'dc bS bS e H q 1B#1y1H vWx Cr r r Va ` S `i Q Qx c a c ` a c q ` ur dpx c H dspX e q Ba 1H` q i g u'vc a H Bi x Wx i H H H H H H H H i H vx H WWx g g w H H i i g
2 A R F 1SQ&# 6QPIH9B&# ! ! F ( G ! 2 % 0 F 2 DEDB9@9&81"5 C 7 A 7 ! 7 # 7 0 % % 5 "643 # 2 0 # ( ' % # ! 1)"&$"
US ` a X 1WBa e rV U ` Wz W g sBdr r c rq aq Q ` e a c ` a cq ` y ldpvc dspX y bl#Ty 1V rS V e a `a Q g V 'Q g W g g g H 1H H w w H Ti w x vYTWg G H xHi w g vg i H H H x x hIG i H
B1 (o, at/p ) E B // (B1 (o, at/p ) - o) E B
e Q c ` X j qQ c q sq 5y vsq g H 1 CYoQ i H ` V H a w y t @H H
SvrWfq 1V 1Q g c e a S j X Qq BsvU X r e r c ` CW#RP lQ y a g w g H IG e X W@fS ` bH CbS Bi r c `i y e q5 r S c e Csr c H c a cq e H X W` r cH i q 5y sy S ` X j H `H r vr X H g Wsp$ g vc x p` H q 1H P H i c Q G Q c ` y eq ` S y V a H vx H vx g i x i H lT H H H x H x H H H
B1
H vx
H
g Ti
r X y H
Q c ra r c vCBpva H e c a X X ` Q a g vp` fdc pTQ lQ i H x H
H Wx
H
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
k
a k - {0}
r2
} } ~} Fr3|
H
H
a
H Wx
aq ` fp$
x
q c Wt BsvU uc Q sp` X Qq r X ry ` g Q c r e r c ` vpHP y Bi sy x i H H H g )Wg e a c dc Wc q 1TspdYQ Wc e Q V aqy ` V q e lQ H H Hi i TY YH WYH x H IG
B k
Spm(k{U, U
-1
^ , W }/(W - aU ))k{U,U -1 } C
d
w (B, O)
W
c 7g
e
Q Q `y S ` Tby
B1 - o k
B{a-t/p S}
h : B{a-t/p S}
// Qrig (X) 1 D ^B k
// B1 k
S c r e aq j Q n vCCp` Y)y pg
q c r c ` WCWp1S
X Qq BWU
h -1 (B1 ) E k
// h -1 (B1 - o) E
T
(B{a-t/p S}, O) |S(x)||a|-t/p
U.S p/d w-t B1
` gi g
H
Q c ` vpve H
He
Q c ra h vTpr Cg
H
C g w o r e H Wx YCWx WhH iH i g e Q Q TQ e ` BWU Bsc oQ q X Qq r X r Qq a x H x H g Ti x H x
" ! $#
h
Sm/Z
T an
A
PreShv(SmAf/Z, M)
k
Q
B
Spectan (PreShv(SmRig/k, M)) T 1
// SmAf/Z
T an
T
a
RigSHM (k)
(Rig , Rig ) : Spect (PreShv(Sm/k, M)) T
x B{a B1 (o, at/p ) B
PreShv(SmRig/k, M)
T
T SmRig/k
(M, E) E T A1 /GmZ = Cof(GmZ A1 ) PreShv(Sm/Z, M) Z Z T T A1 /GmZ Z SmAf/Z Sm/Z Spec(Z) // A1 /GmZ PreShv(SmAf/Z, M) T T Z T = a T a // a (A1 /GmZ ) = A1 /GmZ a T Z Z PreShv(SmAf/Z, M)
c
-t/p
PreShv(SmRig/k, M) // Sm/k // Sm/Z
RigSHM (k) k RigSH(k) RigDA(k)
Rig : SHM (k) S} c A
c
M RigDAQ (k)
(A1 )an /(Gmk )an k
a T a
// RigSH (k) . M // B
// Spectan (PreShv(SmRig/k, M)) T a (c)
Spectan (PreShv(SmRig/k, M)) T M M (T an )
p'
Y
CDsb$ p')Y 1q) 'b b
En (B1 ) X // En (X)
5v
y y x #$v$C )b
Y8b I F c mY #5Y'Yi b a Cc q H
B # b B Yb vR W 5v r vY Y f5bY@bkb' b p $ $ c e p ) 'Y 5 u m d Y Y' ID c #YB s R Cksb$ r'Fb ) vl v # u s qY b'YuRY ' $ Y$ ) )h # ) m pbR CkRY$ @ # 'Y 5 p v v p m c YYdC RY Y'rbY ' kC$p Rb$ ) wb v#Yqb hv#$#F#'vY v c YTYB Y ' TC l B rr'b $ Y Y' 8 a 78 p8opr3p Qa c e t Q U e r qTd` Ta Q Q e d` q sBpBq Bb` 1ky e a c Q c rq a ` Q V a S ` S ` H Q H BsvU BsP Q y X e q 1BpYy Y i Bd jq H g Q e r r Va `S ` e a Va c X QqH r Xx r Q q q a ` Ti#Ta i YH Q X X rS a H H H i H i G TWg H i g Bi t x H
nN k k 2 nN PreShv(SmRig/k, M) M T qc En B1
q q
H
H Wx
U ` 'S C c sq yy ra r X r q TpvU q H 1B#x1sy t H dc f` r Va `S ` a x y g H i t H H
e fQ aS c r vC e p` sWe y X X `q r i H q 1WYH vCQ H H i H w q 1BpYWx ` i uS c r e a Vx r V a ` i S x H H i G
pN RigSHM (k)
Q
Hi BWx
a S
x Ti
a Q
Wx
` a dc H
X
H C
qa ` T#y
Hi ThH
Q YP y ` S ry Q a S olQ lQ q 1WfQ Wp'oQ aQ e a eV e y e `a H g x H ` i U ` 'S C c sq yy e S y Q vc ` q V 7e q Q c q mr z vvx z sir Bdr H r H i x j a a eH V e c x g Bi c B q a q Q x t H f H x H H x H x H x x w g i w F i
q
Q c ` e y eq ` S y vpvlQ Wsp$rQ vc p` q 1P V e q a ` T#'a YV YS c X X rS ` H H H x H x i Ti x X Qq WU X y ` r a e c q vsq y y a ` c c rq a ` #vsBpX v` a c 7e rQ c j H g Ti H i H x g H BWx Hi i i H x Qv`We c e a c e t rS a qC` vpYqdc X BS Vr qa U a Cc X Csy X S ` c Q j 8g a v H p` X j H i q Q X q QWU YV rq X y a c d` q q a &H a S i e Q Q U e r Hc q q y Wg BBQ H ` H pWg c Bg x H a r c H i g ei a c i e fH r S a Tdp` Wvbrdc X x BS Vr H dH c H X YCs` y X H c Q g t iSC H x H dCTS RoQ q aH U a a a c r X c rq a Q P i H H Wg H H H i g Bi x H H H Wx Ti H H x H
SmRigqc /k
cpl
H
Q c ` X j sq psq y H Y H iH
Q e Wc X Q BQ Q BsWU k1V Qq aq e X Qq r X Q e dr Ysy d` q Q VQ X rS ` a c H x H H H g Bi i vg H H g x Pq Q B` sq Tbsq ` P y YBT` eq V a V ` qS rQQ ` XV USQ YW'TP c q x H i i H x t x
k X k
Q TCr Q CCr Ba q Q Q q ry r i H ` CbS aS Xg r QcrBar g TY h Hi H g a S c r y Q c V eq Q c vC qvr Bvr v H i i x WYH
(X/ /())
6 4" 9" " ! " ! 8751 8D3$)$#1d$# A 8kv 9 "
cpl : Sm/k T qc
// SmRigqc /k
T
// Sm/k
T qc T qc
// Sm/Z .
B1 k
H
rQ ` Dsy
Q syp`c ) g g sBdr r r ` g { c rq aq Q ` y e i w w g vg i H Qdr sqWc sg Q e H g Q Q e aq H H Q c ` vWe e la 1e BsssBa q Cc X ` j q eV ` V Qqyq a x x x H x i G H
1 cst /Bk
e
PreShv(SmRig/k, M)
cst SmRig/k
PreShv(SmRig /k, M) // SmRigqc /k
qc
B1 k B1 k
T qc
B1
(A1 )an k // (Gmk )an
q
B1
Spectqc (PreShv(SmRigqc /k, M)) T
Spectqc (PreShv(SmRig/k, M)) T
Spect (PreShv(SmRig/k, M)). T
cst
cst
T an T qc -
B1
// T an
cplf
cst
cst
.
// Rig (-)
Spectqc (PreShv(SmRigqc /k, M)) T
g
e
Spectqc (PreShv(SmRigqc /k, M)) T
PreShv(SmRigqc /k, M)
Susp qc T
Susp qc T
Susp an T
X
// Spectqc (PreShv(SmRigqc /k, M)) T
(En )n
// Spect (PreShv(SmRig/k, M)) T
Spectqc (PreShv(SmRig/k, M)) T 22
nN
B1
Spectqc (PreShv(SmRigqc /k, M)) T
T an T qc -
En (X) Ob(M)
T
X
nN
En
Spectqc (PreShv(SmRigqc /k, M)) T // Hom (T qc , En+1 ) g B1 k
#v
PreShv(SmRigqc /k, M)
1 cst /Bk
CC g
w o
PreShv(SmRig/k, M)
cst
E W
p u j WP'uy
t f v t w u r t PhPWiR
We
v t w e | e r p u j PtpDWP'uy
t Pf
t v v t w u r e t r e t v p t e t r tv y e y t s f p y f t e w s u e s hChuiPl@Pf 7WfChu#'P{Pf 7kqB1)uPo PC}1BCWPuBuPt s e j t t u s f egg f t e f mv yv s u j p s p j e r y s j t 1lCPWChqP~PnqupuWBl'1tuWiPf p j e e y w | g r t e s e s t t v j e y f y e e j y t p B1Pu)n 5kWuPWuqChury BDBBlWR Rp)RUPxB3Rn8PYq
-(-1) -(n) ~ k = k(()) ~ k
#Crx5 1q1qW)PnBuWBlb) y eg y ug e p t y y s f e j t g y e u{PC f s f y eg sv t w u r e r j t r e s e v j e 1qtqPWCDW}uW)W13 s p u t t r y e e y s e t g t r e j t eg p v f m u f dPPCt1Bu1hv @CCdlWq#hCup e t f y PPD1e s f e j t g eg e f m e w s u y e r v y y f p eg f j u r g WBlbhfPn#C dHWu1P3 )) RiufhPCl8 j t r t e v s j Dn1qBW1u Wy e sv pgg s j f P quhqquWr ye ye uWpD1Ux e f mv t t p j e yv s y PnPPWHBqPBCHBqv t f p t e sv pgg s s p P3#P uqhC BC )Y kx s f P) y s f e j t g y e g e j t e s s f j u s s p v t w u r t 'PWurCb}1qlWuBPW1udhPWR u uPt 6 4 ! ( t e eg v y p s e nWHWqPHqq'WBy g ev s y t e v f r s e WqBC1B@hPuPdUU u t pse uPk'WBy e Wp Uq)37#0
-(-1) Susp an (-) Susq an (-) T T (X Acst )(-p)[-2p] (0 an (T an ) -)[-1] T pN Sus1 an ( T
cst )[2]
iYih3 e s fg j t j s f e w s u uPqlClPCD HHWu1PB o eg qWe ug f t pv s s f e j t g eg y s g p e yvgv f t hPCPuBPWurCbhuCh~BhqBR e e y yvg e e j p v y p f lBBhWBlHClj Bnm o e rv v s u uv s p PqChuBWhu# e f s f BPBtP) s e s t eo r z f y g v e s v p g g s j f r e f t u j t t g e 1BuCCHDHBq3#P BqhCBClPP0ClCPWu r y e y eg s p e u s r t e t e y ev t w tv y e y j e p e ug f t pv s ev s u p y f y p 'lpCh1BWW1qPq}HC1CWqPCPBuqu1BWj Ctqg y t S Qa V ` X V T S Q I G B8b7YYWU8RPHF l W'dWP'l '## r# CCq l qqi5lW$qrqPH1W)WClHWHWlCP#'Drl l$1flDlq'ih#$rs$rCu#'0h P#$##Ylhfu# hl$rrllCB#'qCu u## WfPl#PC U$U7 yv j u s y fg p g f y u s t f p t e w s j yv r y e t v p fg p p y e f m e egg f t e f mv yv s u j p s p j eg p r y ~qW1uPqP#'iPB1ukP}Cululuq1qBh#Du{Pnl{qhnRPnqupuWBl'1~qnWu 71e
B1 Susp an (Qrig (X, f ) Acst ) r T k T RigSHM (k) X ~ ~ k k = k(()) RigSHM (k) ~ k f (X, O ) p N r N - {0} Susp an (Y Acst ) Ob(T) T Susp an T : RigSHeff (k) M AE k / RigSH (k) M
e f mys PnuuC
f p e v t s p f p e v t v f p e j yv pg u s e r y eg v pg y e j t eg p v f m u o y e r v f r t o qP@1ChPxluq$luPdD1qPh$~1WWq#qPn1t1HiPPhv p t t v p s g y t p s pg t v v y s pg s p t p s sqBBCbpuud7CP ChuiBC)uPfq~)1nuPw t e eg p y t e e j pg e s e r s f e j t g t f y n1}1hu{C1HWlCq}1BH{P1BlbxP{pe
B1 PreShv(SmRigqc /k, M)
0 (A,(X,-))
iYih3 v v t u r pg r vvv t v r t j pg e r y s g #P htqBhPwWHq{ePqqbhqBhvWHq{'qp o e f j r p f us g p qgPW1u#'igyB~eCehWjhg o qgPpuy~usPurjuByHqgPfCy'Cusy1uBlj1)1usBn1y1HPPuHyhWusWPqBtW{eHq1lPC e e f f s p s t p w e s e y y e t e e yv s eg s e u rv y j r pg e j s f f y g y g p y e t e p v f HzB711eqP'pyt1ljWqg#hCmu y g j u s f e j t y e y egg pg p e f s e wv s e e y e s g s e t s f eheur1eulr@#dyuWBlbg{1r@hqqv}$qgPnm@WhB1ukdB1uhvU11PhvHuWeu{sP j s j e p f y u eg uP''We'1usuhgPB1usqk tP eBn'Bhgqu1BWjtuiule)uuPC t p s w ev s u p e v e f s f egg e s f qqWuPB
LocB1 -st ColimI LocB1 -st / LocB1 -st ColimI I B1 / Ens
e u y j pg y e s f e j t g e y e t p s g w y e v vg j v f p e f 1nHWq1WBlb11uCuui1BhHhqWf@BD}Clj
homRigSHM (k) (Susp qc (X Acst ), -) : Spectqc (PreShv(SmRigqc /k, M)) T T / Ab
sP1BlbRhPn1BunHRP'WWqPpYC1ulHWnBuWp1WPnWuBnH')p 7l1}HClfR f e j t g eg e f m s e s t e r g p s u t u y fg y e g y j p j t y y e y e j e f m s e s t s p r s r e j t e j t y e y eg s p e u s r t e t e t e e j p j y 'lpC{1q{H1BWW{nWHWur}Clt1e yhq71HHBx1CWqPPquB3qBC1B#1qPnUWuBnHP8nWq#hCU1qWWC CpH}uCqx8b7YYWU8RPHF ev t w tv y e y j e p e ug f t pv s ev s u p j pg e f m s e s t e r t eg p v f m u y e gv e p s pv r eg s p S Qa V ` X V T S Q I G W}l'lHlCP#' rW$irtW$r## Y '## Hrlhtt5lW$tlh@P 1W)WClWHWlCP#''l l$1{r}#Wtu##}h#$r}$r#nB#'fU x$UpPP iYih3 pg y t p r y e j p v e t y y e t p s g w y e v vg j y eg e f m ~P~BrCPl~CC~Bn'Bh1uiHqhCl{h}Pntlo e o t e e pv r u v eg f j u r v j e n1}1BqH}qqPW1)l BuqqlurCHlj CnPqChuBWhu# e y yvg e e j p v y p f m e rv v s u uv s p
1 cst /Bk cst
y e f s y e r ev s u p 11)Pu~huCWBWj e t eg p v f m u t e eg v pg y @nWq#hC{C1HWqPh$~pe e f m v pg f C@h$r e f s f y e t p s g w y e v vg j v f p t e f uuPC1Bnuui1uiHqhCl}Pn1BD}Clj
T qc
y s f e j t g y eg e f m s e wv s u uWBlb1qC1hB1n}e uP1}uqpRC7luq$luP1uPhPq$l1hqPn1~RBD}ClHqv o n{uBq1qCblu8C11uCuuix1uiHqhClj t e p s v y t v p y e t v f p e j y v p g u s e r y w s y e g v p g e j t e g p v f m u e e f j | g f m p s z f y g v y v g e f y y e to p s g w y e v v g v f p t e u j s g y p e f j e nWH1l'bCsutHWiPt s f e j t g eg e f m yv f e s j e P1BlbqqsPnuqPqBurBy t f es j te y e yesje PquWWfp1uBl1)y e r e v vg iuiHqhClj e t f m sv pg j v pg e f y p y t gv e e t e f e s e t f e r t p e r t v s e y y p e s wv t s e r pg eg f e c S Qa V ` X V T S Q I G Pf nqlki$hfCBd1e h1W)WCP1BiPPi1Chu1uxB1hPuWP0qihP1d8b7YYWU8RPHF
Homg (T qc , -)
E )E
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
" ! $#
nZ
RigSHM (k) = HoB1 -st (Spectqc (PreShv(SmRig/k, M))) T Acst )
Spectqc (PreShv(SmRigqc /k, M)) T
homHo(PreShv(SmRigqc /k,M)) (X T qc, -) : PreShv(SmRigqc /k, M)
B1
Susp an (X Acst ) T
X
~ k
k B1 k
k
LocB1 -st
X
(-1) -(-n) n N Susp+q (- -) T an
k
/ Spectqc (PreShv(SmRigqc /k, M)) T
r N - {0}
RigSHM (k)
f (X, O)
Evp
Susp (X Acst )
X
- (-1)
/ Ab
AE
Gmk ~
o
o
e yv s eg s p }BPPuH~h " t ev nWqc 51
Qgm (X, f ) r
X[T, T -1, V ]/(V r - f.T )
/ Spec(k[T, T -1]) = Gm~ ~ k Susp qc (X T
Cp
$5x
Cp
CCp
e ts p j pg p t e w s u y s p Pt'hRqHWuW )Y k
K)
y s f e j t g e r e y v so y v t f e s v f s y t j e r p s z f y g s g g p f r t f e r 'PWurCb~HBqPBCHBqkPBqPuBW@HuBUCbYqPWHp e g Wp 5 uBrBupy g ps fj fs e yv s HBqPBC} e r e y y v g e g v j e s p u j t f y Pxpe BPn31BuCCHPCHP vy e f m ses t e r j t r z fy g e r eg p rf t e e ug f t pv s PqPRPCHW1hPquB o BBhxWiurpuPW'By ev s u p y f y e huCWBWj BdPt f t e s g y s g v f e g p s f j f s y e yv s t e y yvg o P{nWHBCbRuBbs{@CuBlH{'lpC~1qY1R3 p f r y e r t p e r p y e y e g s o Wx'PBlBupHHPPuHBy Buqq@le gv j s o p u j iurepuP1P'By t Pf s f PC y eg s p e u s r t e t e e ugof t pv s ev s u p y f y pg j e p e rv j t f j yv r y s f eg e y e y yv hxWuWCPW1hPCPuBqu1BWj C~lRWp'PWqluhg y p j 1uP'By e ygv y eg s p e u s r t e t e e ug f t pv s ev s u p y f y pg e f m v p y t t v v y s s p r~$hBH1q~DWu1P1D1hPCPuB~qBCWu#1j Bq}Pn~huxCn P PqBhBC)uY )Y }x e r y p g f y u s y e r s f y t p y e s e rv p s e f e s )) '#uhPB1u1PPnuC)Wu~q'CPWuP e t f vj e t t r | by e % % s p uj t t t p uu e E 6 P8WqvDPPPstRYPr1nxqDa@lCPPDBWRWb Y3)$" DhCbqYi1uhPB1ut 8b7YYWU8RPHF sv f v p h e f r y e p g f y u s e S Qa V ` X V T S Q I G rlC #xl#C 'h krx5~'lq@WrP@lW'{luW ltu #@rh{P'B#YlnnW Cd'hn#$r$rC#' #WhWW$0 #Y hdl$rrlWBud#'0 Y ex$U f72 g iYih3 y ev s u p j e r e j t eg p v f m u e t f y WhuCWBW{{lWq#hC{PP1e e f m v p f r t p e tv p eg f j u p g f u s e y s v f p r y e r t e e r p egg f t e f mv yv s u j p s p j e r y s j t f s f y y e y yv C@h$gintWBnq}hPClo ruhPyB}Y1uBbgqf~1{nWulHC}qqP@PnqupuWBl'1}uWxPPBuhg yCH1PB1Ry$iuHyh1Cyu1r lHP deHHWq@eqxrh C~yqBu1q~1C$ p u y y e r gv eg y f y e v j eg g s p t v p t y eg j e p g p rf t s f e j t g eg s p PRHP1Blb{q s p e p g e e t S Qa V ` X V T S Q I G e r e g e p rv j f j e j e @phv 5~j@elt5HllepPWltr {Chv lYlq 8b7YYWU8RPHF Pr#ln##iWH rCkhlCrW$ )'HP$r#n'x hC@lW$}#'kBlC$rP#' B5W'Ph hV #HT WP'W Y ex$U pPd d D #}8 l W'f luW t p v f y t p y f p e sv pgg s j eg y t {n#qPnutRuhqquW@hB
cst )
egg f t e f mv yv s u j p s p j eg p qhnPnBquWur'1{hCWu 7r e g j e w s j yv r t v p fg p e r f p e t t p t r y f y y e r f p t ev v y y f p f e t t v j t g y e f mv v y t e j y e t p u y eg j | j y e WRleqPHWtBWBWBsChu#q#tPs1Pif 7RBBis1hUBBCBtPPqBlb1PnhuH'C1lu1P#WhWl1r e yv p s g eg e f m y t s t y f t e jv r t e g y t p eg p s t e f mv s u g p u uv s p e t f y t p HqgBCbqPn8uBnH8tc CWqW)Pp 53RquPCuPWChBWhu#PP8Pr qBWHqlPnq)tPf t v y t e v r j e r tv t tW3)'jqWevHSuYaqq)uqx)bUWh}PP}HClxWu}WWqe 511PnBulH1uh'Bx1PBp e v t y f v b yv s p ` ev g pg s f g p j g f p e t t p e j y f t e v f e j v y e f mv s u u y v p s y e r s f y v t w e | X 6 4 ! ( e y p u y rWP'juy1eRu1PsP'B$hBHuP1Blb1q{B1ClR )Y DR YUq) 37#0 r y f y y e r f p y e j s ygv y s f e j t g y eg u s u rv y t j y t p y f t y t p T 9 V Q t v v t w u W# UPqqBhPWP )) R #W}r#l S $W1W' lP'1BYu 5H~WW)l@#H W$
SHM (k) 1
Rig
P 1W#'h l@
/ RigSH (k) M
R
G H P W'W rWx 9
7 uW#P#'q '#h i#$WuWl W@hB5Wr'x#$WuWpB){h CH}#Bi#qPBlC$rP# C D EFluW #Y l$r#nuu}H')WhW1D)rr
~ F : QUSHM (k) SHM (Gmk ) ~ 1 ~ k R : RigSHM (k) Spec(k)
j
C 1W#P#'@lW)#l8 B hDl$rrllCB#' 9 WdWr x$8 7531)' 8 6 4 2 0 ( @ d#'h}PH5W$WA1' llBUDHl$rWl}rCH)#'dhrl e s v p j e j e r g p v j tv s p g f y u }uBhP'@W@hWhuP#'iBs eg v jv % qWqCx& e Pr t v j e y pg e t p r t y e s s j y p u j y e r e yv s e qBl1Bqqk{nP)BuW@W'BddHBqPBCHhg e t t uPhR C@5Wut'WBW)WC # ! }HW$'"irl r'l WDD' $WWbCDrCHhlWll )i o W' y t e v t v y p f unWu)hPf Bnm u #rqrql'~#W f5WWbCdl8 W luW u## Hrlhss$WWbCqlhk#}W#YlnnWDWql$1 rDWtPRqlCr hn$r5d$r#nB#' #Wfh hWu0 x$88 u t t e eg v y p s e y BPnWHWqPHhB}u1We e r y p r te u r e t p u j CW)W{P}W'By
f (X, O ) A E ~ ~ G(k) Ob(QUSHM (k)) ~ QUSHM (k)
ej lP
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
r=1
: Spec(k)
Gmk ~
o
~ k
/ Gm~ k
~ QUSHM (k) SHM (Gmk ) ~ p gm ~ k SusT (Qr (X, f ) A) p N r N - {0} X ~ QUSHM (k) SHM (Gmk ) ~ Susp (Qgm (X, f ) A) r T
k
f
/ QUSH (k) ~ M
/ SHM (k)
Rig
Qgm (X)
/ RigSH (k) M
F
SHM (k)
= i j
I
WW)l@#H W$ n#Cp
SHM (K)
f
Sus0 (Qgm (X, f ) T r
f
f
X
cst ) SHM (Gmk ) ~
SHM (k)
K
Rig
Rig
/ RigSH (k) M
X
K
Hom(f
nZ
F
~ k
X , -)
R
/ QUSH (k) SHM (Gm~ ) ~ M k
/ QUSH (k) SHM (Gm~ ) ~ M k
Hom(f
t e yg e s f p t y e yv s y 1q1BPu#}BPPuHBqv 7r e yv s yv t r e y yv r t gv j e s y HPPuHCf 7qBC)xphBl1puP1e e HH
f A Ob(SHM (K))
Hom(f!
X , A)
f Hom(
X, f
!
A)
f! f ! A
f! f !
f (X, O )
f!
/ Spec(k ) o
A
X,
f
X
M
f f !
i
-.
r N - {0}
K
~ Spec(k)
f
A
X (n)
M(Qgeo (X, f )) = Sus0 (Qgeo (X, f ) r T r
1
q
Hom(f!
/ SHM (k) ~
/ SHM (k) ~
X
M
q : Gmk ~
X,
X
X
~ k = k(())
K
) A. Qgm (X, f ) r Gmk ~ K / Spec(k) ~ F
Q RCp
CCp
e f mys PnuuC
E
iYih3 u f s y e e eg e PCB1{HHWq{P e f m v f r u r t e t o PniPP1qW es r te t BWWe e y e HH)lu}HCy v f p e f P@BPtHWj e f m yv f y ev t w tv y e y s p eg p CuPPbD1qPq1HHB{huy e r e ug f t pv s ev s u p y f y e t f e e wv t e r y @WqPCPBuqBC1B#1j CtP}thBnWv W @BnWq#qPn1u e e t eg p v f m ye v f m 1Pn$ e r egg ev t e y y e e p g v y tv c eg w r w t e e tv eg y qh1huC1u1DHqv 5YqCh1~nWHWPqWqP~pe s f e j t g eg e f m u s t j t r y t p y f PWurCbtqDC}BunHtRCRC e f m v f r u r t e t y e y v f p e f PnhP1Pk1fRluH}Co @utHWj e f Pnm yb@PP~BCl1{qhukbt1qPq1HHB{hPule v g e t f e s j t e e yvgv f t y ev t w tv y e y s p eg p y t p te t j CWunl) o {1hPCPuBqu1BWj C@Ppe R7 e r e ug f t pv s ev s u p y f y e t f y f s f eg v y s e tv t p r t e CBP)qPhB'WqxCp~nP1Bs y e r eu s WHubg Ph1hqu1BWj CDqg e r e tv eg ev s u p y f y p e t t BPdCC u t t r t p PPCnu eg v y s e tv y e e yv s g qPq'WCh1CH}BPPuHH o P We f s f ye se y es f r t uPC11HuWBBup 7qP e f Pnm v f r u r t e t y e y s p eg p y y hP1R11HHBhPupe e v s u p y f y p g v v s p y j p o j t huCWu#Wj BDqb{$~urCHWnBy We e f m e v s p y e PnlB$~b~}HCy v f p e f utHClj e vvv s p }HCxlqh$ t ev t j egg e ug f t pv s ev s u p y f y e t f y n1huCClhqCWqPPquBthuCWu#Wj BPPDpe e f m sv pg j y e g gv j e v y PntqlD18phBl1p Pt1e
TA = T T B Ob(T) F (G) B1 A TA F (A) T homT (A, B[n]) TB AG F T TB = T B Ob(T) F (T)ess = T
1
CCp o e yv s eg e f m yg e HBqPBCHq{PnWB ye ye r eu s 'lpC1{1}ubg e r e tv eg ev s u p y f y p @h1hPhuCWBWj B}g e t t BPk s f y e t p y z f y t y y egg f m y t s t y e sv p y y e j u t t e e sv pg j t y vvv e v y t v v r t j y e PC311uCBdBnBx1qh1e n3uBnCq11uquBl13nWH1BqlnBqqb}$qBhCl1U 8b7YYWU8RPHF S Qa V ` X V T S Q I G
TA A, B G nZ T homT (A, B[n]) / homT (F (A), F (B)[n]) B / homT (F (A), F (B)[n])
T hW$r'l#'l Hr### S l P V T ! t5lCP#'t'WBlYln}HW# r"q'l qllWl S V T rDWdtD}#' WW)lh V E r#lb#Wl$WRl##r{P#$#Y#'trq{W@ 'UWl l$r#nB#' C')WhW1)l hn#$r}llY#ld WH # ! CH5lCP#'H'W#BW)WCtdl$r"x WDrW#YlnnWWHWlCP#' r$@rDW@'##tl1hn$r5Dr$rCu'fW W W$#Y 3$x$872
F (G) F T T T RigSHM (k) F T F : T /T
$55@q
C
s f e j t g f p t p v f y e eg eg s e f mvg r u t e j t r y e t WBlbfnqPB@H}1h~qWPnhPp 7@Bn1BRPpfR nqn # t$W#'{'W#BW)WC~Ctl$r! sr V # ! $W''WBlYlnCHW$'"q$##W WW)l78 $x$8f72 t p v f y e eg eg t ev t e w s u eg t p yvgv f t {C#hPB@HHWq{hnWqBCfCPnP HHWu1PB~qnuqqhuq
Susp an (Qrig (X, f ) Acst ) r T
y t p r e yv s yv t f j e p t v v y P~}BPPuHBqkPrCp}1kqBhBC)}Clj o s p y s g p v f r u t f p e v t s p f p e v t eg v pg e j t eg p v f m d'qqi1r R1ChPf 1ChiqPh$so WWq#qPn1u e t f y e e f m e w s u eg s p t ev g PPf1#CCpC#Ch CHWu1PBihnWqn~ y t p s w j v f p e j yv pg u s e s t e y eg v pg y e j t eg p v f m 1un'Pl@luq$luPBuC11qPq$U1lWq#hCu y eg e s e y u s gv e j f p e t eggv f e r s f e j t g t f y 1qCB1BPqP'CRD1hqqPRPWurCbt31e e f m yv f PnBP
Susp an T Susp an (Qrig (X, f ) Acst ) r T B RigSHM (k) / Susp an (Qan (X, f ) Acst ) . r T B1
|y y esj e BBurBy
e r e yv s t f v f r t HHPPuHhP@hPqv
e y yv p u j Cuqqg W'By
t Pf
s p u t t r y {PPP1e t ev c n1hC
e lpP le e ug f t pv s ev s u p j pg e r y s f e p s u t u y e WqPCPBuqqBCWu#1ifuP1Bu1PWCr s p e p g y t g f jg p xqv dqPl
e ygv yg e e r y eg e v e y s ev e s y eg e j t p y yv p t t j e s t e y W$hBHq1BhPH1Wu1hH1BPhClBuqPCl{BuP}1e R) sv t e s p r e t e s p p t yv p s e t f j f p t t v f m e y eg p yvg p f r t e e s qPWuPp 7BnWudB'RPln YPnWhuqqPnWH1BBbg y t p r ygv y e r v t s f g g f m yv f $hBHiPuPCbqv nBPP t py f g ey y f t e v g s j e u v t f e t y s p us r te te te ej p e y u v s pg f tv y y e WuBhBqPqhg nuDp hetWBP@P {uhsqpqguWRUn uiPlpPdCBWWn1}1BlHWj e r f u s sv f e r e s e y u s y f y t e eg p s u t u y fg f egg f t e f mv yv s u j p s p j e r y s t r e s j e y eg y BsPBy1u~hCPCHHuWB1uB71nWHWq'WW@hxCbqhqP}PnqupB1Bl'W~uWj Pf HuBl1)q@pe e f mys PnuuChg v p e p ej e f m y t us}pysgWlPnDB y t p r y P}1e e r eg p yvg p f r t e e s g e f v y s e r t p e r e y f e t qPBhHC1HWBBbDlpC}PBB}11fBHPW)0 nWeHWPfnusPhWj1p 11qPuhC1HWBBbRCCCy t e m v u y eg p yvg p f r t e e s g j t r t e r y e y e e p y e r yv r y s f eg e f m v y tv p y e y yvg e ygv uWpPu#3{'P1h~Pn7h11Buqql$hB o o o je y 1pusPUpWuP'juy $iuH81q8Wu1P11hPCPuB3huCWu#Wj Bg y e r ygv y eg s p e u s r t e t e e ug f t pv s ev s u p y f y p E 6 ))$" nqPB$u r t p v f y y t e gv eg e f m e s f p y t v f hBHqPnuBP1e Pnm y t p r s g g p f r t f e r PCbP}PxW}Pp e g e y yvg e gv j e s e yv s lpC BBqlhBl1puPHPPuH 6 4 ! ( le t ev s u e t p u j y f s f e f m e s t e eg f r e f e s p WqB1PBWn@WP'uuPCPnuBnH{ CDHHWqPRWuP@8 )qUq)s37#0 t Pf iYih3
f# f
u s t u r y e e eg e e y yv uBnC}11{HHWqUPBBqg
f! f ! = f# Th-1 (f )Th(f )f
t v p j wv t e r g g f r Chu#1iunWv Hh}
X Y
E
9" " ! " ! A 855#DC5# B%
k
f# f A
f#
o t v j e s e r e yv s y g u yvgv f y t p y f t y f y y e v qBrpuP{~HPPuHBv {qhuCP1BBr 9 6 4" 2 ( & 3@875310) '%
F f (X, O ) r N - {0} p N
" ! $#
SHparf (Gmk ) ~ M
s f eg pg p PC~qP#
X
S
f
Ho(M)
SHM (Gmk ) ~
f X SHparf (S) M
SHparf (S) M SHM (S)
SHM (S)
~ QUSHM (k) Acst ))
X
A
SHparf (S) M
S
X
/S f: X M(X) = f# S
~ k
o
Rig (Susp (Qgm (X, f ) Acst )) r T
AE
Qrig (X, f ) r
Rig (Susp (Qgeo (X, f ) Acst ) r T
F(Susp (Qgm (X, f ) r T
/ Qan (X, f ) r
Susp an (Qan (X, f ) Acst ). r T
T an
o
o
y e f mv s u 1CuBlH e f mv t t p j e s e f t v y s e g s p e s f PnPWHBBWCPqBu1}Hqv Y1BuBPp t CCp
F
/ Rig (Susp (Qgm (X, f ) Acst )) . r T
~ G(k)
~ QUSHM (k)
T
nZ F
CCp
CCp
G
G
T
T
TA
G
A
F (T)ess F F
F
AG
X
e r t v j e s pg y PChurpuPqpe
eg v y s e tv y e e qPq'WCh1n Cpr e yv s g e f m s e s t f pg g }BPPuHH {PnWuBnH$7 t v j t r g pv t ev y e qBrpCp nxC1hu 1qC r e yv s y HBqPBC}Cv
(1)
s f PBy S8ba7YYWU8RPHF Q V ` X VT S Q I G hW$r'l#rr
h
W#
) r# )f#$r'WH}
9
l$rlr'DrBr'
/ A1 o ~ k o
H}#BW$rl
Gmk ~
luW
iYhq8 g 8
e r j e sv r s f e j p t f j t r y }r1uqDP1Bl$g PdHpe
e s fg j t j s f CBhWWP) e s r t e s e r y s g p z f u1B{uCh}Py y e f mv y ev s u p j y eg y t p WPnqPCB}CP~1qu1BWthDR
d=s
u t t r ye p u jy e P1}WP'uY
C ')Wh##lh)}rr e eg eg p t HHWqdqk0R
W
s p { o p u j W'By eg y t s u rv y t g f t t t s ev t e t quWPqBPPWqBn1f hc 8b7YYWU8RPHF v S Qa V ` X V T S Q I G hW#qC t'}h{PdrPHWqrl 1 Cp ltWB #t@ r7 l$r#nB#' $#uW0h $r#dBWDWB)shqWBHs # U$$U 72
d Gmk ~ s r
t p v f @n#hy s f e j t g eg s e t e r p s e t y t v y fg t j e r y p v f r t t pg e j e f m y s g e e v f y pg y t p PWurCbhDWuPdPquulHCCU1qBbW@CDhPhv kWldCu'qtl8BhPBs@R
o y y s y e g t e e f m v t f s e s u r v y t j e e t w p s e y t 1eHBqvPBCHHPhn1}1PnqPWuWlH{P1}u~BCP o s f PC e | f r y p u j p1P'By ys e r p sWeWPHWu@~BBiPnqc j pg e s e e v f m v y tv e t eg p v f m n1hqPn1u y 1e e f m y t g e p PnCh1p s p u t t RPr y1e e r eg p rf t e s f j f s y pg s f e y s e t P@hPHPH@uPuruBdRPC~Bu1qv ) e f m y e u rv y eg s e v w s f y t y f c t tv PnWhRWPP)CnP7PqBy s p le v By s p WlCq}1BD{BBiPnm s e j pg e s e e v f f C s e y f y f e t g e f m v t g p rf t s f e j t g t f y WBC)PPBW)C ~PnhCfCCPRHPWurCbP1e Pnm e f u t t PPtCrnpu rb# P {H1BCWPY )Y tq$ e w s u y sv s f y e g p r f t e s f j f s y p g s f e g v y s e t v y PB@hPRPC}tBPuruBdPC{qPq'WChpe e e sv pgg s sv lpP Prb RuqhCBC)o )Y @hC$ e sv s j u f e t Bqul1RW)kRY y e y yvg y e t z p y p u j 11BuqqP#1P'By 1r ye le s f ygv j e v t P)R$iur1p PnCy We
y e yv s y eg e f m s e wv s u j t }BPPuHH1q{Pn1hB1n~r f pg gv y ev s u p j e r e j t eg p v f m u e t f y e g f m s e s t s f y e y v f p e f $hWhuCWu#WHHl1h#hPn}P1qv UC{1BuCCHRPC)11HHBi}utHWj s f e j t g eg e f m sv pg j y e g P1Brb}h}Pn~hqW{1
F
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
~ (p, q) N2 (s, t) (N - {0})2 (A, B) E2 X Y k g (Y, O ) Susp (-) = (Susp cst ) Sus0 (-) T T T SHM (-) (Sus1 cst )p T F p=0 q=0 p p-q q=0 Sus0 (T )p T A1 A1 cst /Gm cst T j i ~ k Y k Ak Gmk i+j =p ~ ~ ~
X[T, T -1, U, U -1 , V, W ] X[T, T -1, V, V -1 , W ] X[T, T -1, U, U -1 , V ] = = rd - U r T f ) r - T f, V d - W U ) (V (W (W r - T f )
homRigSHM (k) (F(Sus0 (Qgm (X, f ) Acst )), F(Sus0 (Qgm (Y, g) Bcst ))). T r T s
homRigSHM (k) (F(Susp (Qgm (X, f ) Acst )), F(Susq (Qgm (Y, g) Bcst ))) r s T T
homSHM (Gmk ) (Sus0 (Qgm (X, f ) Acst ), Sus0 (Qgm (Y, g) Bcst )) T r T s ~
homSHM (Gmk ) (Susp (Qgm (X, f ) Acst ), Susq (Qgm (Y, g) Bcst )) r s ~ T T
Qgm (X, f ) Acst r
~ A1 = Spec(k[T ]) ~ k
C~ #}Du}#BW$Dh@#BW$r#
e t p v f y t v v y s pg e r t ev e eg f r s u t }un#hkqBhnCBD{nWqtC C}H1hklWunhv
Qgm (X[U, U -1 ], U r f ) Acst rd
Qgm (X, f ) = X[T, V ]/V r - f.T r
Qgm (X, f ) r
j
/ Qgm (X, f ) o r
i
Qgm (X[U, U -1 ], U r f ) rd
X
Qgm (X, f )[V, V -1 ]. r
F
Sus0 (-) T
q
f (X, O ) Susp cst = T
q-p p q Susp ( ) T Y
s
r
9
$x$8
8
Qgm (X, f ) r
Gmk ~
homSHM (Qgm (X,f )) (Acst , [Qgm (Y, g) Gmk Qgm (X, f )] Bcst ) ~ s r r
homSHM (X) (Acst , i j [Qgm (Y, g) Gmk Qgm (X, f )] Bcst ) ~ r s
homSHM (Gmk ) (Qgm (X, f ) Acst , Qgm (Y, g) Bcst ) r s ~
(1)
(2)
(2)
(pr# , pr )
pr
n Cp l#P#' '##
Cp
CC Cp
E
e w s j yv r t v p fg p e r y lululuqkqBq#~pe PnuqPb{1huWBlP~l{qPfqB unWB1uP~nWHWCChCCB e f m y v f y t e v s u e t j t r e e v t w t vp t e y u s e r t e e f m v g 7HutnB@qqU1blu88Pdpt@nBPt v f p s t t y ygv yv g e f y v t w e | e r y p t y e y p u jy y e f r te te te CWP'uthg WBn11hv l h s 0Chur1Bsdr e r t v j e y pg e y t v p t y eg p s e yvgv f t e t v v t w u e r y t e y f p y p u j y e r e p s pv r t f e Chu#'Pxh~'WBhqBHCl8b qBhPWn )Y {PWBd5W'BPHH'qPHH}Clj p s e s u rv y t j t u1B1CldR o o e t z p p u j HH1P'uy eg ev j y y qhWup e y yvg e rf t z uhRPp PnPnUP1Brbxqg t f e v f m s f e j t g e v hc egg e s f p t f e p s s p s p e f e s wv e s e v f m t v j f s y t j e t r tv y e y t s f p y f t pg e qh1BB#1tuuq7u{P~B1hH1BP}PnkqBlBupl@Pf qB1)uPPq1lj s f e r e s e r t j e y f r t v v y s pg y t p r e u t t r egg e j e e f g p t p t v v s j y e r e t f v p s r f t PCr1 CpHuPDWHdClkt CsqBhnCBDP{1PPPqh1l}RPCqqBPhulu1{Ph'CR
r D : SmAfnd/k D d (q , q ) / Sch/k D(X) k k
i i i
s p e yv v r e Bqqt 'WP11q)l1uhC1HPuPqPuPbtYuWuCp CPWbWBCUr1q~hBuD}Clj g p s u t u y p j e g s f j e s v r t e f s p t f s v t s f g eo v y y f u s y p y t t y f t y v g e f y g v p f e p s pv r eg s e yvgv r z f y gv e s j tv p t j t y s f y f y y e v j e y e w s u y eg g f r evg j H}uCRq1qhuf HB)q#ulq#WWe 1PC3r'BBr W{ P WRn P HWuWBxhthPPH$) C@hq'By E 6 4 ! ( pg e r e j t e f m u y t j e t f y e v j e PlWPn1BClPPpnW1C Bqqr PnB1Pu)n 53BhuB1u P ChuiBC)uPRU YqUq)d37#0 e yv v e f m e y w | g y t p y ev p s e y e s t v v y s p
s r s
iYih3 e e f r t v p s y t u r pg e r t w pg s e g f y t 1q P@HHWP )) kqB#'BC}1Hq{PktWuiClj f e s f e j eg eg pg e j e r e f e s e t f s f 1P1BrhhCPWlRWuPPPPn e g e vyy f p j t lpC {Bur o e e f mvg g t f m y s g eg v y s e tv y e y e yv s fPnhPp 50C UCu'qqPhB'Wq}1t1HBqPBC} 1WP1nHWi8R y e j e r e u y j pg s y egg e s f p t y t v p s g y t p s y e r e u y j pg s p v f r tv y e e yv s e j e f m t e e pv r u v p s e wv s u s f e j eg e hqWuPBxChu#ub'B1RWBC)}Cl@iq1HPPuHC}CPWlPnnWH1B#}Hq~u1hB1nP1BrhR Q Cp
q p j homSHM (X) (Acst , q (Z[V, V -1 ] Bcst )) / p i i j / i j 2 -1 q (Z Bcst ) = Z[V, V ] Bcst Ob(SHM (X[V, V -1 ])) / homSH (X) (Acst , i j (Z[V, V -1 ] Bcst )) . M
u j u s y t v p j wv t e rv y eg g f r y e y yvg t y y egg e s evg f jv s p t l1uPRqBWhBn1qR1qqPq 11BBhnBRhqW11hqPlBBx)
j
y p u j CWP'uy
Z[V, V -1 ]
pZ id X
t e eg v y y C1HWqPHhBpe le t e rv u e y yvg e yv s t r e y yv r t nWWuhR}BPPuHPf 7~nBR
X k Y [R]/(Rs - ~ f ) [V, V -1 ]. g
e r e y y pg j p ulDqg
ePrsPfWeurj$gs1hvH1eBsP~qPBChur1pBDP p e eg s f y t v j e s pg y t v p j wv t e rv y eg g f r qBWhBn1q1qhPPs o p u j WP'uy qg e e t t BPdR le
X k Y [R]/(Rs - ~
e g p u y C11e
p u j WP'uy
e
E
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
r = sd
~ k
o
e f m g e s ev t C8WB1hunWe 5g
" ! $#
~ Z k /X Z f T ) = X[V, V -1 ]
X k Y [R]/(Rd - (f /g)) h R pZ : ~ X gm gm -1 gm -1 Qr (X, f ) Gmk Qr (Y, g) = Z[V, V ] Qr (X, f ) = X[T, T , V ]/(V p - ~ Qgm (X, f ) Gmk Qgm (Y, g) r ~ r
X k Y [T, T -1 , V, W ] X k Y [V, V -1 , W, W -1 ] ~ ~ = ds - f T, W s - gT ) (V (gV ds - f W s )
f ) [V, V -1 , W, W -1 ]/(V d - RW ) = g
~ k
X[V, V -1 ]
Z[V, V -1 ]
k
pZ X
o
homSHM (X[V,V -1 ]) (Acst , Z[V, V -1 ] Bcst )
homSHM (X) (Acst , q (Z[V, V -1 ] Bcst ))
y p u j 1P'By
/ homSH (X) (Acst , i j (Z[V, V -1 ] Bcst )) . M
v s j y e hul1u 1#C dr e yv s g y e t e HPPuHC}CP 5P11unWr e r y e v f r u r y e f t y y eg p jv s e y e j w y eg RBhP~1BPBRnB1q1qu1R1P'1qs
(2)
o
e r t ev y u s p j e p s pv r t f p t P1hB1uW~}HuCqP}R
X
X[V, V -1 ]
Qgm (X, f ) Gmk Qgm (Y, g) r ~ s
q
j
j
homSHM (X[V,V -1 ]) (Acst , Z[V, V -1 ] Bcst ).
/ X[V ] o
/ X[V ] o
/ Z[V ] o
* X.
p
/ X[V, V -1 ] .
X
X
Z
X
k
Spec((X, O )) q
e yv s t f t ev t e t p y j t HBqPBCHHPkPnWqBPPlC tlC Cp{CBnHCls n Cp t v j t r ChurCpp
e yv s yv t f v t s f HBqPBCHBqPhPBbg
o
e r e p s pv r eg y t s u rv y t ~HH'@quWB CC Cp
y t B
pg sv hCb
e v vg j pg s p u t t r y BhHhqWHq~PPPpe
Spec(A)U Ob(Spec(A)\(Sm/k))
s{iPPhv p v f r t e p g y Chv 5x1e
e
e f m t ev g PnC1h t v j e s p Chur1pBHqg y e s wv y y s y ev g y e r yv t f y e v y e r e yv s u s eg t p v f '1uWquCuR1qChCCBR1{PPDbH1uiBHDHBqPBC}1B}q~C#hPBy e r eyse t }'WChv o f p e j yv pg u s f r t v jv s y e s p e y yvg e rf t z 1lu$luP}DChurBupB8CuhRP#p t Pf v hc % 8b7YYWU8RPHF S Qa V ` X V T S Q I G
K (uX ) X = Spm(A) k diag u a L
##W drC
/ Sm/k
$W1W' ldWW)lh #) lW)#'1'@$rC#'sP{'lh #
/ SmAfnd/k diag u K
W$)#'H'P llC$1W#$rW) S rf'uWl$BYD Y C D
L
P
WB5)Hr9 ##Wq@rC 8 lC$1Wh$rlY S l# Y'1'h$r#nB#'x)qPi'lq## r'uWl$WY C Y B
K Sm/k
G 5
y e v j e s y t v p s WqBl1puPChu#'Pw y eg e f m v y tv p v f p e j yv pg u s e r y eg v pg y e j t eg p v f m u y eg t e s e y u s ygv s evg f jv s p t e t p v f y v f p e j yv p o hPnBqi1lu$luPH@hh$l1h#hPn{1qRnWCB1B{qqY11hqPWqu 7W pW#n dC#hPBx1WB$g u s y e r y e j e sv r y e p v y eg t BPt1D1ur1uq@1Cq@1q@nBy le y ev yv j y t v p t v f p t e e sv p s t j e f m t v t e p p s eg t WhBqCP'tqB#'Ph{n1}1Bq'BnClPn0ChuC1BB#s'WbhR
diag : PreShv(Sm/D , M) diag diag / PreShv(SmAfnd/k, M) . diag
y s f e j t g y e r e j f p e t v v y j s p t e yv f r tv e y s f e j t g y e uP1Blb'nHfqBhBC)}ClnWBqPPhC ptWn {uP1Blb
diag : SmAfnd/k / Sm/D et diag : SmAfnd/k / Sm/D .
v f p t pv r y s f e j t g y e s f p v f e r t ev t CBuWBlbBukWnWqBPk
y s f e j t g y eg j e p t p y j t uWBlbh1nuC)}Clk eg f j eg ev j y y qPPCl~hRhWup e y yvg e rf t z uhRHP#p
k - D D : Sm/D X = Spm(A) / Sm/D (id : Spec(A) = Spec(A), X) et - D D : Sm/D / Sm/D
t f
je 1p
Cp o o t e rv u g p t pv r s f e j t g t r e y yv r t o e y yvg p u j nWPqWqPP1BlbPf 7@BC)kRCuhDWP'uy t Pf e e y yvg e rf t z W{uhRP#p y eg f j y e r ev s u p j pg y hPlRqu1BWHq1e e f m t v j e y f c e r f u r eg sv y t g e p PnWhPQ dqBl1By C Y }PPPW{qhCb~Ch1p
Y D(X) diag : SmAfnd/k k
j
CC Cp y p u j y e r y e p s pv r y e CWP'u{1HH'r ev j y y p v f m v fg e qlBBCPnPqWWj
/ Sch/k ~ D : SmAfnd/k DiaSch
le
e hvusCWuu#pWjkg{ytpPr}eHHp'sqvPrhtfvyhvpp R3 p p t t o t f e v f m s f e j t g e P0RPnfWBlbshg ev j y y hWup e rf t z RP#p
k X Spec((X, O))
t e eg p u e t t nWHWqnWquPR
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
k
D : SmAfnd/k
SmAfnd/k SmRig/k
diag : PreShv(Sm/D , M)
diag : PreShv(Sm/D, M)
SmAf/k
9
cpl : SmAf/k
u
/ Sch/k
D
~ X = Spec
j
/ k o
/D o
u
(X, O) (X, O )
/ r Rig K .
i
/ PreShv(SmAfnd/k, M)
i
/ PreShv(SmAfnd/k, M)
Sm/D
/ Sm/D
D
~ k.
u
X
(X/ /())
9
[r Rig K](X) = [Rig K](X)
[diag u K](X) = [(uX ) K](D (X)) =
Spm(A)U an Spm(A)\(Sm/k)
Colim
/ cpl L .
K(U ). Colim Sm/D (X) K(U ). u K a : SmAf/k Sm/D (X) / Sm/k (Y, X) X r :
hom(Spec(A), U )
hom(Spm(A), U an )
t v v y s r ChCxChuiBC)uP t v j e v pg e r y s g p eg f j u r Q e e s t e t v p j wv t e r qBl1p PqqP}'qsqPCl1PHH il}C Cp BunWqBWhBnWPqv Q Cp s p es f 1B{uBPp t n Cp
CC Cp
n Cp
8
$x$8f72
Cp CC Cp CC Cp
CC Cp
o
t f e v f P@Pnm
t p v f y e w s u eg u j t t u t p t e tv p f e t {n#qPB~H1BCWPu~h~lP1nWuCq1kR
Rdiag : SHM (D) HoA1 -st Spect (PreShv(Sm/D, M)) T / Ho Spect (PreShv(SmAfnd/k, M)) . diag T
t v v y s s v e v s r o e u v s u r s f e j t g t r e j t e y v v g v f r u r t f p eo v bqqBhBC)uP Y qCbBhu{huWPPWurCbff 7{WWuple 5hPq1qPt s p f p e v t v f p e j yv pg u s e r y eg v pg y e j t eg p v f m u y e r t y y t p s i1qPs1WB$WBPP{1qPh$1WWq#qPn1~1P{nC~unuPw t e eg p y y e e s t e y eg p y y e j t eg p v f m n1}1hu{uWpHuBnW~hu~1WWq#qPn1u y eg e f m u t t r t p f p e v t s p f p e v v f p e j yv pg u s e s t e y eg v pg y e j t eg p v f m u y eg t e e sv pg j e s e y u s s f e j t g e hPnPn' U1ChD0qPt luq$luPuBn1Dhh$@l1hqPn1D1qDn1}1BqlfuWB1uPHPWurCb
diag : Spect (PreShv(Sm/D, M)) T A1 / Spect (PreShv(SmAfnd/k, M)) . diag T
CC Cp
y e s j e y e r y ev s u p j e s t e s f e j t g t r e y yv r t 1uBrB@1qBC1B#1{BuC1P1BrbPf 7{nBqPR
Spectan (PreShv(SmRig/k, M)). T
CC Cp
RigSHM (k) B1
e j f p e t e g g v f e r y e j t e g p v f m r e v f y e t f po t e e t 'nDkWqqhPR P1lWq#hCu @BhP{PHCP1W)W e e t eg p v f m u y e y eg w r e r ev s u p j e e j e r e f mv ev s u p j p eg p y eg p j qBnWq#qPn1HpH1qWHkfqu1BWqBBllqkPnPuHPfhuCWu#WUChukhWhg e v j e s y eg w r e CqBrpuPH1qWPHkr e e y yv luhg es fj fsy e t uPBluBPf r sv t f f e t t p s w j y qPPDW)tC1n'Plxpe e HH o o e rf t z RPp y e j w e r e y y pg j pg t p v f y eg p j P'1 ul}n#qPBRqWqg t Pf je 1Cp e v j e s e s f j f s y pg e r rg e w y f Chur1pBRuPBluBDRPW o e r t v p yvg p j g s p e f t e eg p j qBBhWqqP1BCfq1qg e v j e s e s f j f s y p y e qBl1pBPkuPBluBusr ev s u p j pg v t f t qBCWu#1hPPDR t e e sv pg j n1}1B qlHp o t s evg f jv s p t t v j e y f y pg e r f u r f p u v w p t g e f n11hqPWqufnl P sqBl1By B}qPWPf~P{ 5Pnm y t p Pr e r t p s w j eg w r e CuWDqWHtqg je 1Cp Cb e e s yv y e g f m yv f t p s w j t e e v j e s y tPPuHBR13hv nBPPRCuWnWHWChurpuPR1e y fg e 1qPiR v f p e j yv pg u s e r eg v pg e j t eg p v f m u e t f t ev t v f p e j yv pg u s e 1lu$luPxxqPq$qWWq#qPn1iPnWqBCYP1WB$WBir
cst
v f p e j yv pg u s e r e yv s yv t f v f r u r t e t 1WB$WBPHBqPBCHBqfPiPP11R e y yvg e rf t z BBhHP#p
k
e
y t p r y s f eg p P'P1h}e o f p e j yv pg u s y f y eg e wv t e rv e yv s yv e luq$luP ChhBnWHBqPBCHBql
Spm(A) M
t f s f t p r t yess PP)CPCB1uBClj
iYih3 o o v f p e t t y f y y eg e j e y e s y e rf t z p y e s w g 1Pp C1qurB1uRP#1uPWChp o v pg f r eg f j u r t v j e v e r t j e y p e s w g h$PhldqBl1p PltUuP1qp
~hC1fC{BP1huWbPh{qPq}u{qBlb1r e eg p u f e s f ev s ug tv ev t w tv e s t e r y t v j t g y e e r e yv s t f e e t }BPPuHPlYln s f y g p s t e pg e t z p y PBBCPWqCCP#pe e f Pnm e t r t v ug j pg e r u vg p y s e v t g e r t ev s t v j e v e s wv e s p PPf 7dqBWh}Cl@PBhhB'WChPf n1h#uPqBl1p PuWq}1B{U
(U/ /) hom(Spec(A ), U ) k
y egg e s f p t y t v j e v y e r y s g p eg f j u r e e s t e t v p j wv t e r hqWuPBChur1p PuCh{hPCl#n tlC CpBunWkqBWhBnWPqv
Spm(A)(U/ Ob(Spm(A)\(SmAf/k )) /)
e v vg j pg s p u t t r y uiHqhClH~PPC1e
Spec(A )U Ob(Spec(A )\(SmAf/k ))
t v j e s pg s p v f r t qBl1puPfq~hPhv e r e y s e tv e p g y PBu1qChv pe
e
e f m t ev g PnnWq7P y e s wv y y s y ev g y e r yv t f y e v y e r e yv s u s eg t p v f '1uWquCu1qChCCB1PPPDbH1uiBDHBqPBCHWuPHh~n#hy o f p e j yv pg u s f r t v jv s y e s p e y yvg e rf t z 1WB$WB}P}ChurBupBUCuhRP#p t Pf v iCc %
X = Spm(A) k
U
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
" ! $#
(uX )
[cpl L](X)
[diag u aL](X) = [(uX ) aL](D(X)) =
B1
k
hom(Spf(A ), U/ /)
1
Colim
B1 k
u a L Sm/D(X)
diag A1 D
o
hom(Spm(A), B1 ) = A = hom(Spec(A ), A1 ). k k
t v p j wv t e r g e qB#1hBnWv 5} y eg e y t r v f p e j yv pg u s e r e yv s yv t r e y yv r t htWB1e 7Hluq$luPPHPPuHBdPf 7BC)R
B1 k B1 k
B1 B1 k k
diag GmD
A1 D GmD
cst
diag
A1 k Gmk
y eg v pg y e j t eg p v f m u y eg e s e y u s hPq$l1hqPn1huWB1uP
e HH
cst cst
y p j eg y t p r e f m e s f p y t e yv s yv e 1qPC CpPn~Bu1e HBqPBC}Cl
Cp s p es f 1B{uBPp t CC Cp
A1
diag TD A1 cst /GmZ cst Z r diag TD
A Ob(M)
PreShv(SmAf/Z, M) r cpl T T qc PreShv(SmAfnd/k, M) Nis
diag T
Spect T (PreShv(SmAfnd/k, M)) diag
Spectqc (PreShv(SmRig/k, M)) T L=
r
T an T qc -
Spect T (PreShv(SmAfnd/k, M)) diag
cst cst
/ B1 cst /B1 diag TD k k cpl Tk T = a T T
hom(Spm(A), (U/ ). /)
k k
Colim
K(U ).
B1 U
.
B1 A
diag
Sm/D(X) / Sm/k
/ U A
L(U ).
U
cst
k
a L
t e rv u gv p f j u s s p j eg t p yvgv f t eg v pg e j t eg p v f m u e t f y nWWiu#uPtHWuBW@qCBqhuqqPq$lWq#hC{PP1e
E0 (Qgm (X, f )) r / Rig E0 (Qan (X, f )) r / F0 (Qan (X, f )) r / F0 (Qrig (X, f )) r
e u y j pg e f m s e f s e r j t r p s z f y g e w s u eg s p eg v pg e j t eg p v f m u e t f y 1nHWHq~PnWCPuP{~CHuBP#qP HHWu1PB~qqPq$lWq#hC{PP1e o
F0 (Qan (X, f )) r
w.e.
e j w pg t p s P'1 1n'Pw
y 1e
e f m u t t r t p Pn@Pn'
eg v pg e j t eg p v f m u e t f y e e f m s e s t e r j t r z f y g ChPq$WWq#qPn1~PP1#n1qpdC1BuCCH{~HBU
E0 (Qgm (X, f )) r
y t p r e t p v f y e u y j pg P5Hun#ho~WBC)Hl}e e r t v p jvg p s p o Chu#1qqPP v t w u r e y e j w p hPWB1 Ct'1 8C y t p r t ps PnuPw t e eg p nWHWqP'py j e p eg pv v s qhuB t e eg p C1HWqP'py o e v j e s t v p s w j e t qBl1puPsChu#'PlHPPf v c iCU y t p r t ps Pn'Pw te nWH o o j e p eg pv v s ~hqqBu eg p hPuy t e eg p y e v j e s t v p s w j e t n1}1hu~qBrpuPChu#'Pl{PPf v hc e eg eg sv rC CH}1h{qhC$ nuqqr PChPqBu1hnWq'l1 Cp t p yv v s f y eg v y s e tv t ev p u y e yv s y eg e f m sv e v p y e s gv y ev s u p j e r e j t eg p v f m u e t f y 1HBqPBCHHPhPnxqhu1uqUhuCWBWlWq#hCPPx1e Cx1BuCCHxP)PnxqW)Pp E u e f m s e s t s f e f m y t g e p e j t eg p v f m u e t f y WWq#qPn1{P1e e f m e s t t e s wv e s pg y t p y ev s p y s f ev y fg t e e f e s pg e yv v r t PnuBnC}uWqHWuP@C11qBB'PWqhPWCPWuPtqRhfR
Ga Q ` G S Qa V ` X V T S Q I Pn87CU 8b7YYWU8RPF
e y e w s u y e r evg j y pg e sv f r u r t e C C@l~ C~1HWu1PB}P Hhq'BHuqP1PC1H}Clj y t s t t v j e y p g e r t w p g f m e u t t r p s e y eo t e y e f m v t j e y e s v p t v v g u s e r y f g f j f p e t e e y f e s f e oCBuCCq P Chur1BiPsqie nWPPCs'WBfPHlD1CPP'B@1uhhHqh1BHqP0PCl1)DnWH11BWP g p e v y y e j u t e w s u f r e f e s p t v j e y f y e e j y t p r e u t t r p s e y e w s u f r e f e s p HBhu1W1Pf P HWu1PBHP1Bf87qBl1By BdBBlW{P1PPPf'WB CH1BCWPuxP}CPWuPU
: Rig E
~PPuHBqnWH1PnqPCP1pe e e s yv t e e f mv t t p j y s f e j t g eg e f m e s t v j e WBlbq{Pn{BuCCH7l1Y y t p Pr o e r eg p y t v p yvg p j xhPuqB#uhWhg e t f y e e f m e s z e w s u PPpWPhCpPniHB#p C C0HWu1PB g s f p t e yv s yv t f v f r u r t 1eBPu#qHBqPBCHBqhPR o 6 4 ! ( eg v t e w s u eg s p g p j qyhvc Cn P HWu1PBsh1hg t e eg p y y e e r e s e s ev t s e r e n1}1hu1H11q0sBtWHd1hB1s qUq) 37#0 &$x$8 f7 hWr{')WhW1 Yrl WPhCp u# q ! W$rl#"~Bl' #{# d
/ Rdiag Ri Li Rj u E . / LE Ho Spect T (PreShv(SmAfnd/k, M)) diag T B1 E Rig B1 Rig E B1 L
e r e jw e t f y P~'1 Ppe R PhCp
e u y j pg s e s g y s g p f e t D1nHWHq1}ub'q1kR
Rj u E / Ri Li Rj u E .
o o y t p r y e f mv s u | y y e s j e P1PnBulHBur1)By e r e yv s t f p t s p e s f y e s j e P~HBqPBCHDk~Bup DPWBur1)By e r e yv s yv t f e HPPuHBHHWj s e s u rv y t j f e t 1B1Cl1 t e rv u e yv s e C1n1~}BPPuH~hg o nuqqhuh11PnBulH{BurBy t p yvgv f t y e f mv s u | y y e s j e e r e yv s yv t f p t e e HPPuHBkhC P i}H1hg o eg s p t p s w j t e e v j e s e f mv s u | y e s j e qxPn'PWnWHWChur1pB{PnquBW}B{Bur1)By t Pf v hc euivf@~n y pg pv
E = (En )nN Ob Spect (PreShv(Sm/k, M)) T diag TD / diag TD r Rig E diag u E B1 T r T an = diag TD r Rig E diag u E T L : SHM (k) SHM (D) RigSHM (k) / RigSH (k) M
s e t t j v t s f g e w s u e t eggv f e r y e j t eg p v f m PfWurjCbgffRtrhPuPb~n P HHWuWB~PWC CphWqhqPR l1hqPn1u 7r o o eeBnt1hgpqvPfnm1unt1}1PnmqPWjy1eHv1hg e e e f v t t p f pj t e eg p y y e y e r e tv eg ev s u p y f y p n1}1hu'lpCPqWqPhuCWBWj B@U WHWhW'~5lW$tlH$r#nuu lh@PWr#lClW Ck#'1q
SHM (k)
u
ChCp
W'8
$x$8 f7
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
)
F
diag TD
A
1
Sus0 (Qgm (Y, g) Bcst ) T s
B
1
F 0 (A, -)
Spect (PreShv(Sm/Gmk , M)) ~ T 1
r Rig E
RigSHM (k)
/ SHM (D )
s
diag u E
Rig E0 (Qrig (X, f )) r Rig E0 (Qan (X, f )) r
/ E = (En )nN
r
B
i i j
F
/ Rig E0 (Qan (X, f )) r
/ SHM (D)
/ Rdiag Rj u E
B1
F = (Fn )nN SpectT an (PreShv(SmRig/k, M))
M
Rdiag
/ F0 (Qrig (X, f )) r
Rig E
/ Ho Spect (PreShv(SmAfnd/k, M)) diag T
/ F0 (Qrig (X, f )), r
/ F0 (Qan (X, f )) r Ho Spect T (PreShv(SmAfnd/k, M)) diag / F0 (Qan (X, f )) . r / F0 B1 A1 E
nhCp
ChCp
ChCp
o o ghB#'tHClRHH'qPP'hDR v p f j e p s pv r t f y s g p p t e rf t z PHP#p qhWBup eg ev j y y p u j H1P'uy t f e v f PD7Pnm s f e j t g t r t e eg p u e y yv r t PWurCbPf 7dC1HWqC1qnBR We y s g p e y t 'qsnR s ev j y y p v p 1hWBufh$g o e v f m y p u j {PnCWP'uy e r e p s pv r e iHH'qhg e t ye t z p y p u jy y e r eu s uPt R11P1P'Bkx1}ubg e r o e tv eg ev s u p y f y pg e t v y u PqWqP@qu1BWj CqHPqB1r e f m y t g e p p evg j y pg e r ev s p e r t j e y pg e e y y p t Pn{qW)PP kqhCP'uqBBHClkk}BBChR E u s f ) eg y t p qxPr l f CB e r t v p jvg p y w s p e yv s yv t f v f r tv egg f m yv f eg v pg e j t eg p v f m u e t f y e e p s pv PChu#WqPPB~}BPPuHBqkPhPh{qh1e UCuPPbdqPq$lWq#hC{PP1#1qpHHH'r e j w pg e f m e s z p v f m t v v y s pg e r t p t e tv p eg f j u r p g f y u s e gv p f j t e e sv pg j y P' PnHBPn@ P kChuiBC)uP{nWBnqqPW1#uhPB1unpiu#'tHWnWHWuhqW1e
A Ob(M) Li Rj (E)|Qgm (X,f ) r
0
o e f mv t t p j e yv s PnqPW~}BPPuH h{PnhuCPU@c DBqhCBClj e p s pv r e eg v y s e tv y HH'q@UqPhB'Wqpe eg e f m v p y t Q e sv pgg s eg s p q}x ChB'WHHhv 5v 1qBWh}Clkk}BPPuHkuBrBfC0CB1uulWqhqHr t v y s e g e r g p u r g t e t v ug j e r e yv s f r e s j e y f p j t r r t y e s s j f evgv f e jw p '1 U
c 2 Qgm (X, f ) r
j
y t ev y u s p j y u s s p j e gv p f j u s s p j t f p t WqBBW1uuWiu#'tHWuuWqP}qR vyy f p ey t BBBC){R p u jy e W'Bh
A = (X, O)
e t p v f y e u y j pg Bn#qPB1nHWq}e o y t p r e s yv y e y f y y e v j Q e u y j pg y t p r e j w e s wv t s e r pg e y e w s u y eg s p PPPuHBpxBBr W#HhCp@1nHW@PP' B1hB1@ P ln P 1HWu1PB1qx
E0 (Qgm (X, f )) r / Rig E0 (Qan (X, f )) r / Rig E0 (Qrig (X, f )) r
)
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
Rig E0 (Qrig (X, f )) r
Ri Li Rj (uQrig (X,f ) ) E r
Rj (uQrig (X,f ) ) E r
k
(uQrig (X,f ) ) E r
0
(D(Qrig (X, f ))) r
0
(D(Qrig (X, f ))). r
0
(D (Qrig (X, f ))) r
(A k k )[V ] ~ / /() (V r - f ) D (Qrig (X, f )) = r
pse t e t 'WBPhhg R
ePfnm8ntWeHWeu#vr1uHHqvxeivusWC~R p w u t o pg e r e s j e y eg y qhBur1)Bqqpe e s w g BWqp
k
e u y j pg s e s u rv y t j e u t e p s y e t WBC)}ClH1B1ClH{PWHu1k
" ! $#
Dr (X, f ) X
~ SmAf/k
~ Sm/k
Q (X) / SmAfnd/k
sch
E0 (Qgm (X, f )) r
Dr (X, f ) = D (Qrig (X, f )) = Dr (X, f )[1/] r
(1)
~ k
D(Qrig (X, f )) r
Q = Q k k (X) Dr (X, f )
c j
X
~ Q : SmAf/k
/ Rig E0 (Qan (X, f )) r
i j = c i j
/ Qgm (X, f ) o r
/ Dr (X, f ) o
~ Q = Q k k
c i i
k
/ Sm/k
Q
rig
(X) X. X
/ i j c
~ SmAf/k
(uQrig (X,f ) ) E r
/ Li Rj (E)|D (X,f ) r
/ Rig E0 (Qrig (X, f )) r
+
Rig E0 (Qrig (X, f )) O r
/ F0 (Qrig (X, f )) . r
k
0
(Dr (X, f ))
0
(1) 0 (A, -)
(X)
i
Ho(M)
q : X
nhCp
y t B
ChCp
Q HhCp
o e rtqBq1Bs0qgqes1n 5dkPPCuC v pg e p r e uy g s p u t t rs je p 1C$ s p u jv r t slqv y e y yvg y p u j 1BuqqHW'By e r e p s p v r t sHH'Pf 7r eBC)vtR y y r s f e j t g f r t v j f s y t j pg t e e wv s y t g e p Q e u y j pg s e v p s e e y e s y f t g WBlbdqBluBWtnWHWCWqBPCh1Pp rRCpt1nHClt1Bh'B@RB1uPY E u e w s u f r e y ev s u p j e r e j t eg p v f m u e t f y #Ch C}H1BCWPufPl1qBCWu#1@{l1hqPn1{Ppe Pnh$n1Bnq~hPCl#uhPB1u{ e f m v pg f r t p t e tv p eg f j u r p g f y u s e
Gmk ~ homRigSHM (k) ((Susp (Qrig (X) A)), E). T R F N
t v p jvg g y s g p e f s e s t qBWhPp 5'q~PCBulufR) We s f PC ys f e j t g y e uWBlb1r e v p s e y t j eggv pg pg e f mvg p v fg t pg e j s f eg v y s e tv y e egg e s f p t t v p s g y t p s e e j e f m s e s t e e y e s g qBuWBCl@qhqH$Dq{CqPP8PhkCPq1lPCnChhBu1qp{hqWuPBChu#ub'B~ulW@PnWuBnH}@B1u o
Rq AE / Rq Rig . ~ X Ob(Sm/k)
s g y t p s e t f v y tv p t ev t b'B{PBhRnWqBCkRY
y e f mv s u | y y e s j e PnquBW}B1uBl1)y
s f t ps PC#1n'Pw s p p v f Pnm o
t e eg p nWH1hPuy
j e p eg pv v s hqqBu
o e u y j pg e s w rv y t j t e t z p p u j WBC)HWqB1ClkPCP#}WP'uy t f o t e eg p y t v p s w j e t nWHWqP'p}qBuPlPf v c hn e t ev nWqB o o e r t ps Cuw t e eg p y e t p s w j t e e v j e s e s j e nWHWqP'pWn'PlC1HWqBl1pBPRBurBy t Pf v hc
E T QUSHM (k) ~ k ~ QUSHM (k) Rig E
Rig
y e u y j v f e r y eg e s t e e yv s yv t f s e v v WBC)HlkWP1quBn1RHBqPBC}CkP1qPle j t r f pg g C$ y ev s u p j e r e j t eg p v f m g g f r huCWu#Wkl1hqPn1u iqPH s f e j t g eg j e p e rv j tf PWurCbq1Cfl5Hlj e f Pnm s e s t e r j t r z f y gv e s fg j t j s f e f m s p e s pg sv WuBnC}t@CHB8qPuPqlCll P HPn'Wuq$ e e s yv t e e f mv t t p {PPuHBqRC1HWPnPPWj y 1e e s j e y eg e y e w s u y eg s p uBl1)DhPC Ctlt C~1HWu1PB1qx {BhudkqBhBC)}CltPPWurCb{qg s p e v s r e t v v y j e r s f e j t g e u t v y u r p t g PChB1qC q
Rq q Rq Rdiag Li Rj Lu (E) Rig E RigSHM (k) q R v Li Rj (E) Rq Rdiag Ri Li Rj Lu (E) q R Li Rj Lw (E) ~ QUSHM (k) Rq L(E). RigSHM (k) L(E)
y t p Pr eg q8
e f m PnvPuHRqvBC1B#1jHqgpPWeuPBp1eHvPPusHCy1Pr e s u p p y t r yg s f t y y yv e s ys f)HuChgpfntWqvBCR e t skeP'pCheqvBhByC)}ClhpushWjBuyp p j f t v j y ev y sPByfp1eWjBv$gWuBsPtfheRPnP1Blbg f y p v f m s fej t o euPt0Rx t ev y qljBByCp eP#pqW'By t z p u j t e v m s e j g PfifCHPf1Brtbdeqg y t g e qW)Pc eg v y s e tv y qPq'WCh1e
~ : SmAf/k / Sm/Q Sm/Q v i j / i j w ~ k X (idX , X)
e f mv t t p j e yv s eg e f m v f r u r t CPCP1~HBqPBCHq{PnhP1PR) t f s p e y p e r t e e t p j e r e w s u eg s p P{HCH@C1HW'dHWu1PB}q{x
~ k.
o e yv s y HPPuHBo v t f t e eg p u p t e y yvg e yv s PnWH1hCW}qCuh~HPPuH e yv s y HBqPBCHBqv t f v f r u t Q e sv pgg s j eg s p P@hPr U@c fBqquldq{x
2 2 k
o y w s t v y s e gv t e y t ev y u s p j y u s s p j uBfChBu1H}qv qP}$~1hB1uW1uBWHe
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
Spect (PreShv(Sm/k, M)) T
DiaSch
(E) = Li Rj (E)
/ RigSH (k) M
w
/F
o
y t p Pr
X
En (Qgm (X))
/ Rig En (Qan (X))
D q
T
o
homSHM (k) (Susp (X Acst ), -) ~ T
homRigSHM (k) (F(Susp (Qgm (X) A)), E) T
homSHM (Gmk ) (Susp (Qgm (X) A), E) ~ T
v
c
Q
Rq
/ SHM (k) ~
i j v i j
j
j
j
/ D q o
/Qo
/ko
B1
i j c i j v
c
v
/ Fn (Qan (X))
i
et
i
i
q E
D q
Spect (PreShv(Sm/Gmk , M)) ~ T B1 F B1
N
Q
~ QUSHM (k)
v
/ q F
/ Fn (Qrig (X))
q
/ SHM (k) . ~
egg e s f p t t v p qqWuPBkqB# e r e yv s t f t e t e yv t p s y y e j w y e @HBqPBCHqPqWnWBqPCu WWP'1 1
E
egg e s f p t t v p s g y t p s pg t ev t j e p t p y j t qh1BPu#qB#ub'B}nWqBCfCPr1CqDCBnHCls e jw e t f e j e r t w te j t r P'1PBP}Cl@R1P8le e j w e t f y s g p t ev t P'PPY'q3C1hu~R o o
cst
y p u j CH1P'uy
/ Gmn+1 ~ k R(n, r)
e r y e p s pv P}HuCr
e wv s u t s p e s f hB1n1BRuBp D
yCpWuP'juy@ery1eHH'@kWqB1BB1uuW{q~}ubR p s pv r e r t ev y u s p j u s s p j eg e s g t ChCp e yv s HPPuH eg t p yvgv f t qCBqhukP
~ QUSHM (k) 2
e yv s HBqPBC}
/ SHM (k) ~
t f v f r u r t PhPq
o ev s u p y f y pg qu1BWj Ce
e r t v jv s y e s pg e e s yv y PtqBlquB1uPBCHBq3pe
cst )
s f e j t g eg PWurCbRqle
e s t e e yv s yv t f e sv f s y t j e e y e s gv e s fg j t j s f uBnWHPPuHBuqPBuplH{B1u7quPhWWP
(E) L(pN )# Rq (R(R )
o e eyse t vy p f m t f y Bu1hv CnHpe
e }HCn
sv t v v t w u r s p p t )) RkhC$ ChuiPlH t By le s f PBy
R Gmk ~
R ( , pN )
e r t v j e s p qBl1pBPhg
(-)r
p u j WP'uy
e hg
s f t v v t w u sv PC3C PQ qBhPW Y qn
E Y
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
R
k
(n, r) Ob( N )
Gmk ~
o
e e s yv t e e f mv t t p j y {PPuHBnWH1PnqPCP1pe e r e yv j u s t v v t w u r pg s f u vgv v yv v BqW1uPdqBhPWtruiqqPhBnqr
" ! $#
R(n, r)
R R (E) = L(pN )# Li Rj R( ) ( ) (pN ) E
e f mv s u | y e s j e PnquBW}BuBl1)y
e t t uPR
p u j W'By
Gmk ~
Qgm (Gmn , 1) r ~ k / Gm~ : Spec(k) k
T
Gmk k (Gmk )n ~ ~ ~
E Ob(Spect (PreShv(Sm/k, M))) T
(R , pN )
pr1
/ Gm~ k
R L(pN )# Li Rj [(R( )
yhgPpByqgpfrntWeHWeusCbgntBy e v
cst )
Sus0 ((R(n, r) Gmk k) ~ T
gWeusPBpe}BPPuHBqkPhPqCPWuP}{@uWPW1B}x f t yv s yv t f v f r u r t e f e s pg e r e p u e r t j e y pg s p etlpChPH}HClj r e t f e sv f e t hCtPW)xR e eg eg s p lP HHWq{hb y e y eg e f m v pg f r t ev s y t p r e yv s y 'lpChRPnh$nWq#BPnrC1qRPHBqPBCHBqv
R [(R( ) cst )
(E)
(pN ) E]
R L(pN )# Rq RRig [(R( )
R L(pN )# Rq (RRig (R( )
(R(R )
cst )
cst ))
R (R( )
R (RRig (R( )
cst ))
(R(R )
L(pN )# Rq (R(R )
qu = (pN )# q ((R )
g e s f p t e yv s yv t f j t r p t 1BB#RHBqPBC}CkPP}qR e yv s yv t f v f r u r t }BPPuHBqkPhPq e yv s yv t f p t e sv pgg s j eg s p HBqPBCHBqHPHP @c RBqqulqxx ChCp
qu
q
~ (pN ) (-) : QUSHM (k)
/ q 1 1
cst )
1
~ 1 : QUSHM (k)
((pN ) RRig E) .
cst )
id : SHM (Gmk ) ~
(RRig (pN ) E) .
(RRig (pN ) E)
Spect (PreShv(Sm/k, HOM(( N )op , M))) T
cst )
cst )
/ Gm~ .
cst ).
(pN ) E]
k
2
o
o
e f m e pyy p f p y t f ms p e CCuBCtUCPnuWp
o
o
t v v t w u PQ sqBhPW tPfuPPPnu t t r t p o ntp#hb C qPP v f y j p gg f eg j t r y qkpe R
qu
/ (pN )# 1 ((R )
cst )
(pN ) (-)
/ (pN )# (1 (R )
(,pN )
~ Spec(k)
R1
1
/ Gm~ . k
1 (R )
/R
/
y e yv s y eg e f m t e egv j p }BPPuH1q~PnnWHWqhW$g e j w e t f y s g p e t t r y f t e y p e r t e e t p j e r e yv s e P' P'q~PPCPCnWH1P'PRHPPuH
N
u : (Spec(k), N )
~ (1, . . . , 1) : Spec(k)
/ u u
~ k
/ (R1 , N ) .
cst
R
/ SHM (k) ~
cst )
/ SHM (k) . ~
cst
cst )
1 (R )
(pN ) 1 (-) .
(pN ) E].
Rig
/
: qu
/ (pN )# ((pN ) 1 (-))
cst cst
1 .
cst cst
ChCp
e r e yv s t f t e y yv t s f {HPPuHkPnWBBPuPbg
C1ChCp
ChCp
o y e g p uo y e r v v s y u u v s p 11quW1quBlqu# e r y ev t w y e v j e s y e v vg y e r s f y y ev t w y e v j e s y e v vg y e j ev t 1qPqBrpuPBh}qq1PBhPqBl1puPBh}qq1WqCn1e qP{RPWWHubg y fg e yg e s y p u j 1P'By y e r ev s u p j pg y t p r e ug f jg p j e v j e s e v vg pg e uv j y y p g ev t e y y e g e s g p u j y eg t p t e s t e y e f t e 1PhuCWu#WDqPR1hlWRqBrpuPBh}qqt@qlBBC)1hunWBB1}ub@WP'uqnWuP1PWu t y y egg y ev t w y e v j e s y e v vg y eg e r w y y nB}1qh11qP~qBl1puP}1uiHqhD1qduC) e e t y e v y e r e yv s t f v p t e y 1W)W03WBhkHBqPBCHP}i$g W}pe e f m y t f ms p e PntCPnuWp
: (Etqc /X, Nis) / (M, Nis) . M X X0
cst
BW~qDPY#C
y e vy e e y s us t f v f r v f t t y e fe t e Bh@r{HBqvPBC}1BfPRiPPthWefqvBnWjpsPWurjCbg) ev y qljByBCp e UvPfnm s f e j t g t WBlbPf 7r eBC)vHRCr'q1e}usb8yCH1uP'uy1qxPfB$k'qWehvCq)ugr y y r t yg o g p j y eg s y jv t yv e r e g p e t hvPPft}tR sPfByhgpulu1e e y e f m yg e Pnxq1B WqBn1u1e yg ev t e y y yg s y p j WeHubg1uP'By y e t y e eg t ev s u p p 1qgnByuWpyhnCr{huCWBWjg e BttR enkR 8b7YYWU8RPHF y t S Qa V ` X V T S Q I G ! #Y#7RC}W$rW)WC Wuu#nW S '## rrDr#lBYRh##'
F (U ) F|U U U : M (P) : q X0 = Spf((X, O )) X M / Etqc /X U U / 1 (R ) F / RNis (U, F|U ) M F 1 (Etaf /X, Nis) / (Etaf /U, Nis) X0 r
hW$#l')Wh##W1D)rl W$W}r#@8 r{ WuWpB) l 5W$lA1'h #q#dl#W}#Wk#9 W'l W W$r)'B)khsW}@ r #'1q #W Wi# @ $r#nB#'D)~P#Ct'lh rq#''W#l5W)H @ #Y #r#Y@C$) #7 $Uf72 s f y y e t z p eo y e g p u y g e s PB1HlhulWHubg o o o y p u j CH1P'uy y e r e tv eg 1}PqWqP~e v s u p y f y p vyy f p e t t g e s g p u j t f y Pope v c y e rf t z )WHP#p o 1CCxPnm ye r t y v f o huCWBWj Bg o u}BPs)WHBCbHWP'uy y eg p u y e rv v s y u uv s p 11q'l81qu1ulqB# y e r e tv eg ev s u p y f y p h1hPqBC1B#1j Bqg e t t e rv v s u uv s p BP@RquBWhu# PPfo e t v c u vg p j iCuiq1qg e r e s w v s uWuiulj t r tv y e y t s f p y f t pg e j s f e w s u eg s e s t e r e y s e y t t v j e y f y e e j e r e y e s eg y t p Pf 7hC3CBxCP#PWWPC CH1BCWPuhWuBnC}PBC)uPB}CUqBry BBBllPupBqR iYih3 p g f y u s eg eg v y s e t 1#'iB{qfC DPqPq'WChv 1e y e f m us t p t s PnuBnHDR e g p u ye e e )CW1lpCW s p s e j pg e s f e t gv p f j u s s p j eg s p WWPHWutW)h8r1ChCphBuD}CluBWdq{xU v c h3 e r y s f e p s u t u y eg s f y s e fg p u r e e t e z f y g eg v y s e tv y e e f m s e s t e r u yv p t p t e tv p y e g uP1Bu1PWCfhPB1Pq#WsW)WWqHBqPq'WCnqkpd#qpPnf1BunHxPxBqknWBnqf1~ gv p f j y phBuD}Cl1e
1 q Susp ((X[W ]/W r - f ) Acst ) T 1 e r E Nis k X X k X k k F k
e yv s yv t r e y yv r t Q t v v y s s p HBqPBC}CPf RBC)CC# UPChuiBC)u Y R
/ Gm~ k
er
t r e y yv r t Pf 7@BC)kRY
e j t p y yv f pg e t v p ug lBuPP}qHkqB#WqWu 5g
id / q q / qu q
sv eg v y s e tv y qCbtqPq'WCnqpe
o es te y p u j uBnWCH1P'uy e r y e p s pv r e r e yv s y 1HH'qP~P~HBqPBCHBqv y t t uPWe v c g p s u t u y p j f p y t y y p h'WWCWqCux Q e e l1qUP@HHWP )) e f m v pg f r y s g p eg f j u r p g f y u s e u v t e rv s f e j t g f r u v t e r g y Ch$'q~qPW1#uhPB1uRUPuiuC1WBlbf~BhBnWPqv 51e
r N Gmk ~ id / q q er : Gmk ~ / qu q
e u y j pg e f m e s t v j e 1nHWHq~Pn{BunHl
q / qu
p t f s nm
u v t r e yv s e BhPPf 7}BPPuHU
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
q
/ (R )
egg e s f p t t v p s g y t p s e t f v f r t qqWuPBdqBBCbpuu~PPhPhv eePfnmhgPp 5@C UCu'qqPq'WCnqxpe Pnt')p 73uBnCq E u v g t f m y s g eg v y s e tv y e f m rs r y t s t e p u e wv s p f m e t f y t p r eg v y s e tv y H)uW~}1huBn@PPP~hPqBu1h1e e f m pses t t Pnt'WuBnC}kR
q (pN )# q (pN )
/ q (pN )# q (((R )
cst )
(pN ) (-)) .
1 (R )
cst
/
e yv s f p t v j e y e t f y }BPPuHqsChur1BDP1e
~ QUSHM (k) E = Susp (Qgm (X, f ) Acst ) r T p SusT (Qgm (X[W ]/W r - f ) Acst )
u s s p j eg e f m t e egv j pg e wv s u t uBW{q{Pnn1}1hqlC$iuWR) s p t v p jvg v g f p fqBWhPqBhPD@g j e p
qu
t e rv u e yv s eg y C1n1RHBqPBCH{h1e
e u y j pg v y tv 1nHClHq5Bqc uhPuBClPC{BqqqBf e sv f s y t j s f u yvgv e yv s eg e f m sv pg j y e g }BPPuH@qtPnql~p8
cst
(M, E) U Ob(Etaf /X) U Mdl(U )
mr : N N
(er R , pN ) : R
q
/ 1
r
Nis
Spf((U, O )) qu
cst
Etaf /X
/ qu e r
/ Gm~ k
/ Gm~ k
/ 1
/ 1 q
Etaf /X et
Etaf /X (P) id
R (id mr )
X
cst
X
qu e r
e E = r q
cst
C1ChCp
t 1e e r egg e s g t v ug j pg y }hqWHubsChulqPHW1e e f m ge PnxWB s f y gv j e s PBiur1pB g ev t e y y e eg w r f 1hunWBB@h1HRCBWe
X X Mdl(X) P
eg pv j u y e s w p qlB@BDBy o gv j e s p u j iurpuPWP'uy t f e yv v e g P}BqW3Y o e y yvg e rf t z BBq{HP#p PBP)@qPq${WWq#qPn1tPR1e f s f eg v pg e j t eg p v f m u e t f y
Spec(A )
ChCp o o e j w pg e f m s e s t f pg gv e eg eg s p g p j P'1 {PnWuBnH$7qPC P HHHWq{hx5Whg y e g f m sv e e y e s gv t p s 1Uohv nhC}{B1uhn'Pw y 1e e f m s e s t s f v y tv c f p e v t s p f p e v t t p s w t e e v j e s y Pn1BunH)ChH1qPd1ChPCunWHWChurpuP1e esje uBl1)y eg t v j f s y t j s p hCqBluBWx E u vv e vvv v y u uv s s y eg t e e v y y e j j f y e s t u r t y ev s p yv s t e e f e s pg e yv v r t rh$tlhqb3rb~1ulqBPCB1qnWH1qu1WWB{BuCCHWqRWhuuBW~WuP}q~BqqR 8b7YYWU8RPHF S Qa V ` X V T S Q I G
u'W#l5W)C}ql$l'YlhWt)rr
Bl'
8
# ''l l$WYChW$#luYlq##ltYrl r#@h V T V CWWbC lxl#ul rl T E V u## rr@ll$rYuBYlht rW ! W$r5WH#" Blp'Rll#l'Wu)3WWbC Y x$U 7 e w s u eg s eg f s g e r e uggv p u r y fg t pg e t f v jv CH}1BCWPu~h1hPDBCb~{Wqhq'lhfCl$~P7lx%
T
iYhq8 y e t p s g w y e v vg j s p WBn'Bh1uiHqhCl e r y eg v pg y e j t eg p v f m u y e r u vgv p y pg e r y s g p eg f j u r p g f y u 1qPq$l1hqPn1@BhqhPuHq{'q{qPW1#'iBs e eg v pg e j t eg p v f m u e t f y ChPq$l1h#hPnPPpe e j w pg P' f s f e f m v f r u r t CBP)ChP1Pn u uv s s pg s p BWhuPCBPtqx o y t p r e yv s yv t f j t r HPPuHBHqp R3 t e r t ps @Cuw t e e v j e s t e e j pg e C1HWqBl1pBP@n1}1lCqPH1Bs t f y Pkpe o e f m v t vv e v s p Cqh lqb0lfbhx o s f y y e v j e s y eg w r y e r e u s PBfqBl1puP1qWPH1P1}ubg e r e t p s g w hBnuuilj ev s u p y f p huCWBWj By qg u t p t y fyye v BPhC1u1Pr f s f eg v pg e j t eg p v f m u e t f y uP)qqPh$kl1h#hPnqPP1e
Mdl(U ) (P) Nis U = Spm(A) Ob(Etaf /X) Spf((U, O )) G|U U Mdl (U ) F (U ) M F (U ) /
UMdl (U )
e yv s eg e f m s e s t e u t e p s y e t s p v f p e j yv pg u s e r eg v pg e j t eg p v f m u e t f y }BPPuH{h{C1BuCCHD{P1H'1kCl1qp@xn71WB$WBP{P~qPq$lWq#hC{PP1e e f m s e s t e r p s z f y gv vv s p CWuBnC}~}'HPqrh${x t e rv u e yv s t f p t e t e rv u t v y fg j t g s p e u t t n1q1RHBqPBC}P}fRCBnWPqWfqhWhv ~WPr y e v y e r e j t eg p v f m BhtP@l1hqPn1u g je 1p e e t e r u j u s t v y y f j yv r pg e f mvg p t {BnWP1lBChBuluqPq@CqPPqR o o e s vg g s j e g s p t p s nn P Ctuhqop quW@hCuw j t r y e g e tv e s y e s t e y s P1Urunh1BupB$xWup'We 7us P~ulqBuP}q~hB1 t r e r u uv s s pg e wv s u s e v go f j v s p t y e t p s g w y e v v g j s p e g p 11hqPlBBx711un'BhR1BhHhqW{HqPo 'py e f m e PnHBs t v j e t e y s t r e r u uv s s pg e wv s u ouCCHxl17Wup'We 7us P}ulqBPCBf}iuW ye y t ps 1tun'Pw t e e v j e s v f p e j yv pg u s y e r e y y pg j o pg e f m yv f t p s nWHWChur1pB1WB$WBPD1PfBBCqWiPnuqP8Cuw e f m t ev g t p s PnHC1h31n'Pw y 1e t e e v j e s y nWHWChur1pB@pe e f m v f r u t o y t p s CthPr 71un'Pw t e e v j e s y e y eg t e s e y u s y e j e sv r y e p v y e nWH1qBrpuP~uWp@1qnWCB1BWBl1uh{Hq{hg e f m u t t r t p Pn{PPCn' yg e s g y p u j WHBCbCWP'uy e r e yv s eg s p v f r tv e j e sv r e p v s f e j t g e {HBqPBC}~qhPh~Bl1uhP~CqPWurCbRqg j e p
(U /U) ( H)|U Nis ( H)|U = G=a F ( H)|U
U UMdl(U )/U
C1ChCp
M |U U Et/U
t e e pv r u v v f r u r t eg f s g pg e nWHWu#1HHqiPP1kPlCWChCpqPtubD{P e t v jv s y e s p kqBlquB1u}By e t t o BPfC ye U1usb o s f PBy f1WBy$o WBPkP~Ptnt' l p e j v pg us t f u t r p e r y fyye r f p u1q$ e r g e s g eg w r t WHub~qWHkPf e e j p v y p f m eg p WRur}Clj BC@q'lu uBWhup# uv s ePPf t c pj ivCg1hg y e g m sv e y s g t p s w t e e v j e s y p)qv fnqteRupeBYn1n'Pn1}1qBl1puPpe o e f m v p y t y e t p s g w y e v vg Pnhun11Bnuui1uiHqhClj s p ePun'sPo yuWp1e}BhvhqP'pqg1WChCp}pxlqbtuBns WhqqBu r y t p w e y r u gv p y p y e s vv y t s t y eg pv v s y e v j e s y t v p s w j y eg j t 1ChurpuPChu#'PWh{r e eyu y g p usWB1usPgWehWjqg y1uPBljuByt1q@PC'pnqe0WqghqvPfR PDP1BrbhPt1e e s f f s y eg s f e j f te g e r s f e j t g t f y e f m v pg f r eg f j u r v tv e PnDh$hqPCl1P@bfnq)d o t ps 1nuPw t e e v j e s y nWH1qBrpuP1e vv qb o e t f y e Q e yv s eg v PP1#tqpHBqPBC}{qf$ eg v pg e j t eg p v f m ChPq$RlWq#hCu y e t p v f y y u uv s s y eg p t ~1uC#hPB1ulqBPCB1qDqR
H H
Q ChCp o e yv s eg y s g p y t s u rv y t t p s HPPuH{quChuW1nuPw t e e v j e s nWH1qBrpuP j e p o eg pv v s hqqBu e v j e s t v p s w j e t f v yv j t qBl1pBP{Chu#'PlP8h'DR y t p r y s f eg p U'PWq#e s f y f p e j yv pg u s t PB@luq$luPtPf hc v eg v y s e tv y ChhBu1q1e u v t f j e r e yv s eg e f m v t s evg f jv s p t yg ev t e y y e yg e s g y eg w r y e r e t p s g w j ev s u p j p uiPW}BPPuHhChC}C#11hqPWqu#1q1huC1u1WHubh1Hx1uCuuilqBCWu#1g j e p nWChCp s f Py
H M Mdl(U ) / id G Nis Etqc /X [ H](U ) = Colim H(U) G M G
UMdl(U )
p t f m t e egv j pg e s t t k nC1HWqhW$BunHfC
f p e j yv pg u s t f u t t r t p 1WB$WBPnu
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
B1
X q
" ! $#
Nis
E F = Rdiag Ri Li Rj u E
Nis
Nis H
Nis
k
U
Colim
Nis
/ (Etaf /X, Nis) a : (Etqc /X, Nis) / a H f: F
Colim H(U)
Nis
/ H .
X0
(U /U) H|U
k
Spect (PreShv(Sm/k)) T pN
RNis (U, FU ) / H(U)
/U U (U /U) H|U
Nis
Nis
Mdl (U )
Nis
H(U)
U
Nis
diag T
Fp
Nis
Fp
/ hom(B1 , Fp ) k
PreShv(SmAfnd/k, M)
Fp
/ Hom (diag T, Fp+1 ) g
C1hWHWWu# erPHk#Wp S
V
Spf(A )
Fp
Nis
diag T
k
Nis
Fp (X) F
X = Spm(A) / Spec(A ) p: P
/ RNis (X, (Fp )|X )
T
G
X Etqc /X
H|U Ho(M) M /H
H
f
e f m s e s t e r t e eg p s u t u y fg z f y g eg v y s e tv y PnWuBnH~nWHWq'WP1hPHBUqPq'WCh1e
Spec(A ), [i i j g E]p |Etaf /Spec(A )
s e tv s e s f WPqHB1BPCr
p u j WP'uy
g e s f p t e yv s eg e f m s e s t e y s g p e y e s y f t g 1BPu#~HBqPBC}{q~PnWuBnH}uCh{B1uPUhv f r g p s f j f s y e yv s e P'PBlBup{HBqPBC}{qg BPHk 5k$~1ChCpWBC)HW}1C~PqW5Hlj u t p t g e u y j pg j e p e rv j tf
Spec(A ), [i i j g E]p |Etaf /Spec(A )
e u y j pg e f m s evg f jv s p t e v f r u r t y eg w r e r y e yv s v f p eg v p 1nHWHq{PnWqqPlBBk1iPP1qR11qWH{1HBqPBC}f@qPqB#}Clj t e e sv pg j y e e yv s yv e e yv s y g e yv s HPPuH o nWHWuqlR1{HBqPBC}CWYW1qdHPPuHBv q tY yv t f t e e pv r u v v f r u r t t e f s p s p t e f s p e y yvg y x~n1}1BqH}q{i1hR1nWHPCB~~C1HPuHuh{1e e f m v pg f r eg f j u r e yv s yv s ev t s e r e Pn{h$xHhlYHPPuHC~WqPB1
diag [m i i j u E]p P m i i j u E v ([i j f E]p )Etaf /P
i i j c v f E i j v f E v i j f E
t f Q e sv pgg s j f r v f r u r t P3U@c kuhqquW}hP0R
y e yv s y r e ts p j pg y s g p p t 1HPPuHCv Pt'q'qHRY e yv s y HBqPBCHBqv g p u r t e egg e s g t v ug j e r e yv s f r e s j e y f p r t y e s s qv g Wdqh1HBCbhqBlqPHWfdHBqPBCH0uBl1)0)BuWj
c i j
e gv p f j e p s pv r t f p t hBuD}Cl@HH'qsiRY
ev j y y hWBup
o y w s t v y s e gv t e y t ev y u s p j y u s s p uBPqBu1}Hqv qPH$~1hB1uW1uBWj e v f m p u j y e r e p s pv r e PnWP'uDP@}HuCtqg y t B
I : Etaf /P / Sch/k (cp ) ((Fp )|X )
ev j y y qlup
e v f m g p t pv r s f e j t g e C7CP1Brbhg
eg f s g pg y s g p p t htubDq'qHkR) e t t BR s p u jv r t ~lqv
/ Spec(k ) Et /P
af
e r e p s pv r eg y s g p e s w rv y t j t {HH'qDh'q{uWBWsRY
ev j y y hWup
e v f m s f e j t g e 8PnRWBlb@hg
eg p s f j f s y t v j e s p q'PBluBfqBl1pBPHg
[i i j (f E)]p |Etaf /P
e yv s yv t f e sv f s y t j p t pg e j s f eg v pg e j t eg p v f m u e t f y }BPPuHBqkPuhPuBClH#CPWWPqPq$lWq#hC{PP1e
Fp (X) / (X, (Fp )|X ) (P, (cp ) ((Fp )|X )) / RNis (P, (cp ) ((Fp )|X ))
e u y j pg e f m s e s t e u t e p s y e t egv j p s t j y 1nHWHq~PnWuBnC}}~PWHu1ChqBl'BnW1e e f m t p f ms p e PnnC'1Bs o t y f y y e v j t v j e y pg e r t v v y s opg e r e f e s pg y t p r y e sv p y y e j u t y t ev r u s tv y eg y f p s e f s s f e j e CWCu1Pr lt qChurDqR{1q kqBhBC)uP@RP1BP@qP1uhBu1W1PunWq1uh1qCB'WCPuBPWurhg v y f p e j yv p 1WB$g f p e j yv pg u s t f m y t g e p p v f p e j yv p 1WB$WBf n@qW)Pe71WB$g e gej e r e fes e t f s f t f v r y Pthtpe s f PBy o hg7pq1ldWuPdPo P{ y eg e s t e y eg p j hBunW}1q1qg y e j t eg p v f m u y eg e s e y u s 1WWq#qPn1HhfuWB1uP e f m v f r u r t e t PnhP1P1R e u s eg t v y s e g s p v f r t 1}uWbq'WHHqv dhPhv e r t y e s s j y e v y e r e yv s e j fCB1uul1BhBt@HPPuH{W e yv s y g pv f t e v y tv p y e v HPPuHCv H% Un1BCBhBhy eHPPuHqg r e yv s e j t g t f v f r tv egg e s g t v ug j p bDPhPqqqWHBCb@ChulqPHl~8 o
/ (Etaf /P, Nis)
e t t f tv t j t e e sv pg j y e v f BP}RChuCClxnWHWuhqWpPnm o s f y y e t z p y eg p u y p u j 1H1quWUCWP'uy y e r ev s u p j p qBC1B#1~qg
P
6 4" 9" " ! " ! 8751 8D3$)$#H1n$# A 8} 9 "
(idSpec((Y,O)/ , (Y / ) /) /)
t p p e j y t C#HHW3B
f: P
Etaf /P
D m
m : Etaf /P
Fp ()
/ Etaf /X
c i j
Etaf /P
/ SmAfnd/k
i
P
j c
Etaf /X Etaf /X P Etaf /X
Spec(k)
D m
f
c
v
I
P
diagP : Et /P
g
j
j
j
j
/ P, [i i j (f E)]p
/ Spec(k ) o
af
/ D m o
diag [m i i j u E]p . P
/I o
v
/P o
c
f
Y /P
(cP ) t
cP : (Etaf /X, Nis)
/ Sm/D m
(cp ) ((Fp )|X )
i
i
i
i
U/P
~ Spec(k)
D m
I
P
(Y / /)
f
v
|Etaf /P
k
()
()
U
k
o
i i j g E
/ (p ) i i j f E / R P, [i i j (f E)]p / R P, (cp ) ((Fp )|X )
|Etaf /P
Spec(A )
Etaf /P Nis
(Y /P )
y p u j W'By j e p
e t t uPqR
C1ChCp
C1ChCp
/ Etaf /P t
s fe P1B
p 1u
o y e e yv s f 1WPqHPPuHkr e rf t z PHP#p t f t e t v p f Pf1kqB#hg e f m es y e r e t e r u j us e p g y t p r e C{uCtP~unWl1uP@)'lu 5P@HHWj v c v f p e j yv pg u s e r eg v pg e j t eg p v f m u e t f y iCPYP1WB$WB@@qPh$~l1h#hPn{PP1e
k X = Spm(A) (1 )
w.e. c
Q e sv pgg s j eg sv rU@c DBqqul@qq$
hCp o ePfnmqe vYvPfpWjqg t CCy y eg e f m s e s t e e y e s g 1q~Pn1BunHH{B1u7 E u e yv s y g e r t p t e tv p eg f j u r p g f y u s e HPPuHCv {n1Bnq~hPCl#uhPB1u{
Fp B1 Fp
gv p f j e p s pv r eg pg e j s f e s g t e v j e v y iu#uPtHWHH'q@q}q1lPC~HubfR)Chur1p P1e
0 (Spec(A k R/ /()), Spf(A k R/ /()) Bcst , Gp+1 ~ ~
t v p jvg g e t z p p u j Chu#1qqPPp PHD1P'By
t f y Ppe
t v p jvg g j e p e f s e s e y t e v s r e r e s e eg t pg f jg p j t e v j e v y qB#1qqPPp R1~uBlu{BCBhu@{uPt1HqnPl1kPChur1p P1e
0 (GmSpec(A ) , Spm(A)) Bcst (A1 Spec(A ) , Spm(A)) Bcst , Gp+1 / 0 B1 Bcst A , Fp+1 B1 Bcst A
e y yvg e rf t z uhPHP#p
fCBsPf)C~uhDnWqWuW1qPq$l1hqPn1PPpe e f m e sv r e t ev e s v j e eg v pg e j t eg p v f m u e t f y
/ diag Hom (T, Gp+1 ) o g
(1) w.e
e f m sey fy fe t CWBC)PPB1RP
gv p f j e p s pv r t f p t f p e v t s p f p e v t t p s w t e e v j e tv y iu#'tHWHH'qPkDkR1qPf1ChPn'PnWH1qBrp#qpe e f m es y e CuCr v c v f p e j yv pg u s e r eg v pg e j t eg p v f m u e t f y iCC71lu$luPRRqPq$WWq#qPn1P1e
Fp G = Ri Li Rj Lu E / RHom (diag TD , Fp+1 ) g
e yv s eg e f m y t s t HBqPBC}{q~PnuBns E u e yv s yv t f y HPPuHCkPpe e f m v pg f r e e sv pgg s sv Pni$fPl1qtuqhCBC) )Y ~qCb hBl1pBP~HBqPBC}{qPC~B gv j e s e yv s eg s f e y p e r t e e t p j e r e w s u f r t e e pv r u v eg f j u r v j e n1}1P'DP@}1BCWPuq~C1HWBqP1HHh@qPCl1P8l7l s f y e r t v j e s p qChur1pBqg 1C$dqPhB'WqR1e j e p eg v y s e tv y
p p P Spec(A ) p
9" " ! " ! A 855#DC5# B%
9 6 4" 2 ( & 3@875310) '%
" ! $#
G
B1
k
w.e
B1 k B1 k O
Fp
/ Hom (diag T, Fp+1 ) o g O
Homg
diag Gp
diag Homg
A1 D GmD
Rdiag G = F
(1 )
cst
cst
cst
cst
, Fp+1
, Gp+1 .
k
o
t v p jvg g qB#1qqPPp P f ule e f m ses t e r j t r z fy g CWuBnHH7
BE
0
(GmSpec(A ) , Spm(A)) Bcst
(A1 Spec(A ) , Spm(A)) Bcst
, Gp+1
Y = Spec(R)
/ 0
(idSpec(A {T }) , Spm(A{T })) Bcst , Gp+1 . (idSpec(A {T,T -1 }) , Spm(A{T, T -1 })) Bcst
~ k
o
Spec(A k R/ /())[1/] ~
Spec(A k R)[1/] ~
homSHM (Spec(A k R/ /())) (Bcst , Ri Li Rj c E|Spec(A k R)[1/] ). ~ ~
homSHM (Spec(A k R)) (Bcst , Ri Li Rj E|Spec(A k R)[1/] ) ~ ~
0 (Spec(A k R), Spm(A)) Bcst , Gp+1 ~
G
j
j
i j = c i j
/ Spec(A ~ R/ /()) o k
/ Spec(A ~ R) o k
/ hom(B1 , Fp ) k
c
i j c
i
i
~ ~ Spec(A k R).
~ ~ Spec(A k R)
Fp = diag Gp
o
Gp (Spec(A ), X)
/ Gp (Spec(A {T }), X{T }) . Spm(A)
y s g p v s j y e j w p 'qiulu 1hCpdP' U
ChCp
v y e f m s e s t e r g p s u t u y fg j t r p s e y g @C1BuCCH{7'WP1qP~CD'WBU
F %``yV gp&`c&Tx yGTtcpe''w"QewT`UdW&`c`ttEyyca H a X g a F I X F H V a B VY a X F u uY a F q a Fs R I D a F X a F a aYs I B V I eF a F ua DX V &'cbY i `T'u H F q a Fs Y a g F I X F H V B VY a X F u uY "e"1&toWg`c&Tx yTtcpe''wqF B Ttv`eW&`c`tt1w"1c&&"bTtWT@e p`a H Fs R I D a F X F a aYs IY q I a F a F qY Y X a R u i S F "q F u aY D X V u 'Wti Wy'B F DX V u Va i Wy'TWtY I BF u H B B R S Ia %e'&F %tY yV v&gc&F F %%e'e`#b"&''w1F W&TC H I B F u F I RY q u uY Y X x B A X H F I S B VP H Y a BY R I B FY I eWUTB WwTg%&w`"TV B A f d d B VY IY a V D V D Rs q H Y B B R S F I X F H V B VY a X F u u j CTt TtTiTtWwWy)T`X bCF F %tTV Te`c&Tx yTtcpe''tY s S F x &TlR I F DY X I Fs X R Uts w`WwvD F qY Y "bTtWX R u iS @e p`a wdwY B Ty B )k"h'& p`a Fs I q B V Y F D I F q R u iS B B y"QyT` %T'v)&'Wy'& p`a H B V q I B RI F B R s F uX VP R u i S B H IY V wT " Fs h w&"q %e'&F w&wQX e`yq5ctv%&w`"TV I B F u BY Fs D H F I S B VP B H Y a BY R I B FY I B A C F X V S B F F I B B js H F qY Y X R u i S WyU&q`vV TV 7qF %q"bTtWs'& p`a B c&F H Ia I F DY X I Fs ews t`Wt0 F qY Y X I Y X R "bTtWqWUt`v#x F B q B y"'%T` H B V q I B RI Fs h te"q %&&F tets X &WGTB 5WwTy&w`"TECA I B F u BY F D H F I S B VP H Y a BY R I B FY I V B WyU&1`vV TV 1F %'"bTtWq@e p`a F X V S B F F I B B js H F qY Y X R u i S B rkF H Ia I Fs DY X I Fs eww`Wt) &`TT@& pWa s F uX VP a R u i S B H B V q I B RI B T"%vp T a F qY Y X a R u i S C"bTtWdy'& p`a F q H R I B F u q B VP a Fs D u F F a Fs Y SY 3 "q"T`%e'TR Tpw&"UQw5etyV btG70 C a qY Y X a R u i S &&F tyw`pT@e p`a &"w`Tye`#e@bs a F q FY X V I R S R FI V B B A WvqC) I H r"q F F q F ua DX V "r'cbY i `T'u B '"bTtW@epWa H F qY Y X R u i S Fss F D R B V wte)D Ty F qY Y X R u i S "bTtW@epWa B H @ B TV 1%v`r5e"tyw`T@epWa B q I B R I a F qY Y X a R u i S a F q FY X V I R S Rs a B q &"@tWyT&W#&EtrGvR `X I H q V D F wY ""`X Qws R X F I B B A I B F I H u u V S FY H j q a H F I S B VP a Fs I F a F `&WTV @C%&%&W"1yUQwts yV 5o`X &WUTwU wY B &`z'twr&trU'"TR a F IY uYs a Fs I F u q FY X V I R S tWyT&W#&rR I eF S B q R B V TV iT
X = X/ / X
an
X FY u F D &&w'e`X dF
I eF
Fss F X H I R B F I X F H V B VY a X F u uY tw&WW#GC`c&Tx yqywW`&'trF B H %WwTR j H Y a BY U Ut) @T1eWUTwte)D T@`&ca seF'`TV1@e ipS`a uX P R u B Hr B y"C%vp {Uwe@ B V q I B R I X H F I S B VP Fs s F D R R X F B VS F 5qTe&WaqF eeU %'WUpeWUTwqw&GvspeW0 X H F I S B VP Fs s F D D R R X F a 8 6 ! a B TR q F q ws TV 5`X &WUTr"&"Q"Qc%Gt"CA bw)05#" Y H j q a H F I S B VP H F q F q F a V DY q B e pe UT' s p&"1 eyvzr
j
U # W treUe' U`e T@@pb" de1yvzrUe T'@b U w) CUqyvz e pe y @b ' @Uw{ Ud ppe@1 c 7e' o&e pU@1C e0 `7yU@ U`e e E e l # 0 rrckQk5 UTTz'U Uw{ 7 Us"tyw`@& pWa F qY Y X R u i S p 'W%Bv#tY caiyVw`zY B "q'bszYTqBCAGTtsvbYw`WIByV BQIkF "qETtYWIvtsv#'t5Ws TV 1peWy)D D WEyV y FI R x H B YI R I RP F R xY X a F B V R H R x Rs Y a H B B B R X F a V H a B j H I eFF etY"q&F'tY ip`TR TV qF`XTBB H'X HT)DIews D'yVUaWXyUSB @IkF a&F`yTR#wY Waa&Ftvpe &T1&W&h `v)%&tGe`X e`CA B BF u SX B B u V F V F u S D V H a I B x H s R X B a F a i I V D i a Fs q B F D F X B coc G h W
X = (X, X)
d F u h X V i I Fs X F X I B V u yt t y'&Wye `wQ&W`yy'e"q X yGDdy@ue pS`a H V a R i "HvR F q VY SY I a F R "EBTtWIUt`XWc&`X'bs U'`WwY v5 ){G4 @F vtUGw`X #pzcWrtWwY WepeF %bwD vR d F X I D R i n q s R DY S BY D I R Is H a X Fs F I H a B F R X H Ys D BC U`aGTWWa"q USqwd Gu ydBTtYWwY B "qts1wYyTa&F tywY`yR'& ip`a A g H Va F Y V I R X V x qY X a u S a F q Ha &"'X WQY B s`&c'X &WGTP &&ets y vbW`rw`TyeWveS q RX Fa H FIS B V F n a F H B RY X I a FY X V I R a F a B R a Fs wsvqQ`X H&wTR#x X eF`ByP B rFWtY `WcaTVeSaywtvGTV GTF %bY B p&`dts `X Wvr"syw`&Wh`cUeTR H IS V H X HXI B B Vss R a H B H i S F I a H D F H I R B F q B VY I S F a F I I F S a B er0gh r# qQ ~ k}
|w yw xw z{zw0v d d F uhX V i # u t T''e`T& WI H qTe`X sbT&h pTQe&rTts wTeetv#tY %&qF B 'kF q F x H F D Rs F x i S R Y S F n F Y RP F S B Fs R x H H Ia F %W`yy'pe&o H FXI B V u YS F n I H V I H V D F x Y I S F I Y R I g f e d d "T`hX yGTtWU&k Ym 1wvpUhTvlwyk F H FI B F q S D F D RI js a B q XI B V u R B A yr`y&"&e&`X )vpU 7QGvR `W%T'@iC) a R S Fs a B q g f e d d F H V X I F X B V a H a a F Y yehtQTR QTTtTc@Tx yW`U`ETQW`&"q eS F p&bs iS h f R F %bwD vTcyrGTV %pv@eQ0 H Ys D R B V Y a F H a B H X R u F IY V I B R wT%vpX Ia c&F F %1wv"ts tvU&wT#wY %F B c&F H IY RP H q F Y RP F S B Fs R x H H Ia F pR iS h f
(1) Gp (Spec(A ), X) (2) B E
(1)
F I B R x H a F X hY B R u Rs F q i S h f F I I F S X F aY X V I S RP I H F D B `yT#wY WrW&wv@tr"rF pgWWUed&cb`TWUTQ"&GECA
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
k
0 (B, -)
SchRig/k
X = Spf(A )
X/ /
2
X
SchRig/k
k
SchRig/k
/G VarRig/k
SchRig/k
X
X = (X, X) k
/G X k k
k ((X[1/])an , (X/ , jX ) /) jX tf /G SchRig/X Rig : Sch /A
2
k
/G X
k j : X
X/ / X
k
/G X = (X , X )
X
/G X
X
/G Gp (Spec(A [T ]), X)
(X, , ) k /G SchRig/k VarRig/k
/G X/ / k
SchRig/X
k (X , X, idX ) /G SchRig/k SchF/k
k
jX : (X/ /)
X
X
Gp
k
k
k
j
(2)
A1
k
/G X
/G X
SchRig/k
RigSH(-)
X = (X, X, j)
k
k
~ R = k[T ]
(2)
A
/G Gp (Spec(A {T }), X{T }) .
/G X
X
2
k
B E
VarRig/k SchF/k (-)/ : SchRig/k /
X
RigSHM (-)
X = (X, X)
X
RigSHM (-)
X an / /
(X X, X X)
k
X
X
^ 1 2 3
k
j
X = X
(-) : /G SchF/k
X
X
X
k
k X
I PF D
D
F
D H
F
D
U p ppU G`U@UrW" T U U U p G U@ 5UW tUW" 7 7 'e "Ue 5UW UTTz U`e TU@1 v #W q 5UW 'b" e 'ewe7 epU h rpW" hy@ 5UW ## U pl t&'WyW "tyw`CT@e p`a rrckQk5 '"eU as F u X VP a F qY Y X a R u i S F q F u u R X RY q "'T`yvb"wUT g as F u X VP D a F X a F qY Y X a R u i S pt&'WycWg&"bTtWpy'& p`a a Fs H V D e B VY I wQX yG5Tu G u TtWwY B G{G4 @"TTtvGvR 7c&F F q F H Vs R B js I a F I B R x H a B VY I W%v#tY cTtWwY B "R q F &'WtY Wyu W`y&wte`X `#5GTtW#@`TcvpW&tTQpeWbwtW"H ylwyF & q a F ua i DX V F X I B F a Fss F H I R B a B VY I R u X VP a B R X I a Fs a R D R X F aYsY I j B B V Y X cw`TyeWveS H a FY X V I R F q F H I S H X I a Rs I B F u H Y H R X F X qY a B V S B A a F IY I F D I B F u Fss FY I B F a a F a FY X V I R S a F q FY X V I R S R "'WX ``Wcibg%&&F %tB pe`eh tWTeCeWwWU)r%e'etw&w`%eWWCw`Tye`#er"'w`TyeWveEbs Fss F D R B wte)D TCA
3 B
5 3 A @9
k
1 3 53 1 ) 28) 764C20
k
F a V D aY q B A Wy)cb"C Ia kF
F I X F H V B VY a X F u uY `c&Tx yqywW`&'trF B H 5%e'etv%e j q I B F u Fs R F q F XY R I B F u D u V S Fs R X Fss F D R B j H I "`wT`%e'e)'Tetrpetw&GD TTV %peF R X F I B B j H qY Y X R u i S peWTV yV %F tyw`r@e p`a B %&&wte`X `#B H I B F u Fss F H I R F H VS F ts D yUits a F qY Y X a R u i S "tyw`y@e p`a F q F u X FP B VY a X F u uY "Ee'`eywW`&'tF B B TV @%T` H B q I B RI
k
a F qY Y a R u i &"bTt`Xy@e pS`a F q F u X FP B VY a X F u uY "r&`eqTtW`&'trF B H IwY "tY H q B Y B FGkrF qT@uepWa D I a R iS F F X FP B Y a X F uY "qh&'uW&qyVwWpe'u'wFB H HysB eF`W%yuECA5 j B FXI B V B F &'`ed"bTtWX q a u X FP a F qY Y a u i TR@& pSWa yVc'aw'UF a H a Fs I F a F X FP a qY X I Y R "q'e'uW&F"bTtYWha&Wew`Xv#x yca'wFW`%eFyw`Ik Ym EF B'wY "qGwY a H V a Fs X I B B VY S F H I H B B VY I R SY D D j TtWvebws vR 7s F qY Y X Y R "bTt`hWIew`Xv#x F B rIc&F H a Ya u WTF'e u'F Xv'& B BTV q R D F F a u X F a F qY X "qg&'W&PC&"bTtYWCaTR@& ipWa TWr&trUF u S a H V a a Fs I 8 6 F q a u X F as u X P a R u S "d`ePbeF'`TVoT@e ip`a GHTWpwsg`W%BeBTtWIU&k Ym gF B H15w"e'u`TR'& p`a a V a a F F X I F VY S F R sY s F X VP u i S B QIkF btG05l! H a Y B FY a I X R S I a &wW&`cTec&F
X k
a"bTtWdWUt`v#r"gW`v&gtrF %%e'ets 'tcg'W@#QGc`&"q UETtWw"Ter&wWbT`W1R F qY Y X a I Y X R x F q X X R S Fs H I B F u F D uY a F u X R a H a a F Y S B VY IY q B V S F u hY aY V X I y1@U yvz e pe U e bv eeU U"p r
i/ : Z/ / / /G X/ / (Z/ /) Z /G Z (X/ /) k X
p U "hp@U pe'r Uyvz e pe
T '@b h& $ % &@b b ppU U ` U@ e pUhy1p@U pU@r p # # pUTTze y1p@U pe'r U " ! l "r # Uw o ockQkG5 y @b
XTDd&F& Ttc"I%T`eFWUFptYeFcbY F qpwyTtTV)T`Id&tda&&w&FGD TRyyVW&cadFetYU"Vc`TRQwyTtT)T`p&F vpUF R a B Y a q B V X a I i S x B a a FY Vs D V a Fs F s D I B X F a S a a a FY Vs V D V I a S I a YX V I R a F Fw`Tye`#eS&tsX WatYTyVwyGyVWU`X DsB HzYB "@F"bTtYW'yR'& p`a H F s V D I F I q a qY X a u i S F i S x B aY a I B F u F X H V S F X a F q F a a Rs S "q pbY%&F Wt p%e'e`x Te&`@&"sW`TbUR I UF a F I X F x H V a B VY a X F u uY H F q a F WWeTTdywW`&'t"&"ws X D TRCB T"qc&F B V Ia F q Y X u S "bYTtW1R@e ip`a B H 5qpbxeF WbY `I%eF'eF`x TVe&F`CF QI%BTR`WT)tptT@ts 'tcrts 'e"UB j i SY B a a B u X H S X q I X V D uY aY R u F D uY a F D u F F ` br U U eTQw)h` br @U` pW r rckQk5 7 @UW pWpy UUw eh@b T w TTz Uw U`e
a Fs R I I F a F a aYs a F u a D X V u a Fs I B F x X F a D d F D u F js F q a H F I S B VP aY V X I a Fs H I B F u F I R q u u &&tvpUed&WWbwpWbY i `T'Q&tdy&T`eW&`X 1t Gu yts 'e"eF r"QpX e`yptyW`p&trF %y&'eWvtY &''wY F `eTqCGewT`eo"tyw`pT@epWa &"rF tets QtWyT&W#&S yc'bs Y X x B A a Fs R I a F qY Y X a R u i S a F q BY F D FY X V I R a H V a R F u u F q F I V B B A a F a aYs a qY Y X a R u i S 'e g"rWvCG&&WWbw&F tyw`pT@e p`a &"q a F F uX V e'`TP F q BY F D FY X V I R a H V a R "F tets rtWyTe`#&S TWts FI B B V `vV 1Tv "bTtWC@e p`a F qY Y X R u i S B p B y"o%v`r0g ywY B TtWv`%eW&`X D H B V q I B RI F B VY I R I B F a F q I B F u H Y Vs V D V I F X I IY V q a H a a Y "h%&&F %tyTtT)T`'W`U wT"hc`&F q US F u a D X V u Fs H B VY I B F I I R R X FP B V B FY X i I F B a R D I a j 'cbY i `T'tF %TtW%&WcvspeT&w`e `UV hThc&F B
5
3 8 6 ! 8 2 70 e05)0 v G`
2
k
o tG
X
X
k
U
(X, ) X
RigSH(-)
(X , X) X k
X
SmRig/X SchRig/X Nis
i : Z
k Z
(Z, ) (X, )
/G X
i: Z
(X , X)
i
an
( (U i )
(Z/ /)
X Et/X
: Z
jZ
Z
an
/G X
i/ /
i
X/ /
/G X )iI
/G X an
k
/G (X/ /)
k
i : Z
/G X
SmRig/X
Z
jX
R = ( Ui
(Z , Z)
( U i/ /
/G X/ /
k
k
/G X
X
/G X/ )iI /
/G X )iI
k
(X - Z , X/ - Z/ / /) Z X
a B R vq
Dia Dia
X -Z /G X j : X -Z
k
F G
D E
F
C
h
f: G
(
I
1
/G SchRig/k
/G F
(G , J)
J
F : I
G
/G (F , I)
/G SchF/k
: J
+C SchRig/k 3Q (G , J) /G I
RigSH(-)
k
/G (F , I)
J
FSH(-)
1
G
+C
F
/G F
D
f
5`
I
{
F
I
f
{
F 3Q SchF/k X X f : G Dia k X = (X, X ) i: Z SchRig/X k k /G X Et/X F : 1
'
awyTWbY "GwY Fs I B R a H q B FI B B A WTV C
F u u Fs Fs a B V I V @'&wrtdT`v a F u aY D X V u V a &'Wti Wy'TWtY FHybY yV BvR&SF'ucbY i `T'TWbB sR C05"vU`tTU`X hwX WcpeywqTv@thX &WUTqws B a D X V u V aY H B A H R F S aY RP D a Fs H a F a X F x BY F R uY H F I S B VP F X e`ysB gUF H FI S B VP H I F u u R X RY q H I H q q B V Fss F u X VP B VY I D u V S F ''vpTvb"B wY ""yGywte'`Tiyw`Uts 'TU'"q
X &WUTdt yT%tts D TrD5"vU`tTU`X &trX WpWpeTtdTv@t X e`ydws H F I S B VP Fs I B R H Y D R B 5 H R F S aY RP D a Fs H a F a X F x BY F R uY H F I S B VP F FI B B A WTV 1C X H FI S B VP H I eWUTB reF a F qY Y X a R u i S "bTtW0T@& pWa F q F u u R X RY q "d'T`yvb"F B rwY ""y#TtWveS W%tvTR rX e`ydw %v%bwD v@r5 H I H q q B V B VY I R Y I s R B j q H F I S B VP Fs I B R H Ys D R B g X cr`TeeT Ha FX VS BF H V X WY B Ha F D I q Ia GkrF CkF F %%&&wtW@`t"iC{ H I B F u Fs D uY a R XY q B A I H VI H V D "T`X yGY B )kr"Cc&F F D I F q Ia F %0eWI H sF a F u u RX R q F q F ua DX V &''vpTTtY s"s'cbY i `T'u B H 5Y h'T@& pWa j q B H u a R u iS F E''vpTTtY H q F u u RX R q B IY V zyp X R vD B y"T@e p`a B V q a R u iS F q F u u R X RY q F "'T`yvb"qts F I B B A a B R as F u X VP a R u i S WTV Cr&&F @#be'Wyy@e p`a F q F u u R X RY "q''vpTTt"q B H IY V wT
/G SmRig/F an Rig
P
3 C{'b"
b
' U "1 gW p "UU wo e {T q e @b w eUTTz e pU Tg@1 T #ryr'b"
A 8 8 Bo@9Q7
P (cpl , cpl )
I%&`TP B F uX V R u iS @e p`a
k
@'&wrtdT`v F u u Fs Fs a B V I V F %bY TV Ter'cbY i `T'ycbEB T"eF H B B R S F u a D X V u V aY H S B V q I B VY I S B V m q R TtWGTklyF B p`ywT'ECG UF H a X Vs R R B A I XvRDQapzYH"t"vU`tTU`X d&tQX WpWU&`w"QTT'tQUp&WpeTtp&TT@wdpX e`yp&ts I q BY H R F S aY RP D a Fs H a a F I S F XY q a F R uY I F a F a X F x BY a F R uY a H F I S B VP a F I eF FI V B B WvqCA a I BY V m q R a X H F I S B VP q F H V S &`%tTklyQ`eWUTgF gts D yUiB H a H F I S B VP a Fs H I B F u F I RY q u uY Y X x B A F I B F qY x F X h B R u Rs F q F qY Y X R u i S pX e`y&tQF %o%e'e`#b"&''tpF `eyCTW%&"t&p`etY T'bQ""bTtWC@e p`a B H F u u VS sF uX V p''Tee'`TP B e"pv%e`X `kbWTe{y H XF qX R F F IaY a B V S Y H Fs h te"q %e'ew&ws p%e'ey Tts B H 5c%GWtY TV I B F u F BY F D I B F u F B V D j q F a V D a q B
F F/ / k
IcXTRDgW`I"vR j FX H X &WUSTqB Hp`ywTIz"tY H FI B VP a X Vs R Y H q B X e`yF H FI S B VP as F u X VP a R u i S be'WypT@epWa F F "q''u'uvpTvRb"qB HwY "& qyV5 RX Y I H q B X H&WGBTP'tI%vGHybts D vRr6&&F"bTt`XyR'& p`a F I S V Fs B R Y D B 5 a qY Y a u i S F q F u u R X RY q B "''T`yvb"H IY V wT cU' v2 0 e@# U 1 3 Uep Ue4{ d We1` 12 0 e'v "ee wr7 @"v7 5 @" $ ) v e`bzo ' &`g 1wr" Q@U(ypU &
(f, ) C (f, )
"ee
p @ewU`"7 e "
u I F u a B V Y I Y a V D V X H B V R P F u u R s F q F X I B V u q F a H F I B R x H a B V Y I Y a V D V D R s vu "U1t "dyw`zWy)T`r G{G4 'F %yUg'& br"rW`%T'e"rWY %rW%v#tY WqTtWwWy)T`X 'tR B A D a F X a R S Fs H V D F u u F q IY RP B A CT)cWT&1tCX yG1'e h"gzTiC0 X R D iS H R TF pGvy B VY IY a V D u V S q H F I S B V TtWwc%G'yU1F X &WUTP F a V D u V S Rs X R eWy)'TetvD F ts I eF X &WGTrt%v#wcrW`&Twryv@tY c&F H F I S B VP Fs I B R xY H a F a X F x BY F R u js I a V g Da FX cW
f f (f, ) : PreShv(SmF/F , C) (U /F ((j)), j) (U /F ((j)), (j)) /G PreShv(SmF/G , C) (U F ((j)) G (j), j)
X &WUTB wY B "iC)yGu yTtWwY B bdTR 'yUS H FI S B VP H I q B A d B VY I q Rs a B q F u u V g as F u X VP D a F X a F qY Y X a R u i S pt&`TcW@&"bTt`Cy@e p`a F WbY i `T'B H q F ua DX V u IY V wT0 WUts 'Te"U'w`TyeWveF @e B y"@c%GD cCA F I h D u V S V S FY X V I R S B H F B V q F a V D H a B g F a aYs s F u X VP R u i S UWWbw5&'Wyh@& pWa D a F X F a aYs F qY Y cW@c`ttg"bTtWX R u iS @epWa B g W&` H Da FX I UF S F x R Tlog Da FX cW a F H V S a Fs I B V a a I F m V a Fs I B V q FY X V I R S Rs &ts D yU&to%TWo`ek"To&to%T"QtWyT&W#&brg Da FX W&` FI B B V `vV TUg Da FX W&` g as F u X VP D a F X a qY Y X a R u i S `be'`T{W&`'&F tyw`y@e p`a F q F u u R X RY q H "''T`yvb"B ' B TV r%T` B q I B RI I eF a FY X V I R w`TyeWveS wQca a Fs X H a B q F u u V S g as F u X V a FY X V I R &tWyT&W#&S "&`X WUW`k wQye`e tWTeS GT`X vR yV #ebTeW"yvu u "TtWwY B )))4 dTR Q'yU0`be'`TP F q a F H I S H X I a a Fs F H X F X qY a B V a B V H j B a H B aY VP F I H V e B VY I 8 6 ! W&`h&"bTtWdy@e p`a D a F X a F qY Y X a R u i S F q a F u u R X RY q q a F u a D X V "Q&''vpTTt"F Q&'cbY i `T'u `W%edWbY i `T'u wwY B "iCA tG5#" FXI BF a F ua DX V a Fs I q B U U p UT e1@1 1@b rUwoeq e' 5` yvz r e pU T U@1 v # wC '7Ue Uw q E 5UW weT UU ` UT W @
F
D
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
2
DiaSchRig/k
F
D H
SmF/F
2
k
F (i)
DiaSchRig/k
(f, ) : PreShv(SmRig/F , C)
k
DiaSchF/k
2
(U /F ((j)), j)
C
(U , i)
(f, )
U
e p
(P , P ) (P , cpl )
%
$ %#
"
r
rckT!r
cplf
Rig : Sm/F
(B , K) k l K Spec((B (l), O)) (F , I) /G (B, K) (p, ) : (F , I) p(i) (B, K) (B , K)
^ (f , ) : (G / J) /, 2
cpl
cpl Rig
/G (F / I) /,
PreShv(SmRig/(F , I), C)
PreShv(SmRig/(G , J), C)
(f, ) : (G , J)
k
(f,)
PreShv(SmRig/(-), C) : DiaSchRig/k
P Rig (-)/ /
P : SmF/(F / /)
(f, )
/G (F , I)
(F / I) /,
cpl P
cpl
(f, )
1
(f, )
/G PreShv(SmF/(F / I), C) /,
/G PreShv(SmF/(G / J), C). /,
(U, i)
/G PreShv(SmRig/G , C)
k
cpl : SmRig/F
iI
DiaSchF/k
/G SmRig/F
(F , I)
(f, )
(B, K)
cpl : Sm/F k (-)/ / /G Cat . cpl
^ (f ,)
(F , I) i Ob(I) U
cplf
k
P
/G SmF/(F / /) cpl cpl Rig /G SmF/(F / /) k 2 k (F an , I) U k (F , I) 1
@1
SmRig/F F (i)
k (P, cpl)
(f, )#
cpl
%
G
f d u F u hX V i #w G'&Wye d )))4 E"1yx e`X qtg&Wws cGTUhI e)eWU&t1t5&ttvpU"'"rts gX T9 F q F H F D Rs X F I H a B V S H F D X H F I S Fs Fs asY R I q F q a H D H V X e`yTybts D vR &%e'eB TWtT`''e u H F I S B VP H R F H Y D j a I B F u F B V aY R X F u F kz`&kmTWCyeS"CpTtv'ts Te&"r#pzs W&`'F wTB tWyT&W#&Etg%TWa UF PY I S F V X D a R H q a X Vs R F H V S q I R I H a X Fs R FY X V I R S Rs I B V I V a R S F s a B ye@wTR t&UgeF W&T'"'"WrwY q Rs F S X Y X x F q I H a s I 8 F B Fs RY xY &UF{9r5 eawTttWX`I a F xY I S F m V Y u F a a B VY I R TtWkv`X D 'eWoGTtW#pX Ud&tdUrTtWU&kv`X D '&cTtWv`X UdwyW&cWD V S a Fs I F a F xY I S F m V Y u F a a B VY I R V S a Fs F x X F a X X &WUTCF H FI S B VP a F x IS Fm VX &&ywY`&kTWD Y'eW5a&`X WIUW`Ik&tX W T"r`W%&UyUrWgC&&&tvbw`WI u F a F H S H X a a Fs H a S B V q F X I B F S B V S F a B A a Fs RY xY X a F xY I S F m BY a B VY I R &yw`&kvw{yw`#pX Uwe5&yw`k#w{GTtW#pX US V S a Fs I F a F xY I S F m BY a B VY I R V F q I BF aw'Tx`eW&`X D Fs F X F a F %WbY H D@`XwTteS1c&yTtWU&k#t@WWGW`kib%v`eeTeyw`c&c`TR j #y7 Gu G@&Wye d G{G4 s"1%e'u H a F Y Rs I a F F xY I S F m BY F X H I S H X I a Rs I B R B X F S B V S B VY I X F a a f u F u hX V i & $5 0 3 2 0 ( & $ " F I RY q u uY F V S q F xY S F V D a F X H I S H X I a a F q a R S Wvt"'tts HTe&"QaTtWIU&kmv`X QWWGW`kC&"yehF ewTe"ts a Fs R S V a F H I S H X I a a Fs X R D F S B F u u V S B &`X WU`Wc&tTrU&'yUsCA '641)'%#! eU Upe CUW"7ep @ U eW w"dT U 0 U $ @"v 5 @"T7 e`bzU Uep w"dT1 7e 0 b op 7epe {rUW"U@ @U eWo $ ) '"T7 5 @"v7 Upb U U p Uep w"dT1 U 0 Uypp 7epe { 1eeUW"U@ @U eWo %# $ rq9p 8 U`e T1U@1 v #'T@pb" T U@
F
W UTTz
D
F
yppUT
U UT e' 5UW yvz@ e pU p ee{`pUT edUw UW"Uob
k
Tuet"uF'ueh`XTVeid ){G4 @F qgFttseFUSFy'& bQkyx e`X byy"%v#wcr'e`T& WgtCA d s H F u u Rs I a F F H F D Rs I B V q I B R xY H a F u h X V i I Fs R B W h7 U 1 W e`Cie) ' CU rT o ockk5 7pye)
B
1
F q a F H I S H X I a aY V X I a Fs Y a BY R I B FY I V B A "&`X WU`WcbT`WCwWwTg%&w`"TC
i Ob(I) U B U
I F g F a aYs s F u X VP R u i S elc`bwe'`T@e p`a
(B1 , i) U
X
F I B R xY H a B VY I `%v#wciTtWwY B "'twvCA q Rs IY RP B F xY I S F m V Y u F a F H I S H X I a Rs gYYY TtWU&kv`X D '&cr`X WUW`k@bqbtt F tsvR&S"Vws Fs R S V tv&"ws F xY I S F m BY F H I S H X I a Rs gYY yw`&kvwg`X WUW`k@bqbt F S V tsvR&"ws F xY I S F m V D F H I S H X I a Rs gY TtWU&kv`X g`X WUW`k@bqb a Fs q V r&teh 'u D a F X F a aYs qY Y X R u i S cWW`tt1F tyw`@epWa B H
F (i) (B1 , i) U Acst B1 B1 F (i)
g W&F` Da X S x R FTlQg Da FX W&` a F ua DX V &'cbY i `T'u a"q F F a a Rs Rs I B R xY H a a H I S H X I a Y V X I a F X F aYs R Vs I H F B j H F X I B V u d B VY IY a V D V D Rs q F I B F qY x B VY I R aYs R WWyteSst%v#wc&F`X WU`WchabT`Wh&USeWttv&S"w'"e)DTV %q`W%y0vt t Td yw`zWy)T`X sbF qW%&"t&TtW#`bwT`X eyR B F"bTtW@@& pWa qY Y X R u i S I H VI H V "T`X yGD I UF F qY Y X R u i S "bTtW'& p`a F ws a B VI V GTWT Fs R S V wTe"ws F xY I S F m V Y u F a F H I S H X I a Rs gYYY TtWU&kv`X D '&cr`X WUW`k@bqbtt Fs R S V wTe"ws F xY I S F m BY F H I S H X I a Rs gYY yw`&kvwg`X WUW`k@bqbt Fs R S V tv&"ws F xY I S F m V D F H I S H X I a Rs gY TtWU&kv`X g`X WUW`k@bqb a Fs R S V wTe"ws a F X H I S H X Ia aY V X WWGW`kQbT`WI a F a X Vs I B FY I B A g t u B VY I wsC`ywTRg%&w`"TVC5UvTt u TtWwY B ){G4 tTVy1&tveS"ts XY x a Fs R V F F eSBetsvR#xtY %&s"T%e'&TtW#be`Cc`&F q Ug&teh ''"q H H R I B F u F xY I Rs F X a H a a Y S a Fs q V u F a F H S H X a aY V X I a F X F aYs R S Vs H F D B A i SY x B a &`X WIU`WIcbT`W&tseWbwTe"wgI"&GsCpbeF WtY "1wYyTtT)TWI@W'F"Cg F q F Vs V D V R a q Da FX W&` FY X V I R S Rs a B V a a tWyTe`#&byc`bY B H I H VI H D "yWdX yVGQg W&` Da FX B VY I SY X I a F X a h D R a F xY I S F m V D a B VY I R yw`bW`kWp&`X TpTtWU&kv`X dyw`#pX Uga&F I%eF et%&"q Y %CF p&Gfw%TWCayw`#pX U&ts{ VS q B B B FY x F H a i S h a Fs I B V a B VY I R VS a F V F xY I S F m V Y u F a F H I S H X I a Rs gYYY TtWkv`X D 'eW1`X ``Wcqbqbtt I BF H X X R D I BF &%e'u yWTRvd%e'u HyWTQ&eB T"%TVWdTtWv`X eS&tqV X R a F B V q I B a a B VY I R V a Fs F xY I S F m BY F H I S H X I a Rs gYY yw`&kvwg`X WUW`k@bqbt %e'u T`vTRdy&'u T`vRQe B yV"%TWQaTtWv`X &tqV I B F H X R X D I B F H X a F B q I B V a B VY I R a Fs F xY I S F m V D F H I S H X I a Rs gY TtWU&kv`X g`X WUW`k@bqb XY V x g u I F d u a B VY I Cv GUrDwGu GQyw`zY B ){G4 rtTy FY X V I R S Rs X H u I H F D B w`TyeWveibhtY B 11"&GCA a Fs q V u F q a F H I S H X I a aY V X I q &teh 'q"&`X ``WcbT`WF g Da FX W&`
(W, Cof inj , Fibinj ) (W, Cof proj , Fibproj ) PreShv(SmRig/F , M) (W, Cof s-pr , Fibs-pr ) SmRig/F (i) PreShv(SmF/F , M) SmF/F (i)
3 UC)@b
b
U p "ee tp e )y s U 'b t `@y e pe'yU@1 v #qT'b
Rig
1 7 "1 1W p s Q yhppUT 8 8 @9Q7
o
3 8 6 ! 8 2 70 e05)0 v G`
2
B1 k
(B, K)
k
Nis
SmRig/F
PreShv(Sm/(F , I), C)
PreShv(Sm/(G , J), C)
Nis
(f an , ) : (G an , J) 2
Nis
(f, ) : (G , J)
(f,)
(WNis , Cof s-pr , Fibs-pr-Nis ) ^ (Spm(k{t}), Spf(k {t})) B 1 = B 1 k X k
/G (U , i) Acst
Nis
(WB1 , Cof inj , Fibinj-B1 )
(WNis , Cof inj , Fibinj-Nis )
(WB1 , Cof proj , Fibproj-B1 )
(WNis , Cof proj , Fibproj-Nis )
SmF/F
(WB1 , Cof s-pr , Fibs-pr-B1 )
RigSH(-)
/G (F , I) /G (F an , I)
Nis
Rig
Rig
Acst
/G PreShv(SmRig/(F an , I), C)
/G PreShv(SmRig/(G an , J), C).
A Ob(M)
B1
F
D
%
F
F
FSHeff (F ) M
M
(f, )
1
/G (U, i) Acst
k
U
!
((f, ) , (f, ) ) (f# , f ) f
!
(f , f ) (# , ) ((f, )# , (f, ) ) ( , )
(f, )
(f, )
I
J f#
Nis
Nis
e
(f an ,)
i Ob(I)
k
Nis
I
D D
RigSHeff (F ) M F X
o tG
wh"p1t
U w"dT U 0 b oT peb W) 7epe { 1eegW"e7UTU"#pT@w e hU e` e )pUW"Uq @U 7 eWp $ ) '"T7 5 @"v7 Upb U U p Uep
!
!
U t"pT1 U UypT peb W) $ wh"p1t U 7epe { 1eegW"e7UTU"#pT@we hU e `e )pUW"Uq @U 7 eWp %# 8 rq9p T U@ W UTTz U`e T1U@1 v #'T@pb"
F
D
a FXIS F Da a F &&`WU&)c&"q a YX R &FtWyVT&WI#&S1"HTR X &FWSGTPpwsd`"q&FWU XTV)D H I B V F FX B I H BTtWU&kTWX QF pts h`TCbQcbtw`"1y# VY I S F m V D q F H u X VP Rs F aYsY I H B V a B VY I R X D V a F TtWv`&Gy&"q ayeShtsX HyGDHytYyVUFWI TWIBT`vV Tuvt t r'u'eF G{G4 @F"qCIUFtwGu y'e`TeWE"ts Te&"QGTWW"q U#`ws W&`rF R F V F a P H V a VI u F u d d F uhX V iI H q F H VS q a H Vaa F YS I RI Ha X 5UW yhppUT 1 e 1 7UW h U gv r rckQk5 s U e' W UTTzE e pe p eeippUT etU iUw
B
1
F I B R xY H a B VY I `%v#wciTtWwY B "'twvCA q Rs IY RP B F R I a Fs R S V ws T`ctv&"ws F xY I S F m V Y u F a F H I S H X I a Rs gYYY TtWU&kv`X D '&cr`X WUW`k@bqbtt F R I a Fs R S V ws T`ctv&"ws F xY I S F m BY F H I S H X I a Rs gYY yw`&kvwg`X WUW`k@bqbt F R I a Fs R S V ts T`cgtv&"ws F xY I S F m V D F H I S H X I a Rs gY TtWU&kv`X g`X WUW`k@bqb rg Da FX cW X c&te""'g"Q&`X WU`WcQtyW`{ctv%&w`"TV H a a Fs h q V u F q a F H I S H X I a aY V X I Y a BY R I B FY I B A a Fs R S V C&&tv&ts a F H I S H X I a a F S q B js F q F I H q q F R I a F xY I S F m V D Fs q V u F q F H I S H X I a R a F q &`X ``WchUqF F H "WwY ""qts vpkqTtWkv`X teh 'q"`X WUW`kW "g Da FX cW XY V x a F xY I S F m V D a F H I S H X I a a Fs H V D F i S H R X H u a X Vs R I H F D B A g f u F u u tY B 1pTtv"&GsC#Te G1''eF )))4 wyT1TtWkv`X C&`X WUW`kgwX T)hpv% B Fss H F q X H F I S B VP etwtY C "eWUTB H Ia F c&rg Da FX W&` X e`yt%&%vpX U%e'&TtWkv`X D 'eWQy&'ets eWdc&og H F I S B VP Fs I B R V S I B F u F xY I S F m V Y u F a I B F u F H F a I a F Da FX cW I Fm js H B F ek"TV 7F %q&wY g Da FX W&`
M(F ) = Spectan (PreShv(SmRig/F , M)) T M(F ) T an B1 T for B1 (WB1 -st , Cof inj , Fibinj-B1 -st ) (WB1 -st , Cof proj , Fibproj-B1 -st ) M(F ) (WB1 -st , Cof s-pr , Fibs-pr-B1 -st ) M(F ) M(F ) = Spectfor (PreShv(SmF/F , M)) . T T an - B1
g cW Da FX g as F u X VP D a F X a F qY Y X a R u i S pt&'WycWr&"bTt`y@e p`a F q F u u R X RY q H H V D a X Vs R F a V D B "r''vpTvb"EB X yGppTtvrWy)qCA F xY I S F m BY B VY I R yTtWk#tqyw`#pX erF B c&gW&wW&"WWUUrF %`tH D VS H I a F F X hY B X F q F I I F S H aY F "q F %bY T`v'TV p`X erts %etv#tY %&r%e'ets tv c&F H D VI V u i F VS R I B Fs R x H I B F u F Y RP I a F H X F H X R u F X F q I H a sY I F 8 F B yeypv@&WQ"o'"Wtv&e)1'r5 g Da FX W&`
(A1 )an k (Gmk )an
F %vpX Uy&'eTtWU&kTWX D &c %e'eeTbs 'e`hB c&g q I B R V S I B F u F xY I S F m V Y u F a I B F u F S R D u F X H I a F
Da FX cW
a F a X F x BY a F R uY a X H F I S B VP a F q F a V D u V S Rs X R cpeywdTv@td`eWUTp"reWy)'TU'bQvD
A1 Z GmZ
a B GvR q
F "q I B R %vpX ed%&&Uyts '&WB yTldT"t t ysTtWU&Wa TWtpTR VS I B F u FS R D u FX H a B V x R a H V B d B VY I S F a H V a Rs a B UU 7epe { 1eere1 7epe {rUW"U@ @U eW1 p h1 w"dTQ
Rig : PreShv(Sm/F , M) T
F q F a X F x BY F R u js "gcpey%tgTT'tY 7pg
e
%
U@
e pe
U "UU $ ) T'@ 1 v # rpUT r`y e pe y1@1 T # eU e`e ) eUoU1U 7epe {rUW"7ep @U 7 eW1 p h1 w"dTQ
U $ %# )r rckT!r p"ee U eyvzr e pe T'@1 T # F SY S X F F B F F a aY Rs I a F F x H F D Rs I B V q a H V a a Y S I R Is H a X Fs I B F u Fs R R B yUbUpe"UEeeW`tTtc&gTe`X tQ%T"dyc`&F q U#pzcWrtQy&'etvy&CA
B B
1
% A
|w yw xw z{zw0v u d u F u hX V i v&w Gg&Wye d% )))4 F q F x H F D Rs a B q F u u V S I B F u F I S F F X I B V u q F a Y S F n a F Y RP a F S B Fs R x H "ETe`X t'TR s''TU%e '&WyR UW`yy'e" iWUews tvU&wT#wY % a F q H &"X Wa F q a i S h f a Fs I B FY V x B "@F p&G&t'%&wyTeF s j H Y B VY I R aYs R S V tY %owyF yw`#`ttv&ts R bs I B Faa R D iS H R %&c`TEF pGvy B FssY H &wtwC "pX e`yyV iF %&F `eyx F q a H FI S B VP a B H X YX F I a F X sY X BY u X F I H V Wc&`tr&&F t`e`'X T
B1
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
F
cpl : PreShv(SmRig/F , M)
P : PreShv(SmF/(F / /), M)
k
F
k
/G PreShv(SmF/(F / /), M)
/G PreShv(SmRig/F , M)
k
U
PreShv(Sm/Z, M)
Da FX W&`
T an
T for
/G PreShv(Sm/k , M)
/] /G PreShv(Sm/k , M) [-/ /G PreShv(SmF/k , M)
g
PreShv(Sm/Z, M)
T an
T for
F % %&&Wvt"'toF `ey1CA H I B F u F I RY q u uY Y X x B Da FX W&`
B1 cst k B1 cst k GmZ cst
T
Rig
A1 Z GmZ
/G A1 Z
FI B B `vV ECA
M(F )
cst
k
/G PreShv(SmRig/k , M)
/G PreShv(SmRig/F an , M)
cst
cst
F
B1 k
F
D
UTwe ie ) E `"7e7ib 7sUTe)
F
%
D
RigSHM (F ) F
((f, ) , (f, ) ) (f# , f ) f
FSHM (F ) F
F
f#
(f, )
1
M
f# (f T -)
T f#
k
F
cst
/G PreShv(SmRig/F , M)
cst
/G PreShv(SmF/F , M) .
PreShv(Sm/Z, M)
f#
B1 B1
D
T for -
cst
cst
U 'ew pU1e' e pe yhp@U pU@1rUwr" v p r eUw U1Uw Upe UW"U 'e UW@ p @1 w"d yC U r ockT!r "eU eUTTzq e pe ysp'e pe'q
%
D a F X a R S Fs a B q F u u q I a F B F s T)cWdyertdvR '& rF Qc&E& g
X R D Tr B y"k B V q Ia F s
Da FX W&`
B VY a Hs S B js F I BY V D I a TtctUGwY 7"ye`ytT)c&F
/X, /X.
F %``yV yGW&`pTetdTR rwT`dy&'etvy&kCTTtvGvrwh`T1R H aX D a F X a R S Fs a B q FY R X x I B F u Fs R I a F F H Vs R B R Fs H u X VP
F at F (U /X) =
X R D B B V q Ia vg T"c&F H R F S aY RP T&U`bvU`X D H "q aY I H R F S aY RP Wt`"vU`tTF vB Uk"yB pkg s R IFm V H Ia F Da FX W&` Y a I B F u F H F a I F Y a H R F S aY R cdy&'ets eWpeGWvU`tTP B pkF WewyU"UdUF H Ia F I hs D u V S V S I F I h D u V S FY X V I R S F H a B R q a H Fs R Wews 'yUdw`TyeWveB v `X &wT#x g Da FX cW X Wa H H R F S aY RP X D v&eWbveWB TF t0e'Wy#W&`r"bTtW@epWa qY x gs F u X VP D a F X F qY Y X R u i S Fs X R D I B F a D F X H R F S aY RP D Fs tTrW%eW&`X &WEvU`tTU`X htpg Da FX W&` I F F qY x H R F S aY RP D F Uh"tEvU`tTU`X ts &ylQg S F x R Da FX W&` F Y X S Rs X R ts w`UtCvD FX q B F B F e`"&T &g Da FX W&` X WptTywyGyWts H a FY Vs V D V I R FI B B V `vV 1T#g &'Wy&cW s F uX VP Da FX F qY Y X R u i S "tyw`C'& p`a B d B TV %T` H B q I B RI
3 B
a h D F ua i DX V u Va &`X 'WtY WyycbY
aPY I R I H u u V S a X X R S a F q I wW#p"h'yUp&`WTe"UF
a X H F I S B VP a F q a X Vs R IY H q q B F B `eWUTp"d`ywTz"&"i&qCA eU Upe ) 1eee Upe )UW"UEysU"#pTstU iU Upe )hW"e7 'e UW p 1 w"d T'
U eU U@ U`e T1' v # "C 1eeUU 7epe {UW"UyU"v`yqtU U 7epe
/G Spectan (PreShv(SmRig/F , M)) T k
% 6
"UU 7 eUW)y e pU y@1 v # $ ) eU Upe 'e 7 U` p q1 t"p y {UW"Us
p"ee F pTy iS H R F q haa V "c`y)D
(f, )
U eyvzr
(f, )
% A
e pe
T'@1 T #
k
I B R x H a I R I H a X Fs u d d B VY IY a V D V D Rs F q I H q q B A r%v#tY WQv`ws W&`rw1etu TTtWwc%GyWX br"wY "& C) I BY V m q %tTklyR B dWWTq"c`y)D H Yaa H R F q haa V I B F H X R X R D I B F H X R F a aYs I a %e'u T`v@T1%e'u yWTqWWbtc&F F %WpTV H aX F IY V X WwT`"q I BY V m q R %wyklTB H X e`yws H FI S B VP F g as F u X VP D a F X a qY Y X a R u i S `be'`T%W&`g&F tyw`oy@e p`a F q a F u u R X RY q F q F u a D X V u I H V I H V D a Hs D "p&''vpTTt"r"g'cbY i `T'Q"yWdX yG&wrF
RigSHM : DiaRigSch/k (f, ) f /G TR et FSHM : DiaSchF/k (f, ) /G TR .
a I B RY X R x R X I B V S a H F I S B V `%vb`v#l``%TUdpX e`yP "&"WwT%etW"yqyV 7 tu T''e`T& WtQv % H F q Y a BY R I B FY I V B d d F u h X V i I Fs X R FY S V a a tU"WWyR Y % H I B RY X R x R X I B V S H F I S B V %vbWT#l``yyUdX e`yP B qe%&&wte`X `#qe`U ea H B F I B F u Fss F H I R B q B F I j F H F X I B V u d B VY IY a V D V D Rs q FY I X R D F X hY u F D yW`%T'" Gu ysTtWwWy)T`X bCF tWWvCW&w'&WX hR U peb W)pUwg"prUw 7epe g`"7 CUThU"v`yhwe g 7epe {rUW"U' @ U 7 eW w"d y 7e 0 oU $ @"v 5 @"T7 e`bzU Uep
(f , f ) (# , ) ((f, )# , (f, ) ) ( , ) B1 (f, )
3 8 6 ! 8 2 70 e05)0 v G`
(f, )#
(f, )
2
(f, )
F
RigSH(-)
M(-)
k
U
$ %#
C
r
rckT!r
F
cpl : Spectan (PreShv(SmRig/F , M)) T
Rig : SHM (F )
P : Spectfor (PreShv(SmF/(F / /), M)) T
Rig : Spect (PreShv(Sm/F , M)) T
k
cpl
/G RigSH (F an ) , M
P : FSHM (F / /)
P
Shvt (SmRig/X, C) PreShv(SmRig/X, C)
5)
c{Q o
RigSHM (G , J)
cpl
83
8)
s F X H I R B F u a i D X V u V aY H F X V S B F R B &WW#GC'WtY WyycbEB WyU&1iCA I UF g e e d d TTlwyk
k /X
8)
RigSHM (F , I)
C
(f,)
t
b
/G X i : Z /G Shvt (SmRig/X, M)
X /X
cpl
/G FSHM (F / I) /,
/G FSHM (G / J) /,
RigSH(-)
bF (U /X) = F (U /X) F (/X)
F
^ (f ,)
SmRig/X F (/X)
X
F (U /X) si U /X si U /X
cpl : RigSHM (F )
/G Spectan (PreShv(SmRig/F an , M)) T
FSH(-)
/G Spectfor (PreShv(SmF/(F / /), M)) T
t
/G RigSH (F ) . M
Shv t (SmF/X, C) PreShv(SmF/X, C)
SmRig/X /X /X SmF/X F (/X)
SHM (F , I)
(f,)
SHM (G , J)
C
2
2
k
Rig
Rig
/G FSHM (F / /)
/G RigSH (F an , I) M
/G RigSH (G an , J). M
2
SmF/X
(f an ,)
a t F
i : Nis
F IY V X `zyW"q
Shvt (SmRig/Z, M)
% P
I BY V m q R H I F u q %wyklTiB e'"TR
B1
k
g t e d d yvlwyk
U
)
I
o tG
&qu#qghusu7eEguH7g7uhwue7g &Hu&EhuH7g uPh rbi i i p i p r p o p o v g PXfq`) i f"&Xfbubvf&X g ri PXlXwwv Y v&X a@ 'xlffv g | uX b bp b` &Eqa eX s e p &X U{qPX flX h g d g ai ` a ` d a h h Xdd v a ` a c a i d X d m av X X ai p c v i a v X ` wvqrbb`)b`d h q`ffd%a i qewv wX a g g X h g `c X Y d ` d vq%u&eXReX'a &X 'b` &X v i a a d X g fey&X a d `c d r i i i rbi p X wv Y swPc" Xv ` Xwwv yX a g'v g Y x d a v i g Xd c a `Y X %PX PEh PhqUsV p i i p cv i a v i qPX fsX i wqbb)bfd vh b)&r qa i i qec Xv r ` ` ` g ` X d h g ` i X d h d d ` c Xv h ` r %a fwPfqb&fd c bPPsfgsfwv bec b` a ` X c X v h d X X d v ` i r r p d a ` %bfHv g & evX v p b` X X d h
X wv
////
Y w&X
k p i i i p i @r U"0k @X fbfd ` wwbi i a ffX qwh a i ` v d a Xr c X X v v rv qf pc r i o p i a g d eX h f1w e fPX g X X a g ev eh za g Xc Y dv q` X a a ff&h b` 1 %&X wPffwv fa U5RRPHEC 'A c X d a X g X X d h d F DT I S Q I G F D B eew P ' wPf e b qe h y hwP Pq sw qe e # #@ 1 q1 e 1 qe " ' 5ef 5e fj q h P @ #R RU5v @e k i @r P'i p rt r i r i @r p i Rw0 X h v i p sX v i wPXi qU X d h d X r h `Yi qp pc &X i sV i c X i i p f&wa @r X c X bi m "g X a a &EX bh a X v rv qf pc g qwd fw g a `v cv v d vh sv g wsq& Xd av X p c
i B1 (B1 Acst ) X V m // i (B1 ) i (B1 Acst ) Z V k
1 ^ 1 ^ i ((BZ Z BZ Z V ) Acst )
r i i r i r k p p o X ` Pfq) Pc r v%p a &f&v qf` ` ` sp wPX a v%Pf"qa ` d a X hp a X X d p X X d g di a X v d e ` p X q s"qi fb` ` p i &X i p c x r )p b` d X d X k fwf%PX v dv d a i a `v d c X X g d X pc q& V v e&w wv g X e Y Xc a X i X g Pf%q` X b` '"f hXhd a c a d g X P&vwh vfwv g w bfha eX ` h X X Xd av X a &X g d
B1
1 i (BV
r p i r i p i p p X g Pf%q` q` "g 5 %PX w fqwP f hXhd a c a d Y d a X v iv h d X iv h X X g v b"qf%PX efh bffPfv d a ` u X d a c a `v g c k k r i i i i p p i @r i h & v e&w v g X e Y Xc a X i &X wusX bfh h g X v v rv qf pc a g d eX c X &qi X bUY h ` k i p r p i i r X &q&xp &k bq%i &X 'r X g &fqq` X p c X X v `f a h Xhd a X r '"o fPw v i x&PP v g d g p v u i X e Y k X c a X i X X i h X &qf&fh X g i P'%b` i h b r h rfdr a Xh gp ` r g ` vc qbffX P1 @X X X hv ` i X wvwqbUswe f&X m g Pfv Y X a Xc X g ` &vh b) a `v qwd d g ` Pc sX %PX d a i p 'r r p i p i X fd v v X g `c w qe d fV qPqwv&X g h X X ev h X i av d X g Pqi bh q` "g R hX g ` c a d Y
aNis i 0 (A[-n], H) aNis i i aNis i B1 // V A A Ob(M) 0 (A[-n], i H) // aNis i 0 (A[-n], K) t
k
i r 'r i i p i k i f Ywfd d" v ` h c a d a Xv Xc aX i PP v g X w " X c ` i peX id%Pbk ` i g ei cfk h ag X g a Xc aX i &PP v g @wwqbPX X Xv ` i a X a `hd a ` dv q` w "` X c X aX i &PcP v g @X h f@& " g X c ` g i qf%b}~ X qg t }Pwi v g | p c r x a X p p i p i i o n Ey X h &fdhfb`U{a gzaPXv jcfb` dgX g X c a Y e a X gdc g d fh hhf xX w v l&PP v g 1&1PlbfhX lqwd h d X e Y Xc aX i X d X X iv d c av a `v e k r i p 'r r p p `eywXbPf&h c X i h X h &fecbU'jX g %PX d v g X d a `Y X d a X v dv lg a k p i @r i i r i i X bfh i v rv i p k qcf p d"bdwh b)srX wv Yec&'r iv g s) sa gvgX p g `r g ` e X r &t w a X X d v X g X d X v X e Y Xi c a X i i f r d a @q&X k Pfefywwv EPP v g X ggX a d m g Pfv Y Xc X X v pfq` h a qX g u X X v fyX brfh v v qf pc d bfh b)bfd h feX a g g ` d g ` a `v e d `c i i k p r r r i p i p o n E5 i d &X X g Pd X m"g Xev Y c X &X v fq` &%aPX Xfhc eyfb` gX p h X d d mX c a d m g Pfv PflX Xc Yh t i p i i i k t i p i bfhcX vEfbvfd h dbeyX bvfd fd g XPfv Y X iv d a a ` e `c a ` v c d ` &Xsqaq h fjbifec&X v'fX gfdhffhg @X bfd h f esV g X vd a h c g d i a `v e d `c X X ` fhqeca&X P` sa h X d a X p p h &febUsX g g a g X d c a `Y X dv i p p t p #bi r p "qi h a b` fwbh qb` a X dv ` x d av a eX g d
H(U ). Nis i Nis Shvt (SmRig/Z, M) X i : Shvt (SmRig/Z, M) i! H : (V Z) SmRig/Z i! f (U ) X // Shvt (SmRig/X, M) i : Shvt (SmRig/Z, M) (W, Cof inj , Fibinj ) U Shvt (-, M) nZ // Shvt (SmRig/X, M) t
^ U X ZV (SmRig/X)/V
X dv ` fwbh
x d av y%wq`
a seX g d
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
i : PreShv(SmRig/Z, M)
// PreShv(SmRig/X, M)
p
t p ubi
r p "qi
i p h a q` a
i i i g Pfv PfswwP" X c Y h X Xv c `
i i g Pfv Pfh a v g Xc Y g x h g h g Xd c a `Y X V F DT I S Q I G F D B PfebUHWU5RRPHEC 'A
!
b i! inc
i
B1 V
A
Acst )
H
SmRig/X
B
1
i
p
i 0 (A[-n], H)
p : i (B1 Acst ) V
V
s sp
Z
// i 0 (A[-n], K)
// 0 (A[-n], i K).
iH
!
AE
// i (V Acst )
Lim
t
H
B1 k
X
k
L(i F )(U ) =
i: Z
(U i U )i Cov Nis (U )
Colim
// X aNis i
Eq
// X i: Z // i aNis
CovNis (U )
r bi
p
^ U Z U )
U
CovNis (U ) ^ U X Z
^ U X Z
// CovNis (U Z) ^X
i
^ F (U i X Z)
^ (V j U X Z)i F
(U i U)i
i,j
(U i U )i Nis t // K B1 k
^ ^ F ((U i U U j )X Z) B
1 m
Li
k
// i ((B1 V ) Acst ) ^ Z Z
f : H U
H(U)
u : H
SmRig/Z
^ (U i X Z t // K // K(U) // K U
#
v x wq E@
{we qP " t 6 w q w # x
// X
^ X Z)
qP )vw # # #5vpqi 01yUH 0 i X e Y Xc aX v efP i vv d g w"'a % a
U
s e w @ @ h &c ec a & Y & ( )gf7db`)XW k iv wg X a &X g d
B1
i
sh P t leq yb 6 1wP PE " @ qP " r 6 7 b 'P pk i rbi i V I "P n F d bffX vPaqwvfq)bh &bi gqfh xvX a& g ` c `v d ` ` h X ` h X a ` u wf@bu X h ` h fv wqfq` u fHf4 qa Xh f r i r oi X sX g Pf%b` "g 5"PX P&f"g hXhd a d Y v u hXd Xhd X r v h p fq` r
U Acst // (i Z ) Acst
p
i r r a Xv d c ` c d a Pffh "Pv%&X X g bfb` ` &X v `d p d
U Acst
Xh f
X
h #@# i r ! d i g "%a v sX h X i X p c a g x a `v d c a qPbsq&X 1}qwhq`
i r 'r i r r bi r p i @r p h Yv d ` h X fa u qf&X i pc i i i wfd i "qi fh ywsX g @f%q` k sX & pg i PX r h d a pq` hqPX r qUY k f&UsX bf i @ h ` X v r v h I f @r'r ec h c X r x a F d d cxg e X V"u fHf4 %a " a ffw)PX bUY X v h ` pc qf X h X v v rv f1ffwX qwh pc qf a y g Xh f&U c X bi i i UQ "P r r p i oi i r bi T I SQ "P R I F f&U Xh g c X X V u fHf4 'X X qb a wh X & g r ` a h X fa r p r i i r p i i bi o p p oi hb`& pwGu fFHf4 jX d%a&X fwv fbUPf"qfd Pbv f}a P" bPwu EC jX bi &fh &f" X d h d X v `Y X d g ` h X i g Xh `X p G F D B X g X h Xd dv wbr & A ewe byf ef @ 9 # h Hv q ) qP @ " R 2 0 31yUH 5)' %# ( & $ 8 7 6 5 4 # i i k p i i i p i i i o p m Xc &X fqb&fd c wa Xv h ` r X wv i xp q` g eswfd a hd a v cv X vv g th b` i p d X ub& fi p a "g eo v Y r r h qff&beyPX g v g sw%&X a ` X h h ` c X X g a d h X &bi X X q&"b"&P)P}a pc Xd g `d u dX X Xd f"g ` ec
Xh f
i i k h fzg ev Y g Xc
a g
Z
dv wb`
i
i i k p i i p i m Xc "g ev Y X r o fh yV r a h c X g' i x qh &w r w i r r r rX Xh d a wv %PX P&bi qqwf&X ` c X d h X g ` a `v h v d X X h X a `v h P}P PUY qwfqPX v a }X g
j
Xh f
i p 'i i r r Xh f a X h br v i Rw0 o p c a a a ` c r r p i i eq` P sX X g v ba r sX X v p b` feeyg &Pqs X wqywPfe&"g v a h ` v d X c X h Xp d v h i ) a v ` X h g X d c X m X r p i X v i p k n d PX X b` P&X a a vc h qf g ` d a %PX X Pq"g ev Y v i a ` u Xc
aNis (i F ) Eq RigSch/k
i
p i p g b` fq) fa a a X ` g r r o i p p v h ` c X v c a ` r fq` v E&qr i i X P7epc h b` fa v%PX i w Pfb` s ` v g d a X r c ai l o a g h g ` u d Xh b) &fh sfd g
F
r p x X p b` gX hi ` v d u v i a a d bX b` v&X X hp ` i g qbsp X g %&w d ar X v i r yi i i o g av fa ywq` a X &'v g eX d X & g ` d X v X g v g Efh & X %&X fi qefh d d a Xh g `c X k i g i fbfyh p fqfec vr f ph ` d Y X i X h d ` rx ai fi X @ h a x a `v d cv h d X d a bfef&h r 'qq` a yqr b k &X g'h ` d X %i PX fi qe&h r d a Xh g `c X i p p i k p i i X r wv Y i @ V d a %PX r i bP&}wqa w"g P@PX bU X h g ` c X h X d v v d a X h ` Y i q p X @r '&X kbfj&fPwh i pc i PX v r v h d v d X c g i hX a PPX h X d g X a ` Xc hXhd a f"&f qyu PPxPf%q` h b g ` X g &fd X X d eX qwh v rv q pc a g d g ` d g ` X dv m "bjh qfg"eX
k j U
f
3 8 6 ! 8 2 70P05 0Hf
(U j U )j
2
k ^ {(U - U X Z) U } Nis ^ ^ = (U - U X Z)X Z Z ^XZ U L(i F )(U )
^ ^ (V j U X Z)i Cov(U X Z)
t
Colim
U
// X oo
RigSH(-)
L(i F )(U ) = i aNis F
Uj
i
F (V i )
Z
i SmF/X
j
^ V j = U j X Z
SchF/k
////
i,j
U
^ ^ F (V i U
j
^ U X Z
U
// X oo
XZ
F
^ V j ) = LF (U X Z).
/Z
t
Vj
U = U U/ / ^ U X Z
L(i F )
SmRig/X
y
j i i F =
X
X
X
M
F
j# j F
t
t
i i F
j# j F
X
i
U Acst
// X Acst
// (i Z ) Acst
Z
[X
TX
^ ^ homZ (P X Z, X X Z)
e
hw P qP ) { P rEP # 6 qe @ " @P #Rp " e qA y @ P w eb v u
TX
,s
X
Acst
X
U ] Acst
TX ,s (P ) =
,s
// X Acst
homX (P , X ) hom (P Z,X ^ Z X
X
// (i Z ) Acst
B1
X
s : Z
B1
SmRig/X
// i i F
// F
t
[X
U Acst
s: Z
U
F = at (X Acst )
U] Acst
^ P X Z
^ P X Z = , ^ P X Z = .
// X
// X Acst
// (i Z ) Acst
// Z
s
Z
// X Z ^X
F = at (X Acst )
i LF
TX ,s
s
SmF/X
homX (P, X ) homZ (PX Z,X X Z) ^ ^
TX ,s (P) =
^ PX Z = , ^ PX Z = .
g c X bi i
T g i io io p p r r o p i f h X d a r p a `v v qwd c c a fb` wb` dv X &d v h ` d d i sX g &p fh%b` X i qr r dv p pfyX i X v p b` fi & a q` %a P r v X a d v ` X q%awqfvwvh sPa ab` &swv p PfyX a X d X X Y X X d p c v i i f v X d v a d av ` X iv i h X a ` c X Y X g g ` X v h X i a i r q%&X 'ryfk w}g q%wqfsi X bfd f&qeywv sa vh b)w& e qPqwv d a ` X h e X %bf1fdywX g h&dPqx X '"f X @ &X i bUY i qbi X vh i d g ai h ` p c r a g c X) I &bi #P X g fqi ) fa X ` g u 'r i r n ` v iv h d c b5 wf g X & a h g a vc wfb` P&X 1fv wqfqesvh '&bi qqwfqPX X h ` h ` c X X d h X g ` a `v h v p i i p i X d&X a g X g Pd X X d h X g ` X av v ` i g h X g ` h d d g X a ` u X d d fPqi bbr k wfqba y&bi qfw"P)xq%qw P&X X g a b` %a d a d r i p i p i p o o % d a q&X Xbfe&)fh iv d c X X deX h Xd ` a a &fjb` g X ` @ a &X qqwv g &%f "&fxb%&&w effh a d a ` h X ` g d g X a ` u h Xv g cv d d PX X fPqi b` dhX g i p i i r p io p p &'xi r t r &Xqr vbq&X h av ` i dPX h X &ec fq` "P) a Xh c a d gX x @X g w v a `v g c qwfev X V i p X g %&w d a X v i i p X a P%bec g p i X g &fd a X d a ` a X &bi q` br wfqbi a g d h X g X av v ` dv wb`
d eX d g `d h g ` "qf}b p r r p i i q R s 8 lX &h e X "&X w#& ea X m g X u c X c X g Pd X
//
X d v m X v fPu
V Flt(p)
X v m X fd"ep v X g fb` fbfq) c a a ` ` a ` d X r av qfeb wf&X X h ` X iv d c g qbV qwh&X
R g i p oi I h X dv h a `v cv d av ` X X h e d X p i p p qqb)w&vqfwh qfw f&i bwvi %bfw i gjX f&sX g Pwf&bi i g ` d g ` Xi e v h X i a v d a ` p bfd r bi sX g u r X v u X I q b d & v h ` X e v i d a q%PX i bP& i X bbb)i bd i Eb)}i bfd yfiv "g ev vf i re qPq%E%&wPX X hr g ` c X h X v r ` ` ` i h g ` p a ` v i d X c Y i o h X v h X i a vp d a X v i x p g i i 8 g@X ab` v d v h Xhd a a wfjbfd g X v iv h d a `v c h PX v c v m g X EX v i v h g ig w w X w""&X wX g v wfd p fhchf&Xh i Eg &f%bi ` id iX w p i X i k g a a d q` gX X qwvd a ` g r X v h X ` g d g Xp a c X X g %f a X ` g u p i lg i ffg &%f s"P)i a X S` i @ a g r bi r d a q&X h & d X v&sfwv i fa I X v p p b` fwX %bf c X X d h d h ` v d a ` p eX i i i oi Xd a v f%PX i a `v d cv h d X h X u a `v v d X c Y rP'fbr efwka @w@p%a v f@ g r X qwhfgf'wHbfd f"g ep v &h i kg ia ` v @ g c v X d i g g h ` PX bi UY qf pc g r qPE%&X i fi qei fh a g p eX p cv i X a v d a Xh g `c X d X qwlefh i v rv h r d v r Xh g `c Xh i k bP&a g d a %PX i i bP&a h 1X @ r X X h g ` c X hi g ar h X p c vp i a v d a qPX fo %i &X fi qefh X h gr ` c X i c X "bi qf pc a g dv wb` wP" qPx"g Pfv &wh fEX v p b` wwv g w5v g &fqq` v c ` p cv i X a v m X c Y X g h ` v a g d av h Xhd a a
d PX r p p Xhd a c a d ih&f%pb` X q` 1X q&X p %PX r p c v i a v d a@ i d g ` X v v rv "qHG X qwh
I g i i p r r p p m g Pfv PfsX X v p b` Xc Yh h a g dv g Pa aX
i X b v h p i p i r 'r i p i r r r a fd h fw&v P wPfd bfe&fX g X a d X Xv a `v d c X X Yv d d "g `P%a Ph c d aX u eEw eEX v p b` dX X d h a g d a Xv %&wq` i r i X v Y d a i i@r i p p ik i &wr Pc w@r X k P i%P X r R f p d a ` v v v d X v i a ` vi h X X i %bfwyevUEbfdfPf r &HV p r'lX g h i h w"xqyU'wv g h q0 X i ` @ r r X vv d g a ` v X h g ` a a bX @ i &X bUY g g ` a h ` q pc r X a s&X g d ` q c g d%PX fd v a X v1P X a h X ' ab` X g v ba s%PX i bP&'ws%a f"g PeReX a a ` c d a X h g ` c X h X d vv d a X u d X r 'r o p @r i XfwvgX h ev d a `v h qwfqPX v sX h q` a i i ik i r"qi p p ik s H p a 1dgyv g br wfbb{aleX g X X av v ` i g d a g d lgX X g qU h ` V vvv g ev @wPfw xX X c Y X u g X S` @ a u a bX S` @ r r o p @r i i i s X e d fh vw&X a `v h qwf&X v sX h a q` xg ev Pfw sX Xc Y X u g k k i r i @r i r i @r i r"qi p X g qU h ` V vv X @ PX bUY a h ` q pc a g&"v g qr afqbi'a eX d X X v v ` g d u a bX @ &X qUY h ` pc qf a g d &X r o p p Q f X X m X k r r p r p ik p i r i @r p p @w PX X w& d d %a&X X PX %&X Xbr qw wwg ` X ` @ av d a a ` X ` a &X b&' PX bUy g ` X a h `Y pc qf X "&X &fbf%bf m g X g ` d d a ` d PX v i i pk X d Y v g ` r & v &pk v gfd wq'r X ec p p i i r p p a v d a ` X h X ` q`wv bPcy&y&%f b` "&f@a g c a d g X X g qb`)@b`fw%fqfh &f%b` X '"ffb&q&X eS 1X p c h g a v dv ` ` h Xhd a d g v u g ` v c X X wEw i r i p o i i i p i k h x d a %PX X g v q` a Xfq&X v a c dv d a g Pfv Pfh V Xc Y X d i g cdX %a v Pu u u X c e w ywEP"u u g X c d X u k p i 'r p i r Xd fb` aa g q%&X f1brbb)bd h gb)%aPX fi qe&h a g p cv i a v Xv ` ` ` ` d Xh g `c X X bfh v rv pq c dv wb` i p Xi d v ` h d a X &) ewbfPX P"fh a c ` k r r i@r p i r &X %&X X g a v @a U5RRPHEC 'A c X d a v g Xd hd F DT I S Q I G F D B X v g d X h `Y w f wgyPX bU qc p q w " e P w eb v e @ u
y
x 0v q EH
k p i i 'r p &X cXV &X qw v rv h q pc w # E #
^ P X Z
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
^ ^ homZ (PX Z, X X Z)
// Z
s
// X X Z ^
qA
y Y
a TX ,s (V )
Et/Y Et/Y SmRig/X
k
k
X i Z i si
ColimV Flt(p) TX ,s (V ) 0 s s X
TX ,s Acst
Ouv(Y )
X
Z
Z
Z
(ui : X i
Y = Y
X
// X Acst
^ p P(Y X Z) /
k
^ p P(Y X Z) V0 Flt(p)
k
R
// X
// X
R2
s: Z
Y = (Y, )
Y = (Y , Y)
// X )iI
k
Y
X Z s
k
// X s : Z // X Z ^X
B
Colim T yVY X ,s Colim TX
1
Et/Y Y Y
V Flt(p)
s: Z
X0 V
s: Z
TX
,s
X = X X/ /
k
(V )
k
(V) ui Z
,s
Et/Y k s s
s s
X
// X // X Z ^X
// X
&&
a|V0 TX
X.
// T X
// Colim T X ,s (V) k
V Flt(p)
yVY
^ V X Z =
Colim TX ,s (V )
R1
,s
k
.
p P(Y )
Et/Y y Y k Et/Y R2
0
u TX ,s i
s (V0 ) ,s
Xi
X0 s X =X k k X
Y X R1 Y/ = Y / Et/Y Et/Y SmRig/X Et/Y k
k
X
X
Z
V Flt(p) ColimV Flt(p) TX ,s (V )
k
X
X X0
// X X
g p d a ` t %q v &X e d
TX i ,si
X
r r r p i io p i r i i i i X p fq` wX g r Pf%q` r X qb wbr H p v h Xp hXhd a h ` d v r d ` c g X h a r X a X a p ` v g ip X ev h X i a p X qhc i ewqh Pd Pp &bwwPbfd P&h qPqwv d X' g `c &fwrd v a qewi &X X v 'r b` wX g &f%b` X q ` p'"5'r uX qg lPwv i g | X r PfebUEX i &qwP X g v Us&v d wv rp h X h Xhd a c a d g i p c r p x ai X h g Xd c a `Y iv h g X a a r X ` m g %P"g d aXd ` Pc d eX g X d c a `Y X v rv h qh &fhfbU&HV X bf qy&X c y%PX X g v a @v a U5RRPHEC 'A pc d a g Xd hd F DT I S Q I G F D B w P e 5ef b " w P hf hq 7 P4 P b P qe " 5h @e @ 1 #R 0 (c & ( 1yUH 7b))& i p i f ! qwfq)bh sX fv bbevX g &X g eb a `v dv ` ` X h ` h ` c v cv ` i i r Rwy i k X c ` w "w i Xc aX i PPi v g k sX a 5fg &X g v d X g f%bp ` p Xhd a k i p i t n i k o n E vc P&X X c ` w "w Xc aX i PP v g g c a d d h e qqa d d e ` X a s&X g d X g d r @X a i 'fb` &X p p PX b'f%P'X v e&w wv g rX h n d r a X e Y X c a X i i v g a ` ui X h g d c g h d X d d X c x d a bqffhffgjeP}yq&X bfd &f@qfh x &wi v g k | X &X t X X iv X h X dv ` a X d ` X hp g d c g h d 'g ` ffhffg yh r qfn i i r d d eX k X g Pfq) fvd P)q%qw " hX ` g gX a ` u X c ` X iv d c ` X g d c g h d g `p p c r x a X d X ` ih e d d a ` qqa %q i qwhX h Eh hf vh qfX qg 1}Pwv g | X gX X g %f a X ` g dv wb` X e Y X c a X i wwv lPP v g X g d a sgX X X q'X a g pc i i r r r r r Xd h g `c d qa i qe&X h 'X v p b` a vgX c h ` v g d v d a f%PX PfeHX v p b` fw&X X gX dX v h ` v a g d h g r i i p r r i i i p i r 'r i o p o h @r xX p & e i&X k "g ev jiX X v p b` X X a m Xc Y h a g p g P{bfd f"g Pfv @X h &febUY g fd g d X a `v v d X c Y g Xdc a ` X ` Pc Xh i X g &X g h h X h d a r g ` h i c d X X i c ` k d'"fv0&& i "w &PiX P v g i 'wfe&e r g u X c ` c aX i X X d c X d Xc a X i ei &w wv g '&xbPf&h X X i h X p X g Pf%b` h q0 i wv Pjg1w " X g wv 'HV d Y X h 'X qfg {qwfq) ejX h PhqU@ p c r x a `v dv ` `c g Xd c a `Y X c X &qi X g qw&f U5RRPHEC 'A a ` X F DT I S Q I G F D B eEP P h hh fPh hq w ew eP ) w q @e P #
B1 u : PreShv(SmRig/X, M) // PreShv(SmRig/U , M) k
H " seq s sb qP " " l @ b 'P #5 0 1yUH 5v 0 pk i p i i p i i r 'r f g vc f&X eE qwfq)bh sX qi Ph wa a `v dv ` ` X gX vv g d wfd e %a v yX d i g X X V p i p Q i o i p r r i Rw0 "P I a `v dv ` ` h F bfw%fqf"u ))4 'X bPh sX Xpi g X X d ` c h ` d g c a ` p p i i # i ) ep wa i X i ewqb "ei qepc a X &X g w%}qwh eX g X g a a `v d c X d d &X X g Pfq) fsq` "&f}qqf elX bP&f Eh hX ` g c a d g X a ` u X d a `c X a `v d c X qwh&f lX i i i r r R r r p i n X av v ` i br wfbbg e&sX v fq` X X d d X p h X p fq` wv&f& fqewqw "&fa v h X h X h v a ` c h ` d g X d g `d g ` "bvh qf p p i i i r i i p i s o p X g qjX Xdh ` X we &bbPPa fv g @a sh X a X i a ` c X c X g X Xv v d w@ww g a ` u Xh q5b)f ja c X X @X a g X e r r 'r o p i @r i i i r i d a %PX qPPxa g X i X h dv wb` a `v h bf&X v X h a b` xg Pfv b) PEPv h X c Y X u X d X h X k i t p p ik i r i @r i h vqh b` bG g g ` X )%7ywbPfwX ` @ X d X a ` X a a g f&UX @ &X qUY Xh a h ` qf pc a &X g d X g &%f vqb h X ` g h ` "P )}b` d g X a i r i i p i i i r p i o i uqf PPv h %a wfd a X d Xh X d vv g X qPX fwbr wfqb'a h p cv i a v X av v ` i g hX PPc y"P)b#b%PX e&h v) e h Xh d gX a ` u Xd a c X d i r i p i p oi T gjX X qg 'eqfh f&bwa Pv &X p c r x d X X dv ` x X h X i av g a X e d a `v v c ` qwd e"ff sX g q&X fv wPba d a X dv h i r i i i p i r r ! Pfq) Pc yh X ` ` h ` d a d qw wv P a %&f"qffv X v p b` g a Xd e ` v a h X r i p i i i Xd a fb` {a x d a q&X X g v a 'br qwqfj{qwh&fxePe g X a ` ` h X a `v d c X X d d X c u X g qw&f U a ` X h X s @ e'br wfbb{yh a d g X d X av v ` i a g i r i r r p i p n hX PPc @1wv g | Xh x Xdd X d eyX p fq` v h g X Xv d a `v d c X a g w&wh }bfe&f1X yX a q` P&X a vc
1in // V X ^X s Ns X X ^ V X Z y k f Ns k f // U X ^X Z X ai y Z u: U s: Z // X X X f // X // X Z ^X // X . // Bn X ^ k k
n BX
i r 'r i pk i r r i ! Yv d @wfd "g `c v e5g gfbffX efh sV fd "g d X g ` vc h c X X d ` ec
Z
s
X p
X
h v p b` pk
a g
X
d a ` g %bff&X
vc Xh efe
X
m g "&X
X w
i p i @r X wqr a fwsX g i &f&X qwwp v h di X X d v rv h pc qf i i@ri p T dwb` v Xf%a v gXfh&wv a d i xX gjfqPb{bfd w X a 1wv P a x X h X i av a `v cv g d a d i i p p 'i Xd g cia q` hiX ` g i &%f a fbfPwqr a ` dt X a v i X g i v f wfa X d X av v ` i X d w e&br bq&%a h bPh g `c c X X iv d c e d bi bfe&X v E&X X g qbHV h ` X
V y Z
D
3 8 6 ! 8 2 70P05 0Hf
2
RigSH(-)
yY
y Z
r
r
f : V
// X
X
k
p
r
r
p
p
(i , i ) : FSHeff (X) M
Nis
rf : V
V
H
// K f : H u
(WB1 , Cof proj , Fibproj-B1 ) u (f ) B1 f Nis
(cu ) SmRig/X // K
// FSHeff (Z) M
Nis
u
i i
i : Z
Li i at (Z Acst ) PreShv(SmRig/X, M) (cu ) ^ V X Z ^ V X Z // id // X cu
^ f X Z
u
RigSHeff (Z) M i : Z B1 B1 u (a) (cu )
// at (Z Acst ) // X X Z ^ :: (i , i ) : RigSHeff (X) M // V // X ^ VX X s u (f ) (a1 , . . . an ) O(X ) i s V i
rf
q g
// V
B1 H
// RigSHeff (Z) M // V ^ u Nis u
X
Z
Nis
K
X
V
k
|ai | 1
Z
// X
(cu )
a :
Xh f
r i i b8 gX g %PX wec Y &PqEPfe&yV d a Xv h X dv h i h g X d c X X
i :
V SQ g i i @r i p X X g a a `v Ph fd jbfd h `Y hd qUga X a wv g wv g d a
Spectfor (PreShv(SmF/Z, M)) T at (T an ) i (-)
Xh f&U
Xh f
i i @r i p o p X X g a a `v Ph fd xqwd h `Y hd qUgfa fX a g fq)v b X ` a `
// Spect at
Xh f&U
i p i p p i k X p c r x a Xv qg 1}&wwg | X e&w v g X b` wv g w & Xc a X i a g c a d av X c ` i r i i p i X gdc ghd m fh hf"g %PX qwd PEX qfg '}&wwg | X e&w wv g yX a & d a X iv X h p c r x a Xv Xc a X i g d X k r p i @r Xh f&U h g X d c a `Y X h ` X h PfebU1V PX bUY f&UxX qwh v rv pc qf a g k p i p i i i i p i s y Xh f&U dv wb` %PX ef@d w f'X w e 1X bqw a &j e e bqwh fbffPeX gf}@a d a c h d g h X d g r ` X dv d a ` u a `v d c X g ` X d d X c X d X h X i p r i " f X ` h ` c d a Xv p r 'r p i i p r i i p lwrh v bbp P}r g p %PX wPc i Y X g ` c v bP& wX `c P R h f i bqo Pvq` PUlwv wv a p yPX @r` b` k &X qf'r v yev p b` &p P ` X o wbqPfd i c Xv ` i a X c a d X Xh d g d i a d u X r v d a g d X X i S` a h X Xv h ` r s g X g d a &X v u g f5Pfw X E gX a g a d h &febU1wX bh g X d c a `Y X v rv q pc a X g `c g ei a X bfwPwv &qi a `v g c a c X
PreShv(SmF/X, M) k
X ` &%f
SHeff (X ) M
i
r
` ec
i
x d a 1v%PX
r
i n Xh hd a v f%b`
r i X ` Pfq) Pc ` 05#" 8 6 ! a g dv wb` i o ls d fsb` d g h X o p X e h a r d w vfr &Xbiwv&X d fwv a g dq&X ec@XHWU5RRPHEC 'A a ` V F DT I S Q I G F D B w #5 R01yUH 7` ))& (c & ( p i @r i X bh v rv qfywv ~ p c X r
id // i i X
k p i @r i r h PhqU@a we a PX qUY g X d c a `Y g dv a d h ` q pc h Rw0 i r i Q f i i i 1& o s X ev h X i av rd r qPqwgX d g ` i fwv a bPc yX g p X @ p vfv bbew&i fh p yf"g p h p Xp h ` h ` c X u d X h d i X v p b` 1X g " qwfq)bh 'X wv g a X %&X h X a `v dv ` ` d v i d a X fv Xh ee 1qq w " h % f 'bx e qe qP yP " 6 @ 'P i Rwy o s 1 p p c h p c X h d g h X q&X q&vd w f&wb` X vgX v i d
k X k X // FSHeff (X ) = SHeff (X ) M M X // at (Z Acst ) . i : X P : SHeff (X ) M // X i i // RigSHeff (X ) M // id
d &X
p p i r t pk i i k X g g q%&%a h fe%PX bfhX h %bfvffX e"g ev Y dv a ` u d e d `c d a X iv d c ` d a ` g v c m X c
^ at ((X X (X - Z)) Acst ) // at (X Acst ) .
m g "&X
pX
p i X bg p fq` a c X p h X wsX g b` %a a a d
` i fv d
k r r p i i X v p b` h a c d a X i d gX k g X bg q&w wv g p i 'r p o X &fd X v v rv wyX qwh q pc a g X d v m X yf"PHv g Pfq) fsk b` "P)}a hX ` g c a d gX i r i p i @r Xv r ` ` ` h g ` d a wqbb)bd b)U{q&X "&X wX qwh X c ` d X v v rv pc qf d g ` di X "qfg c X k &qi p r @i p i r k i p i @r &X h X g &fqq` X h Xhd a h g v u d ' H"&hc Pc g &X qww ` v v rv h pc qf r i i i u d a q&X fP&h v c g h fvw "w&bf&X g V g X c ` d X a `v d h fffeec g X v d X d
Z k =J I Z B1 X Z X
i i u #" bfw%fqfh vh k a ` v d v ` k ` X g p i 'r i o p a g c e w }a vuwPX fg g g X g X v v rv q pc wsX bh r f%b` r qybg h Pqw i fe r &br &X 'r ` n Xhd a a ` u X vi e r Xi v h d k ` a X X g ` X e Y X c a &pvh qfs v Ee&wrX v g i X a vgb g g d X i k X v wfPqi b%PX fi qe&}a &X k a d dhX g ` d a Xh g `c Xh g d v i i i ` v ec f g X a &X g d d g `d g ` X e Y Xc aX i "bh b)ww v PP v g
B1 Z
r
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
B1
(Z i )iI
^ Z = X X Z
j
B1
at : Spectan (PreShv(SmRig/X, M)) T
i : FSHeff (X) M
at : Spectfor (PreShv(SmF/X, M)) T
X
^ Z X (X - Z)
t
X
// Shvt (SmF/X, M)
i: Z
X
SHeff (X ) M
^ at ((Z X (X - Z)) Acst )
// X
T an i (-)
i : Spectan (PreShv(SmRig/Z, M)) T
// i (T an -)
// X
// i (at (T an ) -) P i at (Z Acst ) Z iJ Z j
i
// Spectfor (PreShv(SmF/X, M)) T k // RigSHeff (X) M i: Z // FSHeff (X) M // Spect at
P j
at : PreShv(SmRig/X, M)
T for i (-)
at (T for ) i (-) // RigSHeff (X ) . M // RigSHeff (X ) M P
X
// Spectan (PreShv(SmRig/X, M)) T
(T an ) (Shv t (SmRig/X, M))
(T for ) (Shv t (SmF/X, M))
// i (T for -)
// Shvt (SmRig/X, M)
// i (at (T for ) -)
X
d a q&X
r
p X r a ` ` X d bbbh vwwv g v v g a q g r 'r v X g g ` sa vh b
X P
r
h XY a `v h PUbqPX Xh f&U
Li i at (Z Acst )
. X M a t : X
F
9 #P4 5v i d g h X w f&sV
# Hq f q A ePh hbf A " Q 7ef s Q b P g q q 1U 0 i i p i i p i p Rw0 Q wv bbPvh E qPq%&X X h ` h ` c X X ev h X i av d X fwv a bPc X g wv g qw w qe d d g ` d Y h ` X g `c
at ((T an )r ) Li i K (Li , i ) // at ((T an )r ) K.
g ` vh b
i r 'r i p p Yv ` i c h c X X h v a ` c p ` u X p c Pfh yEP fbP}b"bq&X qe a `c X t i d p d g ia o X g a d e ` Xhd a X pc X g v g w qeX qfq&X q&g v a g w&X qg X X q V d X pc r pc a X ev h X i av d a ` wwf&bw%q
j [at ((T an )r ) i K] RigSHeff (-) M // j [i (at ((T an )r ) K)] et Li [at ((T an )r ) i K]
d &X
g Xd c a `Y qh PhqU@X
X g `c X bPjV
p p r V X g Pbbh sX 'b` o H 'i X bPfwvh hX i g ` c d g h ` p di X i r r r a r Xh d a fv %PX P'&bi q` bqPX ` c X d h X g a `v h v qfb & v e&w v g a `d ` X e Y Xc a X i
at ((T an )r ) i K j
Q i i g Y v i hp X a ` k wfd f&qec &X %q d a `
B1
v a fv
X & a
h X ya
k i i i i i g bv h }g bv wwv v&PP v g X i a X i a X e Y X c a X i
p i K
i r 'r i p i r t o p Yv d d " `c h c Ph } X dX d e d `c d a X iv d c a i ig p a g fq)v a k ` Q w ev%a h P%PX qwh&r X wv k F hv ` i d g ` d g ` X d X c a X &X wUu p h g i a ` v d c X i g bfe&p f b` u fi Hf4 i swqbU i X k bUY h ` i p Q "bvh qfvwe v&P P fi v g }X p c X X v ` i a X hv ` x X d X h a `v dv ` X g `c X i ak t&X q'&i wqbP1& g{X g t bbi }i f&v bfwfq)b h X i bP& g bv p 'h T fg bv &r %fd r Q ` X i a 'X v i v h bfhX Efbfqi h be&qi ebfhX qwd h bes&lqf&fP p g wv 1Vr qq&X i ec X X i v d c p a v a `uv p d e d ` c "p sd X Q X i v d c a v a ` v e d ` c p X X i @X r h k Xi d Y X g ` v X X hv ` i w'bqU t rX ih X dv ` fwbh %b` x d av a eX g d P 'PfebU1V &X bf h g X d c a `Y X v rv h pc qf X E%PX w &'fwv fa U5RRPHEC 'A c X d a X g X Xd hd F DT I S Q I G F D B q1h lh b qP h h q hf hqse ef qEew ew # e q h # b q Pfy q w b h 4 q 4 # %y9
p K
Q b g HHPP P b P qe " @e ' 1 #5p b qP h q1h lh h q hf hqse ef qEew ew # e q h # b q Pfy q w b h 4 q 4 # %y9
q1 e ' qP "
(WB1 , Cof inj , Fibinj-B1 ) // X i: Z i : Spectt a i : Spectt a
Q g
HPeb y P b @e d a %PX r
@
p REqi
0 ( & 1U5)' $ #
k i r r i 'r i i i r 'r i X g ` g r ` &h qfEX bqw a wiY fd "g v d `c h c X efh yw ! d "%a pi v fX fi qP wwqfb &X bU g h ` p d X p a ` d ` p h ` Y p q pc i i p X r P a fwbh 1Pwv g | X &PP v g &X b` %bf%v g y&%PX PbfwPwH&%bfw ffPbH&X qH&yb` X dv ` x a X Xc aX i c a d a ` X d a v i a `v g c av X d a ` X cv d h X i p c X
Spectan (PreShv(SmRig/Z, M)) T k
i
i p i r 'r i p i k i Xv h ` r c &fqb&fd X wfd "g Yv d `c h c g g evfh ja fw X %Pf"q'a "g ev Y d a Xv d e ` m Xc
&X
p
i i i p p Xv h ` r bb&fd c a v d Xv h ` a X d d fh gX PfqfPfwv g h @b` ` X
3 8 6 ! 8 2 70P05 0Hf
2
B1
B1
B1
(WB1 , Cof inj , Fibinj-B1 )
Spectt a
pN
(T an ) (Shv t (SmRig/Z, M))
(T an ) (Shv t (SmRig/Z, M))
i Susp+1 an ), (at (T an ) K) at (T
Susp+1 an ), (at (T an ) i K) at (T
(T for ) (Shv t (SmF/Z, M))
i: Z
_
RigSH(-)
// X
k
t
k
B1
B1
i
r i
i
t
t
// i (at ((T an )r ) K)
// Spect at
p i K
// Spectan (PreShv(SmRig/X, M)) T
// Spect
// Spect at
// i Susp at
// Susp at
at (T an ) (Shv t (SmRig/X, M))
(T an ) (Shv t (SmRig/X, M))
(T for ) (Shv t (Sm/X, M))
(T an ), (i K)
(T an ), (K).
k
_
K Ob(Shvt (SmRig/Z, M))
// Li [i (at ((T an )r ) K)]
j i = 0
i
p
Li [at ((T an )r ) i K]
&X
p
Q SQ g r r r h X v p b` wX r g ji f%q` x X hX i p Ph d a d a ` X cv d h X i p c X qq& ffPqX q&&yX g &f%b` X h Xhd a h g @"f0
// Li [i (at ((T an )r ) K)] (Li , j )
P
i i i p r o En g a a a u h Xv g cv d a X X h dv g b` Pweffh Pjfv a R S` q` h PhqUgX a g Xd c a `Y a g d
SHM (X )
i
r i i X iq) ` ` p i @r i i p r 'r r Pf@r ec k wh I f n 8 6! F 05#" h gXdc a ` PfebUY a 1wv P b` g h g d a d a X X fh yfa X ` h ` PX qUY q pc a g dv q` r i i i i Rw0 i o 1s d fsq` d g h X X ev h X i av d a qPqww%PX P&X X r d d g ` fwv a qec 1 1fv bbesvh qPqwv X h ` h ` c X X ev h X i a r p p p i i p p i r r o p r d &X d fwv a g X v p b` wg g qbqX qa P a h X X dv a ` u v i d d X fv %PX PV U5RRPHEC 'A Xh d a `c X F DT I S Q I G F D B Pe b'q w " h w %# f 'bx e qe qP yP " 6 @ 'P #5 51yUH 7` ))& 0 (c & ( r i p i p Rw0 d a h%PX wfbP f X v iv h dp X iv h X X g wv r g PU n d Y dX i i i p p i p i i X e gbb a qX w qe bffPff X bP&f V wfd PfqevgX d av X g r ` a ` X g `c a `v d h X a `c X Yv i h X a ` c d g Xd c a `Y h &febUEX qewX g fqq` P&X X g ` c X Xhd a vc
SHM (X ) i X // RigSH (X ) M i : FSHM (X) X k (Li , j ) Lj# j k X // FSHM (X ) = SHeff (X ) M i i X // id // id // Ri Li i : X // . // X id // i i
v h d X iv h flqwP
k i i X v u X fwR&& e g&PP v g d Xc aX i p
! X X i h X w1bPf&h
k a
@e
R SQ g i r r i ` k i ec h c X p @r fh sV i i r i p i i r t Yv d wfd r "g oi i x d e d `c d a X i vi d c ` X h dp c X i a p 1&X g d i dv wb` i PX bUp` y i o qp&X'r n c g %PX qfh 1&fd h Y pc r d a g r h X i bi o qh p g&Xw l%a h P1%Pi X qwhX h jfhff @r p av ` X g X d c X wqjqh Ph&m g v qye&X qw X a ` d X v rv h pc qfyX jw'v qbX fwv e p X c X X d h d a ` u g d ` F DT I S Q I G F D B U5RRPHEC 'A w qP e v yw )w1 5ef " 1q' y @ 1 eP e q q) 'f e hfhq R ' e " qP " #5 ef P 1 q P qP ' 1 R 0 1yUH 7b))& (c & (
d eX
RigSHM (-) FSHM (-)
w)wy0 h g X d c X X Pfe&xHV
i i p r io i i @r p p Xv h ` r c X h g ` dv c ` wbqPfd wvb) "wsX X "m v%a Pf%v `v d a X d a d wv g a i h f%&fh &f&X d a X Xh d d a Xv d e ` %&w"b{a i
F Homg (at (T an ), Fn+1 ) hom(i (-), E)
k r X g fe gxfe&) vhd XhdcX
a yfv g v a
pc X q&f&X
Homg (at (T an )k-1 , i! Homg (at (T an ), -)).
p
p r i d wv g h swX g %PX wec wPba ` X d a Xv Y X dv h i
r i i X ` Pfp q) ec vh PX ` d m g "PX V X h g
Em+n
k p i r io p r r p t p ubi o i p p I Q X h d c Xp fe&) X X X v p b` h g i %wq` d av vh wP a q` i %bfw jX &q& X a d a ` X pc p g qr e oi X bfhc X a `v d d iv g a `v d cv h d X h X %a wwqwhfgfwwwh h ff&fq` g Xh b`
n Fn = Eq Fn
kN
X d g X g dv a d fwv f'a ye
a
a
i k i h r g X i x h &w @X g fe g@fe&) vhd XhdcX
i p i g X i qh &w x a g dv wb`
E
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
at (T an ) Shvt (SmRig/Z, M)
Shvt (SmRig/X, M)
p
i p
a
(Fn )nN
Fn
Homg (at (T an )k , i! (-))
// Hom (at (T an )k , i! Ek+n ) g
Homg (at (T an )k , i! Ek+n )
Homg (at (T an )k-1 , Homg (at (T an ), i! (-)))
1 n m n m+n
// X
Homg (at (T an ), Homg (at (T an )k-1 , i! Ek-1+1+n ))
////
lN
Homg (at (T an )l , i! Homg (at (T an ), El+1+n ))
// X
at (T an )
at (T an )
m+n E
F
k
# P
i g X d c a ` Y X Pwid f&bi hX vh t k h &fhfbUiX g d e g ` X eX q` qxyh jfbhc j X v h ` d Y X
Shvt (SmRig/X, M)
r X g fe g vhd F DT I S Q I G F D B U5RRPHEC 'A
RigSHM (X)
j FSHM (X)
i : Z
j# j E
E
i RigSHM (X)
at (T an )
// i i E
i : Z
// E
Li Ri
B1
k
X
T UQ g i p p i k i p p r i H g r a v d X i vh a 5fwv v g r a %&X r w i g vi a d d a X v r 'r a i bwgi v wqr a fd n ` g bv h g bv Pf&ffh "esq&X X g v b` ` s&X X i a X i a a Xv d c ` c d a `d p d
2
// FSHM (X)
// RigSH (X ) . M // id (Li , j ) P :
I Q g
f h q q d " eb 2P 7 q 5 R w @P qe ebE %# qew e 'q b b ' v qP " 4 ' b #@# 'P R yUH `gc 7c 7Fb h & e hc c k i @r k t r r p i p t i p i i o p q Y v d a ` X cv d h X i p c X dX pc a Xv i dhX h w&X v fhq` X i %p bfi w i ffPqw&X qt w&v%b` PUhX h fi g p rd a r Y v d c ` v rqf i X p Pf&ffh h }a P'r r d c h c r tg oi c ` p wsd "% a h qf v fd g ` a p w e p igfwfhX th wX k v p b` @r a h k qfwp f rwX %PX r br a qX X bPfwbX X d X Yv d c ` i h i g g ` X e d a X pc h ` p d iX u i X d eX a `v h ` qwd P)bwX fq)v ab%Pwfe&X h f g X ` ` u Yv d c ` v qf pc X X v p b` h a g h b)qP effh a g ` u h Xv g cv d i p i bi o p p Rw0 I I f n I f Xh X X g X X h X d v d i X p g X d c X X wh &vh &fe&sw r i rw ` 'v r bhbp ` dPl p Q qwfqo )b fu ))4 i 'X &qi Ph Rwwv&f" r ra ` v d v i ` ` h F R I e Q u PX X `w"m r &igb i 'X d v X h ` X g r bo `Pc& vX w"m p V R lwv bbPswveX p X i o bPfEwwa X gi Xr h i ` h ` c X d h ` p d X g aXd a ` Pqqec ` v d a ` %bf dX e &X v"m XV v%hfgX@X wv"m V ` iv d d ` X x @&X "m w&Pq@X qr y U5RRPHEC 'A `v X h X dv h i dv F DT I S Q I G F D B f bl P w s F a `v d a d x Pq bfwvq e ) u fHf4 P " q % 4 7 @e 9 s #@#
A X RigSHM ((-)an ) : Sch/A RigSHM (X an )
"
e
k
Rw0 p o i ` v h ywv g ws&br d av d X X
y & P qe " a 4 @P #5 p p e d d h vwv g fg `h i o p r r pk r r p r a q& X v p b` w5&ffX PEX v p b` fw g ` X g %&X pc h X u g v c h ` v X ` d a
X = Spf(A ) ^ U X Y
r
U5)' $ # ( & i p r r dv iv dv h d fwfa fX X v p b` h i i Xv wec Y PfPqsh Ph&'V h X dv h i g X d c X X d eX
h qP f e l # ` W
i i i r k i i abwv ~ ` d PX u c X bi h g X v u d sffwhc ec f g ` v i i i i i o p r r r p X h qh gPd P a&brw&vh gfaq`wd g i&w&h wewfh&XbivgXqr a pqc& X v p b` wsX g Pf%q` X H U5RRPHEC 'A X v X v a d X h X hXhd a d g F DT I S Q I G F D B e yewyf 1eP ) % j wlq Ex &Pq h @P b s @
f RigSHeff (-) M U 2 RigSHM (-) Y X pN
g
hq s qe b h " @P R yUH 5v 0 i p i i i @r w)wy0 o p Xv h ` r c r wbqPdo yX PP r v g sX %a Po qv i p X ci a X i d aXd a p 'r r i i p i r i X g ` X EXHV wec Y d Xv fa X iv i h X a ` c X Y g d `c d a t qwd PfbPwv r EX a %PX i qUv g | r hc i Pw%bfueq` P fa i` c a a i rbPr & X 'c i &X n wd a h ` Y d e ` X qeX p q&X g k fqq` P&X Xhd a vc X d h d d X d v ` c fi%a fw1&pv wqei'h 1& e '%bf%a h d X d p d a ` i d e `Pc r deX i u &qi c X kX i i d a %PX X e g d h m g Pfv Pf@X &fhff g Xc Yh Xhdc X X X &g ev bPj@X wPc &u 'X ffi h Y i Xhc a i a `c X X v Yi d X v x r hv d X d hi d d X d v ` c X d k i r t i i p p &%a fw&v r wqeh wr & e g d a %PX qwh&X f"g ev eh swjg v g PX a Pffh y%PX X g v qfb` ` l&X X iv d c ` h m X c Y X X d a a Xv d c d a `d p d h sq%a h d g d e d a ` %bf
RigSHM (X) F (U / /) k T qc SmRigqc /X Acst ) p A RigSHM (X) HoB1 -st (Spectqc (PreShv(SmRigqc /X, M))) T // F ((U / ) /) Susp qc (U T Nis SmRigqc /X U B1
k X k i p i @r p i fh s u wX bfh h c X X v v rv q pc d g ` d g ` v dr X a X d h "bvh b)5feyfPgqPX vWUX efh qf v%Pwf&bi ai V ` hk T d v ` h d a X dv h i k d a qq` d PX d gX h g u X "g c"w v m ` x a `v d cv h d X h X v d a qwhfgf&ffE%PX PfPHfH "w X gX dX v c ` p i r t i i i io p i 'r o X dv fwsX d %a h d%PX qwhc&X fhjg ecv Yeh g g fh f"g e d a X iv d ` X a d Xhd a q` g h X v k i i i r k i p i @r p i i Xc X i PaP v g X ag a ec Pc f g dP5&ffR&X bfR d ` v X X v v rv h q pc Xv h ` r &X fqbPd c a `d ` b U5RRPHEC 'A F DT I S Q I G F D B e w H @q qw b %# " h v & xb qe v q b " P hw' E h b @P x q E)b P " tb 4 b @e #5 yUH 5v k i i i p X d i v g k a ` v d c X g ` i s i r p %a wbfh bf wfa g ` Xv d g d g h h qf&w"wd w f&X g PX g f%b` qbfe&f bflePa Xhd a a ` u a `v d c X g ` X d d X c X a Pa b` p 'i hX Pfv g
Q Q
@X
r
h ` pd b& fg
p
X gX qi Ph
i
8 0 B Q B 3 ' 8 I G I G 0 8 B 3 A 8 3 0 ' % " 2!9R94SR5)H9P4DPHFEDC!1@9765421)(&$#!
3 8 6 ! 8 2 70P05 0Hf
2
Susp ((U X ) Acst ) T
F
SmRigqc /X
SmRigqc /X // (SmRig/X, Nis) (SmRigqc /X, Nis)
f# g Susp ((U X ) Acst ) T
Susp an (U T
S
g f# Susp ((U X ) Acst ) T
X Acst )
Nis
A f
k pN AE
S
^ ^ f# Susp ((U X Y Y ) Acst ) = Susp ((U X Y Y ) Acst ) T T
RigSH(-)
A
F (U )
g Susp ((U X) Acst ) T
k
f
U
H1 Y
X
^ U X Y
X
1
g
RigSHM (X) X RigSHeff (X) M
H2
// X
f# g
// X.
// F (U )
f
SmRigqc /X U
2
AE
// TR
// g f#
2
^ Susp ((U X Y Y ) Acst ). T
2
SmF/(X/ /)
: H1
(f an )! RigSHM (-)
2
U
// H2
Nis
(f an )! k
a u b` e
'X
h ` p G F hv ` i fqP wu )Hf4 sbqU
i r r i X g b` a X v p b` fa p 5fig v a c h ` v g v k i p i Xh X i a v f&bwv g X e&w wv g yX a &X Xc a X i g d X g b` %a d a a d
Th(M ) j i V 2 Th
-1
d PX
r i i p i r r p p XPfq`) ec wh sa b` ` a X g b` a X v p b` fa wv g b"&&X v a c h ` v g d a ` u g r'r i i p t i i sf Xr X }g p w abvfe&f i deX X g i a ` doc X xs vec"`ff Pt Xfhq`fhXbi ish g bfe&X fh v dk c e @r ak ` v d c ` cr X i d r i p k bi ` wG X PcP v g ec" v wa p X aX i Xv ` g Y d vhcX h vf g ` q pc a g g X ev d a h fyfh w%&X w "sw g ` X c ` X a g r p r 'r Rwy hd a fqq` p &sX d X X i X V a ` c X X g ` bP&fywvh q wwf&bgX X ev h X i av d X f%v bPc X g d a g ` i @r r rk X g X g h Xh a ` u qbX v p b` h
(i , i )
p
i r'r i r rk r r Yv di d"g p Ph i o cysXf wd"igabX r v p r hb` ` cr h X i vv ` hPv Xh swvr bo w vfh&Xbiwvvd%ab`f X X r' g ` h X e a d eX X g Pf%b` hXhd a p i X dwv h@"gf)vbuqfwvw c"wX X `"m h Xv d%&X w `eh&bi q`@abvqPX d ` v h a c Xd X g ` h v fab`fdb` FUT5RRPHEC 'A D I S Q IG F D B w qe h
i i j: U // X i : i 2 // i
wPf P q ) qP bj @ e j qe 1 " 7' 4 7 'P #R U 5v i r r p i d i g p X Xi p a ` '` r a ` v g a ` c p o i p i i i i i f"bQ f ! qa wv fsX @i wswqf fbUY beyX rd s d wv a X h Xd ` a a ` &f}bu X d g a `v d a d ` ec rg wv a Peyg qb p X Pc @i x r X c X h ` r qfwv f a k p r bfwv Pr a p X X v pi b` h i a 1b` "m a i & e g&g r v qf` ` qh Php qUY r X X r v p bi ` h &X &efbh &X g PX g PbP h g co a X d X d X `d p gk X d c a ` h d v h ` i h X iv h qw bP&%a wv Hh q0 we X g v qfb` ` 5qh PhqUY X X p fq` a wv &&X a ` g ` a u d d a g ` X d `d p gXdc a ` v h g d Y a X d X g bHfw f&bsa av ` g hh X i i p p r i p i i Rw0 op dv d a d fwf%PX gX u a `v c dv d a v X c ` Pqwd f%PX w&P' g ` qX g ev bPsyX bfywwqh Phwxm g v a u Xhc a i a `c X av ` X g X d c X X
an an f# Rig Susp ((V Y ) Bcst ) = f# Susp an ((V an Y an ) Bcst ) = Susp an ((V an X an ) Bcst ) T T T
p i r i p X ec Pqh Pd P &b&X d `c gX h a r r 'r i r r X ` m "g %&f"g d a Xd ` ec i p p p i Q gya X dv ` fwbh X fh h e q #@#
2
k i i @r p f a d eX u X v bffw q pc a g c a b` wb` dv i i i r p h fjbfd w &fh wfqPb&X g a `v g i X ev h X i av d X g &fqq` X '"R%&&v d wv h Xhd a d g v u X a a p i X g qf%b}~ a `hd a ` deX X g a b` qa d ffw a d X v h b)} wfqPbx&X g ` X ev h X i av d p i i @r i cv&X %PX P%#h d a a a Xv i ` X X h g a a `v &w&ffd fbfd h `Y hd X bUa fv&V U5RRPHEC 'A F DT I S Q I G F D B qP blew ) " wPf v q ) qP q " @e # P 4 5 'P
X Spec(A0 ) SHM (-) f RigSHM ((-)an ) Y f# Rig
bx
q eb@w qP " # a 'P R yUH 5v i r 'r i ! d i g "%a v fyX X X ws a eew qe ew hw ye q b' " h b # s Pew qP f ew eq yew 1h b Eq #@j R " e @ s q ' 'P wPf ' " b P e Pq #@ 9 P4 eq wPf " w qeb l v #@ 7 P4 1 wP fPA 1by f 1qe ewl w 'j @# # " v
f 2 Spf(A ) 2 k f : f X -1 f ?
q R y qP b@ 4 7 5 'P ' P l" v q R 4 7 5 @e
S S X f: Y
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
X : H1 (X)
// H2 (X)
v
'
// Y f
// X
2
f : X f
// f Y
f : f# Y
// X f#
M OX Th(M ) = p# s
wv i eX d Y X d
d PX
j i
: H1
SHM (Y )
0
// H2
Rig f# Susp ((V Y ) Bcst ) = Rig Susp ((V X) Bcst ) = Susp an ((V an X an ) Bcst ). T T T
p : V(M )
Rig
i: Z
// X
f : Y
Rig : SHM (X)
// X
p# s
// Rig f# BE pN
i i
i i
-1 s
(f an ) Rig // X
2
S
// p# s
Th(M )
2
Th-1 (M ) = s! p (M ) // i i // RigSH (X an ) M // Rig f 2
p
i
T P I
r
(M )
Th
-1
?
// p# s . Th(M ) S X f Rig M ? s f# X j i j i 0
pk
v ec
R SQ
dv wb`
p
g f&X
Rw0 pk
q ) l P b 4 5 @e Q Q ei X i p i r r i r o p r r 'i vc h c Pvfh xg bfd "g a `v d ` hX g ` g ` d a ec Pqi bh h b)qq&X P %PX d a g hX g `h j" Pqi bfd
Th(Nt ) S
i r p p r r o p R v h c fh yX dX ` i p wsg u X %e&k 'X b& wu ifHf4 @X feh g sX r X v fq` p v sX rX ru i h ` p G F r rp b i h u d Y a X dr X g v r PxfPfh yX X v p b` i fr` v wbi wwd PlX a Pf%b` y &X h X c X dvv ev `c g hXhd a x a h c a yb` gi a d X u a `v d c g h d a ` c qwhfqeh X v p b` h X g ` dv ev wh b) fwwv fd ec X a x%PfPqi ` g d a X dv h d X X v Pffw h qf g `
? ?
d s
t p #bi i abf eb` `v d c a h P"bHv w&o epc g p a X d e `i g X X d X pc i )7i g br a q&i i a Xv d c h &wf&h fh c qqa a ` d
Ex!
X
h v p b`
&X
a `v dv ` bfw%f
r r 'i i p p i r 'r i r p p i h a ` v dp ` h Xhd a c ai d h g X v i i r r X r br k v g p bfd "g Q ec &f%b` p X b` wv @ "fH wi ` x X ev Pc 1wwd PyX v p b` fv X e g'X g wqbX wec &5 bi `c h ` dv a d hv ` i Xv Y d X X v c X
d
s
r i r r o p X a P h X ' a X v p hi a gi ff1X v fq` a ` v vp h ` dp qwd bhc Y i c&qi X d r r h a d 'r v P X PX p a Pl& PUqwfqPX g d X X h XY a `v h v X a PX g a X k i i i t X jh }ffqfhc xh&X f g a g X v h ` d Y X Yv d c ` h v
f f = (p s! ) (p! s ) f =ps
Xv i g X i %v p f@P p Xd a r sX q&X hP)Rv i g X abfd fqfhc `v v h ` d k i @r pc qf
i r 'r r r 'i i p i i r i p 'r r Yv wfd d `c eX h g X d v m X v d a X r r" g r pbr v ia sg"i e5q }g q&X fd v v i` c h XY a `v h &Uqp wf&X v @X a g PX a X r qp wd o bhc f"g o X a g i p r a ` v vi h ` d Y X h d dv wb` r a&f& gX X av d X g Ph d a hX dv ` ec r d eX r k c a a i i p f%br` X wbr r H Pb` &1va X a `v m a ` c qw"PX a bPX X t v p b` @r fv k a qb r a ffX v p b` h i ` g hp ` X r v h i Yla g X d X Y v d c ` Pwfe&X fh r pc qf r X o p X v p b` r g h a dv r wb` 'r X v wX v p i b` h a g wv fyP &UxqwfqPX X v io g X h X Y a ` v h v p r r X p fq` v h d g `d "b}X g v }fv e&fwev a g V U5RRPHEC 'A g X h c d X dv c F DT I S Q I G F D B X
Q q
q # 6 7 Ph
// f! // X
P qhq 1 R Rew e 6 e
# @ hq @ q ) 4 7
f : f!
-1 i
l w f h v Pq e v P qP @ 1 ew 7 # 9 # # l P b'h @e '
f S i
Pq % " eq ews
f
@#
b@
1
" "
3 8 6 ! 8 2 70P05 0Hf
2
i =
RigSH(-)
S
f: Y
2
i! = i
U
h &c ec a Y & ( )gf7S`& ))XW
s
p
f
f = qt
f
S
2
f# Th-1 (f )
f
f = ps f! f
t
2
r
r
p
r
d ! t!
s!
Th(Nt )
f
Y
Y
t
// d! t!
// f# Th(f )
//
//
//
t
g
g d
s
// X
p q
// X. // t d . # // t d #
f
h " R yUH 5v r p i i p h ` p G F X i gX r i fqP wu ))4 @X bi Ph 'r wfa h X i g ` h d h g X d c X X Yv d PbbfPfe&V wfd "g `c d e&X
// ""
// // // p! s!
d
?
// d! t!
// s! .
S
// f# Th(f ) .
2
i r 'r r r 'i i p r Yv d d g `c PX h br v X bUa X h `Y
i i r pk r 'r ar X v d c h c X p Pf&ffh h yX qUxa ffX esX h `Y g vc ` Pc PX d u fb"0 a ` mv i 'r m "g h ` a d a bq%PX fqfP'h Pfv a vqf&&wq&v"bd X v ` h c x g X v i g dv ` u h Xv d a X d g ` dv ` d wbfgX u X m a ` c d X h X v g rt h d qPX a qeew&w b&f&X u Yv d c ` h d gh&X fgX i i r rk ! f%a v ffPwh bh vww%a wPbX v fq` X d i g d v ` X d dv h i p h a g d eX i r i p t i p r i r i i T u c X &qi X bUY i wX PX bwh p &fh r PbP%PX hc i i egX p h ` d e ` X X@ a X r k a X d a Xd `c d Xv h ` r u i i i fqbPd p c u F a `v d ` ` h d a X X v d X d p X g ` wh b) dv g av a `v g i bfwpv %fqfy u p fHf4 sh q%PPec PXqa P as)%7X 'r Hf im i d d p r n qc p bX fwfi h br &p X bfd wqfh ia ` h v r i ` o o 'X fq)fv b5qw yX g fPhc h c@X p %fp s&X X X ` a ` u X g a v d v h d ` g d X ec eh &fd & Pq&X i h f@X w P bi d `c g X h a r g g iup r i` r t t a b` g qb'w e v h X i a v d h ` X X g &fd%b` X aq` '" qdhc ` c h Xh a c d g d PX X qfg yq%wq` p c r x d av i f&bws&X r'r PsdeX b`sw%r PbPf&h i e X 'a X i h X r d qh PlXg v y&wfd "g&f}qfu g X d Y X h X vv g d X a ` d eX h qf1wv"&X g ` X a d av ` m f}"g q&fd"g d a X `ec d g X d c ra ` Y X qh &fhfbUvV r k i eX u ev h i av h v g c d a X d X g a h X wfqPXbvPXwevfi fh PPUy &X v p b` r r i Xhd a Xwf%PXv X v p b` g a h X d hX w@wv Pf u X u v u X ev h X i a d d b qPqwv@wv P v u d fy&PX i r 'r n i i i ls r rk o a ` u d Xhd b&fh xf"g X e hX i a d wvfqPbvl&X X v p b` h X jX g dwbbi{q`b vqPqwld&X v ` a u X e h X i av d g ` fwv a qec jX ` i i i p p i p r io p i X v ihX `c d qiwd PfabPxgX Xh wv }X g vg qb c}X X "m }X &P)PX a d a ` u dv ` `v u dX X eqPqi%jwbf v h X av dv ` X g wv ' wv d h g r rk p t p ubi i p p u X e h X i av d ` b vqPqwlwvbf Xg h b g ` X v p b` h a g X qwvhcaq` a ` d h 1v g a d r r r i Xh d a wv %PX w r eyP PUY `c X hX X h Xhd a x Xd Xh i g &fqq` wgfwv a U5RRPHEC 'A F DT I S Q I G F D B b @ # " h @ r'r r r o a v f b`Q qhPX vi i a ` d ` qfb &bi q1qwf&X X d h X gr` a `v h v vk X a g i u wv b`hb`Pcwyh we &g v qf` ` PfebUY X Xh X X d X `d p h gXdc a ` e ) l qe q' q hywh7 e ef @P v 4 #
Sch/A A 2 // RigSH ((-)an ) M i: Y Rig // X
i c X bi wwf&b&X X ev h X i av d p r r X v fq` f@a gX p h h g d X g wv d v v " q sew @ le b #@#
f! Rig f
!
1q e e q s UU5)' $ # 2 ( & i p i i i c a h X a v u d i g d g Q fh X X h gi c a v x d v &yX 5g &%i a v d w fsfwPwsffh p q` av ` X e X h g ` d g d X bf&)1&b)1%a j"@&} X h ` h ` c X }fv wqfqeV
f! // f 2 k
)wH0&q &w f " @P a #R i i p ! Xd g vv g wfdi w i io r fwv vp a Q Q yqbi b&qUe"bG a ` a g ` a u v `Y X d g ` X fqP d g bi &f X h ` p X g Xh r i Rw0 T "P T h ` p G iF i Yv d c i pU 1 i 'X fqP wqu ))4 yh UwhXi uqp w& i f&'r PX i b qh p Ph&j"g ubi fv i w%bfwv a ` v d h X X h r wv a h h g ` g X d c X m t p & d a ` g bfd "g a `v d ` ec V g aXd e r r k p X a ` v d c a i i qwhi q` i h 'P"b` p h &X v fq` v &X g fd h v 'X a ` X Xh d Y
v q R e 4 7 5
Pq we T v s F e qwv fe u ))4 a `v d a d
f
t r r k ` g ` a g d X ph h qff X v p'r b` fp` v iw&X p rfv r wv i d g i a Q r h o X p g '"bp` dP &XV X c a a h br v p p 1X i x h ` X g ` c o g p qw r qe& r k efb` p & EX X X d )7X k qr a q& X v p b` X pc h &X Pb"eX a qe}X d X a `v m a ` c r r p p i X v p b` h &X qwv fe a `v d a d u dX 5&eReX a F DT I S Q I G F D B U5RRPHEC 'A w qe h f b e e ew ' ` W
2 2 2 f oo f Ex!
g
' " P b' s P q ) l " # # 1 & 4 5 @e ' #5 U7b))& (c & ( p i p r i r r 'i i p Rw0 i "&X &Pva i q&X bi wPc EPqsX c m g i X c d a r Xv Y X dv h i X h br v p i r r i r r r r 'r a `v d bfd "g ` ec V ffwX v fq` X v p h a g g ` X h XY a `v h qsP &UxqwfqPX v a g d sX gX ` c bw m g X P a h X fa U5RRPHEC 'A F DT I S Q I G F D B
f f! g f
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
j j an (ian )! j Rig = 0 i 1 Rig i! (ian )! an i i
#
1
@
" s # P
g: X g X g -1 (Z) g B
p
SHM (X) // X A
i
X
j
2 f f
// id
-1 i
Rig : SHM (-)
2
SHM (X) A
g f!
j: U
f
(ian )! Rig j
f!
g
Y // X S
Y
// f g !
// g f!
i : Rig i!
i
i
Rig i! j = 0
f!
k
// f!
// X
// X.
g
2
// f g . !
// g f!
// f
A
i!
// (ian )! Rig
j j
(ian )!
i ! j
A
g (Susp (Bcst )) T k Ex ! i
!
(i , j )
f
j
2
pN BE f
f
X g qbHV e h ` i c X y&qi q&X Xd a i
k i p r 'r p i @r Q s F a `v d a d qwv fe u fHf4 ja X ` c v rv h e1&X b q pc i r v i dv ev %&1v wd e'X a E%&Pqi `c g d a X dv h X V
// Spf(k ) oo //
o i f g g (f,pI ) (f,pI ) (g,)
i i i r 'r r r 'i i p p i a Xv d c h c x Yv &wf&h &fh @d d g `c PX h br v a sa b` %a d g a d
f : RigSHM (F )
j
k i i i r r 'i i p i i 'r p a Xv d c h c x d iv g &wf&h w&fh '%a wsX h br v X a &X bf v rv h q pc
X
X
i p h br v a g
B 3 I 3 0 " 8 3 B P CR9" 90 7 5 1 20 75R9
r p i i i r 'r p r r o p r i p Rw0 o Xh X i a f&bv ` wG X PP i v g E&qi qwd "g p Xr c a X i c X a `v d `c PyX X v p r b` fv eX 'r X p Pfi q) ec i gX p h ` d r X ` ` d X X dv wfh i a ` d X g h ` X v v X v y&X &fwv fd c X h X d h d r o p r bUY ff"@qbbfgX g qbV P PUqwfqPX rd g r a ` u X v i d ih ` X h X Y a ` v h &b'beX r vX q&wi oa X v a ffwX v p b` X v h a g wP PU@qwf&X Xv iv g X h XY a `v h v X a h fbhc xb%b qPq%wgX g X v h ` d Y a ` u X ev h X i av d X g PfPqb h PX p lx h X dv h i h g ` X g p r r 'r i @r r r d"qfwX g `d v X a b)X g bg h g h g ` a ` X v X a X e X g ` h d X h a ` X h XY a `v h @Pv wbi qfequ P PU1bqPX X fp b` v h k p t p uqi i p p i i r 'r r r a g X bfei b` a ` v d c a h @l a o wv s g d a `v h r m qwd P)b` "g p "g Xd ` h X h d ai x i Pc o X g i &fqq` 'r f&wwPX v &fr h o b i r X d X h p v u r h a h X d g ` d g h X XhX i a qPqwv p G Xc a i c X ` Pc v h ` h ` i ` w"X PPiX v i g E&i i qwd p d g rq a ` v d w fb` o p X X p fq` fv qb Xbbfea i g `hdXh h 0Pfi qfe&b1f&g 0 w qwh&f gX bE g PX g X v h ` d c X i r dr a s X g a a ` v d c X d h te ` c a h c a yfb` ga d X p p i a ` u qHbX X g wv g bs d a ` d ei X d v %a wi i p i i i r r X X h dv c ` ffww "wX ba fPU X r v h d X d X X h ` h ` X d vv d a h Xv g r h PwbPX v fq` p h a g d X g a `v d c X eX w a qwh&f c X bi fv wqfqec w%a w"g
p
i i o a X v d c h c a g X i o gv Rm X &wf&h i fh i 1"ev ds i @r p i i d g X "&f a p c X r gi p c v i a v X v r ` ` p ` g ` X c r ` d a ` v d 5h @f"g u d Xhd h Xv P i qi f@yh bf qPX fjpX wqbb)bfd sh qfi ' "sgp X qwgr X g a &X g d ` c bw g X P a h X a sfq` u X & whg f&f%a %f PyX fe swvh fe&fh efsa vh X ev d c h hv X d ` `c Xh ` a X g X c a X h g c h X g
Rig i! Susp (Bcst ) T // (ian )! Rig Susp (Bcst ) T Y X Y
i i o x p c h Xr p a jX qqPqc b` g e " wflqa P doi X p i v h d d d v a u b` &&X P&f"g hXd Xhd a X ev h X i av d wfqPbgX
r X g Pf%b` hXhd a i i 'r r p w "g c X m h ` a d a bs%PX bteo 1h Pfv i a 1gX X v ` h c x g X v i g d Xr g p Pfq) f"&f hX ` g d g X d e aX g i eX q` Ps P X g &fqq` X '"fHwf7fv h Xhd a d g v uv a
Y X i
an (ian )! g Rig
!
X d f7X
i o p r r o i X r pc ba q X v fq` v E&qi p h ` c X
d eX
)
3 8 6 ! 8 2 70P05 0Hf
2
RigSH(-)
?
?
p
X
p
i r i i @r i r 'r r r 'i i p i dv ev fwwv fd Pc v%a h "e0d "g ` d v m X Yv d `c PX h br v g a a
o i! j an an ! (q ) (o ) Susp an (Bcst ) T
k
p g &%fr sk fb` h X @ g c a i` gi r b&fh q q pc g ` qPqw&X X ev h X i av d
f
(F , I)
f : Rig f
!
f
f
// (f an )! Rig
i! Susp (Bcst ) = i! p Susp (Bcst ) T T i
k
(ian )! Rig g
g
(f ,pI )
(f ,pI )
(g ,)
g : g f
Spm(k)
Spm(k)
(F , I)
(F , I)
(G , J)
Rig
// RigSH (F ) M
i
Susp (Bcst ) Ob(SHM (X )) T Y =X
j
j
j
// g an (i an )! Rig
q
Y
o
// Spf(k ) oo
f
// (F , I) oo
// (F , I) oo
// (G , J) oo
// Rig i! g
f
// Ad k
// X
i
q o! Susp (Bcst ) T
p
i
i
i
// RigSH (F ) = SHM (F ) . M
~ Spec(k).
~ Spec(k).
(F , I)
(F , I)
(G , J)
?
p
k
p r'ri i p p r r p r X &X h br v X X v p b` X bfwfq) ec h a `v dv ` ` i i @r i p p X X g a a `v w&fh d }qwd h `Y hd bUa f1X a g b` g dv a
// g an Rig i ! .
(f ,pI ) (g ,) (f ,pI )
Q q g
p a
h PhqUa e g X d c a `Y g dv a d
p
r r 'i
dv wb`
// Rig g i !
Th-1 (k d ) f!
f ! = Th-1 (f ) f i
(ian )! Susp an (Bcst ) T k
f i
Ex!
2
X ` Pfq)
r
b g i r p i p 1i ` Pc sX g Pf%b` X P &bw q` hXhd a X h a r g c a hX PfH
r F
b
i r r i Yv d wfd "g `c h c efh }a lq w "
B1
w
i i 'r i o p X X w&fh gd abfd a `v h `Y hd bUa X a g eX d i rr p a gX ` v a ` ecfh& fqec Xv c ` e" i@r i i io p X g b` g h a X qwd f% a a `v c dv d h PhqUh g X d c a `Y X i i # i s a ` u Xd g buqfwv a X g a X g gf&fec v dv h d i h a `v d a PX a qw%PX r
M
r i p r 'r n ) XPfq) ec fbP fb` ` ` a `h v a # @X X X hv ` i vbb k p i @r o p p h k PhqUa g fq)fv a X qwh g Xd c a `Y X ` v rv q pc p i @r p r r bi i p r r x vv g X bf1 v rv h pc qf X X h qr v w'%PX f@gX X d a X v d p i p o p d vwv g v a h PhqUxg i %ffv r a g Xd c a `Y pa X ` i Xv d hX g g X p d ` p p Xi d d X d a X dv cv m i p i wf"g Pfvb` a h fs r fbf sffePc a i g %PX k P "eX h wX c c X &qi X g qbwq` g b e ge&w wv g h ` av u X d Xc a X i X 0&X a g d X p
V g
p i r r r i 1X &ffd lX v p b` w1X g X @ Xh g a h X h
X
~ k.
h ` pd X fqP y
i
i i i r 'r r r 'i i p a &X v d `c X r i p ik p iwf&i h w&h p 'd r "g k PX p 'br i v i wp ic h c x Y v @ d i r h r X qUY}a f h ` Xv ` ep c" oi p x0g eQ XQ wvX S` @ v d X v r a &X bbr &fd c h lPkv w Xv h ` r c av q p pc X Xi h i bs r v d vq` eq` Pbf hqfe c a a a ` ` bPX) PX a ` 'X b& fdwbPf%q` fb` "qwh fbfyfPeyX va h ` p X a `hXhd a g a u a `v d c X g ` X d d X c X dv g X yble A u yb@ # ' qP ew hwPf eq b w " ) b#@ ) ef 6r 74 " 'qP # # Pe P P s # a @e q# 7' " qeb j b qP b 47 " 1 P 6 'P # P #R 0 UU 5v p pk r 'r ! %&X &gfbffX eyX d a v i d X g ` vc X X V Rw0 i Q fv i wqfqesswfd q` v i X h ` h ` c X X v v g a Q h q U u fFHf4 bqU g ` hv ` i gd ghd d hcf &X u dX X &P)Pja F DT I S Q I G F D B U5RRPHEC 'A lb b f ef w 7 w x s w' b #@#
A u " 6 t
P
6 r
r
X
qv A u " # i i a `v dv p ` ` a p ! qwfq)bh X g `c qe& v g 'X X v r 'r X g
h v p b`
P b Q by P s F q e a `v d a d bfwv P u ))4 q Ex & P qP " a 4 'P #5 UU)gf7S`& ))XW h &c ec a Y & ( r r o p i p t p #bi i Q X r v p i b` p a g r ffg %ffv q"bfeb` h v X ` i a ` u a `v d c a h # i i p a h 'X q&X ba q@X X bPfwh 1 qPqws&X g X e d a X r pc h ` p d X X ev h X i av d u X v d bffws&X
g
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
g
r
X = Spf(A )
k
g : f g
// g
gf
2
= Spec(A [1/])
RigSHM ((-)an ) : Sch/A B
j
// B = Spec(A ) oo
// TR
i
= X
Rig
v
1
i i xp y g i p i X eX X g vb` a c@bfd w P&)X h a a `v v v c X P q ) l P @e
4 5
Rig
f
// an Rig f
E Spect (PreShv(Sm/k, M)) T Ho(Spectdiag T (PreShv(SmAfnd/k, M)))
i
Q Q
r
i Xv h ` r wp bqPfi d c t d e ` g eX qa 1 a q` %a d a d
D k
u
an
B1
// an Rig u
k
SHM (-)
X
k Rig : Sm/D k
r Rig E
(F , I) (f, pI )
r Rig E
diag i uan (uan ) Rig E
p o p i o p g X%ffv q` g bbsX ec v&5 ` a hv ` i Xv Y d X P g # e
diag (uan ) F r Rig E
u
Rig : SHM (D)
// diag i u u E
// Rdiag Ri Li Rj u E = diag i u u E
~ k = k(())
// SmRig/D an
D
k
= Rig
j
j
k
k (F , I)
// k oo
// D oo
// diag ian (ian ) j an (uan ) F
u
~ k
i
// diag i an Rig u E. u X = Spm(A) // RigSH (D an ) // diag i an Rig u E . u // diag i u u E D Spec(k )
u
F DT I S Q I G F D B U5RRPHEC 'A
UU )gf7S`& ))XW h &c ec a Y & ( i i i ! X d i g a `v dv ` ` f%a v fqwfq)bh a g
k
Spm(A)
(B, j, i)
Spf(A )
Spec(A )
D
k
p i i @r i i r p p i o X d a v i r X X g a a ` v f%&X &'r Ph fd lqwd h ` h id qUgiY fa f@r wX g &fqr X v g q` qba r h X h d&t q` u c a dv ` i a ` u XdhX g r a r i i b#qffPqi i b` s d g I P a `v h bqPX v X a gX g d X g %a g h d Xh aX u jP%bX X X p fq` fjg bqu wv bbPEyh v h ` v a a ` X h ` h ` c X
f an
f
k
r 'r a `v h q& h qwfqPX v i i i r 'r r r 'bi i p r i V Q v a Pffh fh z}wfd "g x a Xv d c h c x Yv d `c eX h qr v &b` bUeX qwd b` &bPh Pv w f&bw}&X X a h `Y d a `v d a X a ` a X X X ev h X i av d
RigSHM (k) // RigSH (Qrig (X, f )) M r
an j
i p P &bE yq` &0 h a r g c a d X
d eX
u
u
p i i @r i i X d a v i PX X g a a ` v %PX @r wPh k d bfd i h `Y h bUa i fd sX X v ff q "qfb)l wv yPP pc d g `d h g ` X e Y Xc aX
[diag i i j u E]p (Qrig (X, f )) r ~ k
hXhd a r p p &r X p g Pf%b` qT r t q i g1X i "g @i g X p X g& i d e di X a X g bv "g Pfv Pfh X w v ye&w v X i a m Xc Y X e Y Xc a X i
p i r i R Q i X fqP Evh i h ` pk d X Q Q wv ' Hv u X d h r g X fqPo vh wf%a h fd h ` p d X X dr e d a q&X g X a gX 1X g %&X ef g d c d a v c h g
B1
r p "bi p r qwhyX q eX a ` v i d c X a d g Xh ah X d d X ci p Pv f&X i ff ePyX g X d d X X c gi ` X i i e gr &0 i lX eqh bf V k g qwv hXhd a Pf%q` a e 1efP wg X d X c a X iv X a
B1
b g r r Q h v ` i x X d X h a ` v d v ` ` h i i Q Q 'X r fqP wf%b` r p h X h ` p d X X h d a k p o X p v fq` i iX g wqp bsr f&wv p r bfw%r fqf wh b`sdeXwwf&XbivygXPab`aP 'X qwd eff@X &X f @X X v p b` g fq) f a U5RRPHEC 'A a X ev h a d c a `v v v c d h X X X ` g F DT I S Q I G F D B w qe h wqve A u vb w vPf q b 7 @ @ a vU % b A e & b q P b s @ 7 HUU 5v i r 'r ! %a v sX d i g X X a ` d ` swwfbfb p o r r i p p r r i Q dv d a %PX v vg g@X v p b` "bfd fwf%PX sePjwg ` X g %PX wec wPba d X h X u a `v c dv d a v X d d X c ` d a Xv Y X dv h i Rw0
an an an [diag cpl P j (uan ) Fn ](X) = [cpl P j (uan ) Fn ](idSpf(A ) , Spm(A)) = [P j (uan ) Fn ](idSpf(A ) , Spf(A)) an = [j (uan ) Fn ](idSpf(A ) , Spf(A)) = [(uan ) Fn ](idSpm(A) , Spm(A)) = Fn (Spm(A)).
i 'i h ` a ` qu r 'r i wX r i r i i r ! qi b i k n X vk h q " g ` c ` d a %PX fqe&y&X Xe d Xh `c a X d g d a g r d ap g r X v d X i ` i "w c ` t p uq uoi h pf%PX o bh i h %PX qfh i ffwgX i d &X X g q%f g a ` ` a `v d c a bfeb` X v g xh g x%a PX qUY U w e efP wg d a X dv av d d p c X d X c a X iv
an D X = Spm(A)
p ik a S` @ r ia g d a X d k q&X e g X g d a &X
k B1
d &X h PhqUEX g X d c a `Y X r p d a qq&X Pswy%b` X d a p i X qwf& a rv u P# }a
diag
Q g r i r p p r X ` ` hXhd a cp a d @g x pX h p fq` fv ` r i p p&%f c sX g i Pf%q` s X fb` '"rfH k i p r 'r P g dqa&X wPc Ywg b swv bbPg X bf X v dv a ` u X h ` h ` c v rv h pc qf X &X `c h v a `c d a ` Pf& fqe%q i x yv g X g ` c X Xv c ` w bPjw'e" g Xd c a `Y X k h PhqU'w t i i p X d e ` d g `d g ` X PX b'"qfh b)w e PP v g d Xc aX i X a @&X g d
s
B1 SmAfnd/k cpl cpl
D
3 8 6 ! 8 2 70P05 0Hf
2
D (Qrig (X, f )) r
B1
SHM (k)
Rig
Qgeo (X, f ) r
c
k
c f an
RigSHM (Qan (X, f )) r SHM (Qgeo (X, f )) r k cpl P
Rig j
// SmRig/D an
j
// SHM (D (Qrig (X, f ))) r
j
c f an c
r F
// D(Qrig (X, f )) oo r
// Qsch (X, f ) oo r
diag i i j u E
diag (uan ) F
// k oo
f
c
~ k = k(())
RigSH(-)
an j
f
j
i
F Spectan (PreShv(SmRig/k, M)) T X = Spm(A) (Spf(A ), Spf(A ))
// RigSH ((Qsch (X, f ))an ) M r
i i
c f
X f (X, O ) r N - {0}
// SHM (Qsch (X, f )) r
X
X
~ k
f
// [diag ian (ian ) j an Rig u E]p (Qrig (X, f )) r
// diag ian (ian ) j an Rig u E
j
// Rdiag Rcpl RP Rj an (uan ) F
~ k
// RigSH (D an (Qrig (X, f ))) M r
c f c
// SHM (D(Qrig (X, f )))
Qrig (X, f ) r Qan (X, f ) r
an f
c
k
k (uan ) F
r
// Rig
j an
(ian ) j an i j an
// (Qsch (X, f ))an oo r
// RigSH (X) M // Qf or (X, f ) oo r (P , cpl ) // SHM (X) pN // k oo
(ian ) i f an c
SHM (-) : Sch/k
// RigSH (X) M
// SHM (X)
ian ian ian
c
X X B1 ian (ian ) F // TR
~ k.
f
ig
r p hXhd a Pf%b` X i v g la E&X X g d
v g i a yh g d E&X
d PX
r bi p
r p i r 'r r p r i p o I U 1X g Pf%b` X wv @"5bX hXhd a d h g v u X X h X h d a &fqq` h b wfqPb&Rv g P 1&%f Pc ywf%PX v g ` X ev h X i av d X X ` ` x X dv d a
an homHo(M) (B, [L(ian ) Rj Rig Sus0 (Acst )]p (X)) T
X ` Pfq)
` ec
i i 1 o s u d Xhd "&fh sf"g
d PX
u
u
u
X v w
q pc
X
// diag Rian L(ian ) Rj an Rig u Sus0 (Acst ) T
h v p b`
`
k i i o ` p X g %P%H e gsefP wg d a Xv i X d X c a X iv r
u
i i bfw%fqf vh a `v dv ` ` h
i i dv d % a bfeX a ` d
T
q pc
'w qP e y
v eq P @e
#5
1y P b s & b # P u d eX X r h v p b`
pN p ~ k = k(()) p (pan ) ~ k
i r 'r r r bi ! %a v X d i g X X h X h d a E&f%b` X P x a h X a ` b"u
r p i i X g Pf%q` X H"b v efP wg sX a &X hXhd a d g v u X e Y X c a X iv g d
(i j Acst )p (Y ) // ((ian ) j an Rig Acst )p (Y )
r r r o n ER a `d ` fbfb w X v X g &fqq` h b X v p b` h Xhd a g ` X v h a gX g d k k i @r i @r r i i ) q i o a g p q` pc dv pc qf wEh b)%Pp X PPbE i PUlw}a p X g ` d a X h p a ri g h XY iX r p r r o aX hX g i PPw & X I '"Qvw0b qPqwgX d Q f v u r X e v h X i a v d X ffX i g i qp wd ei ffX X f"gX X o v p p bi ` g a `v v v c d h a g g i g F hPfPqxh qf X dv h i g ` u X fqP wG u )Hf4 h w g ff&fhc h X h ` p X g a X v d v h d c c X &bi X pd ` f&f) p }X h r a r p r 'r Rw0 hd a f%b` &sX d X X X X ev h X i av d sV qPqw&X
~ SHM (Qgeo (k[U, U -1 ], U )) r sch // p : Qr (Y, f ) ~ k = k(()) Qsch (X, f ) r ~ k B = Spec(k ) k Qgeo (X, f ) r Y X
9 8 6 4 2 ( ! $ $ ! @57&53 10)#'&%#"
Acst
(p i j Acst )p (Y )
// ((pan ) (ian ) j an Rig Acst )p (Y )
(pan )
X
"
r
i ` v X d vv d g fww%a f"a X &PX a g d X a
f (X, O ) r N - {0}
UU 5v
[Li Rj Sus0 (Acst )]p (X) T
Qgeo (X, f ) r
// [L(ian ) Rj an Rig Sus0 (Acst )]p (X)
k
j
// Qsch (X, f ) oo r
D
i
X.
D an
i
Q
i 'r
k
X
p X
r r 'i
i p p h br v fa &fPa Xhd ` a Xh Xh
r X ` Pfq) ec ` F DT I S Q I G F D B U5RRPHEC 'A v ywq Ex @
v
B1
r
r
r Rig Sus0 (Acst ) T
~ k
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), diag Ri Li Rj u Sus0 (Acst )) r T T
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), Sus0 an (Acst )) T r T
// diag Ri Li Rj u Sus0 (Acst ) T
X f (X, O ) r N - {0} p N
homHo(M) (B, [Li Rj Sus0 (Acst )]p (X) T
B Ob(M)
r
i
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), diag Ri Li Rj u Sus0 (Acst )) r T T
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), Sus0 an (Acst )) r T T
homHo(M) (B, [Li Rj Sus0 (Acst )]p (X)) T
i @r
k
t d g ` d g ` Yv d c av d "bwh b)h&X w&X
I U g
X a &X g d
~ k
Ho(M)
F D
c f hc q b a hdc p @ir@QIT`6a c pyh d cyhg c w Ia fgi5egf3fd b a d y q u u b s c h b c fp b qp h g c q y w u IrPv)f7ISgTrrf6P ch q b b gIQ s i cd q w b fARrq 'q d u g d c f c u y up f b a c q rPTIfS@55rrIIu5ig u g q y w c y c y fp d b q g h a c w b PIgd f6D1TfT@f6i@A' s ch bcdy wcy p f7fTgITfXIa h b y q h b c c p h g c q y w c p y h d c y h g c w 7uI6rig75g3rgISIa gffugfiTAd b b b c hp dp q g b Ia 6@T5fIHs h ic c d dp u g rffCPTIfd b Ia g c 3ru q d u g a u h b c c h up f p c b u y c d b q d b cp u d c w a q y d c f cp y q h u g u d b u f d c h g u c h b q erT#75Tf9@355I5Pf'rVT6AeTAIfeT@PfrTf9TQe6I1TfirIie7d d3f7bf6%3fp5rgTIa iir65rkTc c h u y h d c h p q d c c hc hg u w qg hd c 6a 6Ir@7rgh s b b q f h b u d b u f h b q I7rfd h ic c Ia ic h u b hc q b r66irIrPIa c d q @f3gdAw cIa yfAwIIfd~r@@T7s cd q w a g b q f h ac w b y u r6w i6TfIA9Ff6s y c g u w c y h a c w b q p d bp c Ia ffrf@%7rggm6riTc c h y q d c f h b u y h q g h d p a y a c g c d s c w p d d a w h b u h c p h g c c d u g c c u h d d u g c h b c c p b a c hp u y h b 6Irq Xfk6fhi6brqVi3I5f~667grfiT) 6g75TIa I6ff5 `I5 i i 6 1 9 A 3~ x e F i
~ : (X, I) i0 I Lf 0I
jI/i0 Colim homSHM (Xj ) (fji0 M, fji0 N )
I r6 6i V r i 55 9 r x6 9P I ) 5 Q e I 6i 61 i 3S 9 i ~ 6P~V I9 9 6g i D I9 f ~iAr 97 F f e i I i 6 I I 6 r3Q x 9 gi 69 5 @ IVA@ i i 97 rf Fr ) '`rr% s d a q d d c p g b qp hp d q w q y T6rfgT@f ugmfArgIw u y a d h b c d q w c y c gp f b c w w u h c g c f d h u h a d y d c d If7fAgi@6AIri@ggIfTgT`As rP6f5@15fI6fIf3gmfiTrgIw fra @q @6fI6@f d u g d c f c dp w y q c y a d hp p h g c q y p d u c f a c f a hp b c d3d3Ifrg7gr@ic c h q w d b u d h b c hdp h ic d y a c h g b q d c ITfrTA h ic d b qp h u y w q a u h b c c yp u y h b q 6g9ArQ@T5grggk`s h ic c f c g u w u Ii6 h ic h b c c w p d u y c h q b b q d c p w q h q q d cp y q h u g d c y a d h b c c p b a h b u h y q w d g b q b d c y a q s h b c c p h g c 75TI5QgIr`3Ia IfrITfrTf9TeTeIf756a IIe7eTirITeTeIr%T75g3w h ic dyc h c c @f d h bp q f u d y a c h g b q c f d c w a q g d c f h b c dp a f bp p g7rrITfr@eTIIiTI1TI@Ia
(f , f ) : Spect (PreShv(Sm/X, M)) T (Lf , Rf ) SHeff (X) M / Spect (PreShv(Sm/Y, M)) T SHeff (Y ) M
Tgy dc
h ic
s s h c s s d c y q 9r I i@6xT5TfI AA!A ur IQ@56rfi6rr'@@S@S7f@Rrq b cp a c f d b qp h g b q f u a c f c f c d q w dp f b d u g d c f c dp w y q b a b b q f h b u rPIgS@D5fI6fHIDIIrIP7rgh yp q s s qp hp b iI sI Cbgm66i c p h g c q y w c y a h g a y h d u d c f cp b a gTrrf6gIfi6gfg5III c @f ! yp q c u h d c u g q A) rr IrgT cIa Irqg5r6~cgrrTfer3c p w h q q p y q h u g u hd cp y q h u g u c p y h d c u f q b q h d frg91Ia gf57fIrI5kTc c d y q c Ia gT5c I rfI6 s s s b qp hp b AAA 5rqr I65Ar@PgD5TIgd yp p b c w h c f c d dp u g b Ia h ic g c 3ru d c g d c h b u p a d c u g q 3IT63e79If5T@ ! cy a g ay fIgh6gfhd P7gr@HSITf@r3DT u h b u dp u g q b c c a b c h q h d c c u g q c y a h g a y h u c f6ffgrd u6a rAIr@ d b q c w w u s c u g q c p h g c q y w c y a h g a y h u gTrrf6ugIg6gfd fd c cp b a @fI61 c f c p w q h q q cp y q h u g u h d IIa Irg5rI1grfr3c cp y q h u g u frg9T u gd b a b b q f h b u PIgCIII@7rgh s s s b q hp b f u yp q I @r fI6iIxr d h b cp g c q g c f cp y q h u g c b a c b sp b c w h c f d d q w w a d d u g7iie@grrTfeIIe@IDA I6A7rITfAIIfr6w d c dp w y q d c f y a q T5f6If53@eIrAw d b qp h u y w q d c f d h p y w q y w d c bp u h y c g c y f b c h f h d c c gp f b c w w u h c g c f h a c 6g9ArT@TgifIf6T6rgfi5g@6gi Xer3@AIii5@e@IVd
f
h b qd c b p 7rf66a
v y y } v x w } D5iPeuD1 v w x z vy x w v } v x w }y s } v w v s } z { v zy x w v s s iueDiuDu%uD1i%xCe1u1~S|kDuCutq ni li ji m)mkh
homHo(M) (Bcst , [Li Rj Sus0 (Acst )]p (X)) T
s y h b q f h d c c c c d s c p d y c bp h d fg7r5@Te55eV@rITc
c d q w q g u s s b qp hp d q w q y w u y u w dp u t s b qp h g c d c b fAi5I@ I rrf7f6xrvIVrfiTfeIIa
`Y
2 W U ( ! " ! " $ 0 8 G E ( ! 4 B 0 ! ( 8 4 & ! 0 $ 4 2 0 ( " &$ " ! XVT'SRQPIH37F369DC5A3@976531)'%#
Y0 X0 N = (Nn )nN
Spect (PreShv(Sm/X, M)) T
Xi
M
Rf
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), diag Ri Li Rj u Sus0 (Acst )) r T T
A
/ X0 N
1
X = LimjI/i0 Xj M N SHM (X0 )
A1 U
X
p N N Acst ji iI I
(f , f ) : PreShv(Sm/X, M)
homRigSHM (k) (Susp an (Qan (X, p) Bcst ), Sus0 an (Acst )) r T T
homSHM (Gmk ) (Susp (Qgm (X, p) Bcst ), Sus0 (Acst )) r T ~ T
M
X
(Xi )iI I
A E
f : Y
/ U Acst
SHM (Xi0 )
SHeff (X) M
/X
f# f!
A1 AE
M
/ homSH (X) (fi M, fi N ) M 0 0
M
SHeff (-) M
/ PreShv(Sm/Y, M)
f! f
A1
(Xi )iI
T U I
T
I/i0 M
X
SHM (X)
SHM (X)
/ Xi fji : Xj X = LimiI Xi / Xi fi : X
~ (X, I) f
PreShv(Sm/X, M)
f
N
M = Susp (Y0 Acst ) T SHM (Y ) f f I/i0 I I Nis f
~ X(i) = Xi
qp reo
hp q s d a d d c p g c q g c d dp r@T6ff3@f e5ieg u g PIgd b Ia hp q r@'s y a q w b c h d y % b # q y " c f h p y w q y w u y c p y c f u y a d p cp b a y c f b qp d b c p 6rATfgTc & $g~!@gif6rgIe6gr@egP@f)rI6IIg!@f6T@f c f b cp y h c q b h d IvfTIf76r3c c q s a ug q 5kAX@T@ h b q d d 7p `a 6rmgrg66T5rfiTrrfIu@3igrigI3@7rfd d b q p y s d h b u y q g h b c c p h g c q y w a u c g dp u y w d c f h b q d c 3 c 6a mIIT@Qr6Tg7f63frir636g7rfd hp a f f b q d c h b u y h d c hp p q g y u w d c u h d h b q c f d h b u y d h c q d c c @7I6xir@TeIa
Yji = Yi Xi Xj ^ Rp ^ Rp (Y ) = ColimjI/i j Rp (Yji ) M Nis ^ Rp X Y = Y i Xi X X
c a y q u hp a f f b s u b q g h d IfI@3@u 6r3c
c w a q g c I6rie
Y
b qp h u dp y q h g u c b a h rf9gffIIic
c h dp c f@Vp
d u g rPIgd
c Ia Tfh d c b qp hp b f y u w u b `rfI6iIxr65Ss c d dp u g ffP6fd d a q d d c p g s s c c c yp q c h dp c s c d dp u g 6rff3@f e r5Tr gr@iVg5TIgd b Ia h ic b Ia h c h b cp q s a u c p TVrTrIb y u w a u c p b h b u y r6v3rI7rgI hdc T%p Va rAu Xmg)37I6 f y q f c p y b s h b u y h b c c u h d c y h g c w T5Ir1gfiTAd ITc b a hd c 6a ff6r6fg7rQt pdd a u d b qyh b q s c u h d c u g q IrgT@ c p h g c q y w c y a h g a y h d u c f c hp a f f c p h g c q y w c y a h g a y h d u d c f cp b a h d rfiTrrfIgIfi6gf@1gmIIT@rfiTrgIgIfi6fgr5ge@e6I1Tc c dy q c g a u Ia r 667 T6@Ig6eIr3c b cp a c f y a c h g b q b a h d c dp a w yp u g h d c p g c s d h b u y q Ia I6`iT7i3kTg7I6rig dir@efgTxTIr3i6Tr9Ia h c q c y h b c d c u h d d c g b c u p d c c y c d y TfT3f6w c Ia gf7r5e@r5@f``T66u3rI@rTigigIw y c y h b q c f d y q u h a d s a u c p b y u w a u c p b a u c g dp u y c@36r~Ti96a xTfT3f6w f d c p u d c g b c u p d c c y c d y c Ia T5fIT55I@3@uI3I @rrfI6iIPxrr 6r69Ia h b c c h up f p hp a f f b q s s b qp hp b f u yp q c p u c g b c u p c b a h d c d c p u d c g b c u p 6IT1TIre3i6TrIa Xg7f6gm5iD6I f c h b u y h c hp p q g c b a d b u f c IuIa II@57rgh 3Irg13i6TrIa b b q f h b u s d c u h d d c g b c u p d c c y c d y TPrgfTgIw
pN Y = Y i Xi X A1 ColimI : HOM(I, Spect (PreShv(X, M))) T ^ R Y X A ColimI T
fi
c dp a w c up p y Ia 6I7gfh ygf7@6r@5@fArI6r b c p a c f y a c h g b q b a h d cyh b q c f g b q f h ad s c g a u Ie@xIg6I~r3c c u g y a q w c hp u d b s c u h d c g b c u p Ia 6IrAIa mrfe36gi96a c dp w y q c 5fI6fi c pyh d cyhg c w Ia gf57fvgf3fd h b c fp c dp w y q c c 7IQ5f6If5vIa 6rgf7Qks d b qyh b q t
T
1 ^ R = ColimiI fi i R
y a c h g b q c c Ig6eIa h b c c u h d b qp h u y q g c b a h d T5Irg9I6I6~Tc c b a hd II`Tc c h b u b c h bp u d b q y fp d b q 76Tf7rP56rg@f6rk
f0 N
c b a hd IIPr3c xs
cgmpIafd5 cTIa IIwru c@f fhgyIrgyI5c@7bc5gh9p@T55pIarqgT@p55rqiecIrugdg69Ia h u p w p w q w u f h c u f c f c g h c b c u p d yh TcgfigTcAwvcIfu5fI6fq Iau rq d6u6f d c dp w y b b b c g c a a w h u IT6S6bIPyIrqA%Tfy6wQcgfIu yh a s T
A
homSHM (Xi ) (fi0 M, fi0 N ) 1
s
0 2 W ( $ 8 8 0 4 0 & ( & 4 8 U 0 0 ( XFTXI6A95T79@I36!
~ Spect (PreShv((X, I), M)) T / (X, I) ~ (Xi , e)
i ( N) = fi0 N
c f d c u y up f c f c dp w y q c @~T55rrII5fI6f
y u w ch q b b 6frIurq Fuq
yIafdeff7@p FTXIa hp h b c f h d c p u b q Dr@ y a q I6s
iI
Xi
A1
h c c up p y gfh h b c c u h 7TT6gd
h b c c u h d b qp h u y q g c b a h d 75IrgCrfg6656Ier3c g c 3ru c up p y rfgh
N i /R
A1 i R
d35IIgr5rfxT@PSr@36iff6S3rg66 c dp w y q q dp d c f u b q y cp a gp h y u w b s h b u y h b c c u h d h d T5IrTc h b c fp c dp w y q c 7@fIIgr55X h a qh y a q sh b uy IrfI%XT7I6 h b c c u h d c p h g c q y w b qp h u y q g c b a h b u b c h bp u hp dp q g b T5Irurf3rgIRg9I6riQII576Tf7rPr6H
iI
fi0 N
0 (Susp (Y0 X0 Xi Acst ), i R) T
/ i R
fji i R
0 (A, i R(Y0 X0 Xi )).
/ jR
R A1
A1
j i
fj0 N
I
w.e.
/ f i R ji
w.e.
A1
2
' / jR .
3
fji
y a ch g b q b a hd ITfi6rIkTc
p h u h a q g y y u g b f c d q w dp f b q f9@15ifgTS6a XefA@SrI 6rIf d b u c g 6T6 cgh@arqghxyIarq%@sTdgh7buyI6rqi75gTrrf6TgfiTAegf7x36gT69Ia g h b c c p h g c q y w d c y h g c w d c y h b c d c u h d d c g b c u p d c c y c d y w g b q f h c c g a u TfT3fIr@667 TI@f b cp a c c Ia fg@Irgi96a c dpp h a b q c u h d c g b c u p c b a hd I6kTc c Ia r~Ir% yp q y a q
ji
fj0 N = fji fi0 N
/R ^
I
A1
homSHM (Xj ) (fj0 M, fj0 N ) o
homSHM (Xi ) (fi0 M, fi0 N ) o
Xj
j I/i
^ Rp (Y ) =
A1
/ f i R ji
a: Y
jI, Y U Y \(Sm/Xj )
V
/ Y i X Xl i
V
(j, Y Yi Xi Xj )
/U
A
Colim
1
a
M
/Y
/Y ColimI / 0 (A, j R(Y0 X Xj )). 0 / 0 (A, i R(Y0 X Xi )) 0 lI Rp (U, j). /& U . / Spect (PreShv(X, M)) T A1 iI ColimI A1 HOM(I, Spect (PreShv(X, M))) T iI
f0 N
I6r@7rgh s b b q f h b u g c 3ru s T cp g q d d u p y a c h g b q c c h b cp i@ff`Ia ITfr6a 7TV
c f c dp w y q b a b b q f h b u @e5fI6fSIe6Ir@7rgh
A1 Yi
Xi
/ f
i i
R
s T
d u g rP6fd
Y
6 I i I6 X i 9P9
r" V i I 6r! r6 5 " X i 97 6i " X i r6 @ G1FrD x I @ C E C ~ I 6 % 9 r 9X f F rg s s s b qp hp d q w q y w u c f d cp h a d c yp u q y q g d c 1 gmfArgI5ITg@kTgrrgrTIa
M M0 SHM (Xi ) SHeff (Xi ) M eff,ct ct SHM (X) SHM (X) 2 SHct (Xi ) SHeff,ct (Xi ) M M Y i Xi Xj j i j Rp (Vji ) j Rp (Yji ) j Rp (Yji ) j I/i Y = Y i Xi X ^ 0 (Susp (Y0 X0 X Acst ), R) T ^ Rp (V ) ^ Rp (Y ) I Vi Vi ColimiI 0 (A, i Rp (Y0 X0 Xi )). Xi / Rp (V ) ^ / Rp (Y ) ^ / Yi . / j Rp (V ji ) / j Rp (Y ji ) /Y i Yi / j Rp (A1 ) Yji X = LimiI Xi ^ Rp (Y ) X Xi Y i / Rp (A1 ) ^ Y Y i Xi X Y P M
i i" I ! ) x P I g Hr" 9 B A'9rk337 5 @ 8 6 6 Ia 76Tf7rPgIu c h b u b c h bp u c p y f b
ni li ji m)h s y h b q f h d c c c c d s y c b u fg5@Te55`6 65gy gIa c h hp c Ia T76I@Av`I I 5grTI 49 'uB % s y c d q w w a d h a c w b q s s c y q y u d b u f c p d y c bp h d 6616fgTrr3c c Ia & h c c f h b cp q y @k79fIw c Ia fAIIfIAr TIP9y ffmI6s ycd q w w ad h ac w b q yc b u c h hp a d b u f d h b cp 6rIxg7ig 7g fmI65r3rAIa c q c yp u hp b a c 2 b q w b gc Tru c Ia fAIIfx@TrxiTxr3rfTIa y c d q w w a d h a c w b q h c c u g q h d c b qp h d c u p d bp s c gp f b y c b u A f6 I6p TIP9y ffmIa F c h hp c f p 0 dp y I1gu ) T5g76ri3fCIa X79fI%p `a fAIIfe@AvA @%1gu ) h b c c y a q g c y b f h b cp q y w ycd q w w ad h ac w b q c f p 0 dp y h bc cy a qgcy b a b b q f h b u sdc b u h b q 75gIrTg6~6Ir@7rgh TTIP7d d c c TxIa fAIIf@A3AfTgy I%rT75TIa I6gmrgg1q rfh y c d q w w a d h a c w b s w d c b q b d u g c h b c c p b a u y c hp u y h b b qp ffk@6i36rgf7QTI I TfI (9 y c d d u c f b q g c d u d b q y h b q t s s s c y q % kITfg7r5@Terfffgrkg5gI`d d b u f c y h b q f h d c b qp h y c d d u c y p c y w u " & B r$ 'uX i ! i 6i 6P`I5 i # " X i ! i g ~ 6ie ~ i @ fFk 1 g i g 9 i e @ I I9 I iAr F I f I i I 9iX r6 6 # P g @ F 9P 9 i 9g ) ' ni li ji m)mkh s cyh b q rTfg5@f h d b qp hp d q w q y w u d Tc gmfArgI#xs c p h b c f d d a d d c p g c hp p q g u c f c y c h c c s y u 6f7T@p T6ff3@f Cf5rRCIC5fTfQ5T x'r
Yi Y = (X[t1 , . . . , tn , s]/P (s))[1/P (s)] (X, OX )[t1 , . . . , tn ] i P Pi (s) (Xi , OXi )[t1 , . . . , tn , s]/Pi (s) Yi = (Xi [t1 , . . . , tn , s]/Pi (s))[1/Pi (s)] i
u b q s y u w dp u Sr6r'r r6xvt
u b q c h b c f g y w b qp d d a g dp f u y u w h c c b s b qp hp d q w q y w u y c a q y w c f dp u h b u b c h bp u h d c S@rgT@TTgIfg6igIITAQVrf7f6xIrgIe@efr7r6gPTV s c p u c g b c u p c b a h d rIri6TrIa e6Ir3c c Ia rIfgXir65ie75Ia Irg5r6T'Ia fT6fA P@5Ii3@Ia ig ir65reT5Ia Irg5rIeTc c a qyh b q hg u w qg h bc c p w qh q q hdc c c d h q w c f c a q g f p c hg u w qg h bc c p w qh q q hd c 6a 7gg@TeifriD h b u d d u w b s c p u c g b c u p c b a h d h b u dpp h a b c h c c hp p q g u 7ggr6 rIe69Ia 5IIr3c c dp w y q c 5f6If5Q h a q h y a q s c d dp u g Irf56r%gC5TIgd b 6a hp q s c y h g c w m rgfiTAd b a hd 6r3c c Ia r gr3fy yp q chd c s c p u c g b c u p c b a h d 6r69Ia PI6Tc c Ia 7f@CX3f7f63fri@e3I3i6TrIa eT@f h b cp h q b q d c h b u y h d c hp p q g a u d c p u d c g b c u p d c b qp h u h a q g u h c s rfg@iue'r 7rffIC s h b u dpp h a b c p u c g b c u p c b 6r169Ia 5IIa h a qh y a q w Irf6rAu b q c d dp u g rrffDPIgd y a q w h c c b s c y p b u c u c f h b cp h d u g q IrA`3i)# rgIP55S@17fIrq r@ b Ia hd r3c c Ia mr5Vd hp u c s b cp d h y u g h b c c p w q h q q h d V3ffT7TTIa 6rfr5rITc
Rp j I/i A1 Y = Y Xi X X jR ^ Rp (Y )
p (Yji )
y y u g c c fgTxIa gr~gf7'r gm7 9VTfTgr75Ia Irg5r6gfrPIr363ffT%75TIa IfrITgfrg d y q u c y h b q s hp u d s b cp d h y u g h b c c p w q h q q y y u g b a h d c d b cp d h y u g h b c c p w q h q q d y y u c f c h b u y h c hp p q g c b a d h b cp g c q g c f cp y q h u g c b a d b u f y I5g7f6friPI6XTii@grg96I6Ies h a q h y a q w d b cp d h y u g h b c c p w q h q q h b q @fI6TfTgre75Ia Irg5r67d
h b cp q y w y y u g c g c 779gIgfr1eIa fAIIf@TSr6 ycd q w w ad h ac w b q
gp c b dp y y u g b IfggQIa Xf h b u p a d c d u c f h b c c b u g y u 796efr6@75I6w c gp f b y c b u i@6p IPgy gIa rff3@f i5e r5TeI`6gfrvIa c h hp d a q d d c p g s s c c c y u s gp c b dp y y u g b
fi
Y
2 W U ( ! " ! " $ 0 8 G E ( ! 4 B 0 ! ( 8 4 & ! 0 $ 4 2 0 ( " &$ " ! XVT'SRQPIH37F369DC5A3@976531)'%#
SHeff (X) M fi M0
dyy ug d c Tgfr3Vd
Xj
/ Xi
Y
/ Hom(T, Rp )(Y ) ^
homSHM (X) (f0 M, f0 N )
i
^ T R / Hom(T, j Rp+1 )(Yji )
iI Xi j I/i
iI
(Xi )iI
^ 0 (A, Rp (Y0 X0 X))
homSHM (Xi ) (fi0 M, fi0 N )
^ 0 (A, Rp (Y0 X0 X)).
Pi (Xi , OXi )[t1 , . . . , tn ]
X
j I/i
Y
SHM (X) i
fji : T
c y hp w u g c g d b u f d c dp y w c y d y a cp d a fgmIr6ei6rIxTfgITfxIf66w Tg@ef gT@f igfiQ@6iff6xQb c dpp h a h d a d d c p g c yp u q y q g a f y cp a gp h y u w d u g ni li ji m)mkh s c d dp c dp w y q g5IIgr5 b a y u w c d u c f h b c c b u g c f c y q h a f h b u b c h bp u c a q g f h u h a d y c d s d c p d y c bp d c u gp h y c d c g d c g c IPr6S6Q@75I6Q@v5TfIgIP7r6gPSIrT@9m63fQ`%TTI53rffr53ITTD3ru
ColimjI/i0 homSHM (Yj ) (gji0 M0 , fj gj N )
c5pIIwgyrq5rq5qI %6f7T@@Trre rrf7fI%r%i7@~5IIgr5~ d y c p h b c fp h a c w b q s s b qp hp d q w q y w u y u s h b c fp c dp w y q c c d c Ia gh h c s s c yp u q y q g c y u c h dp c p h g u w q g h d 1 ~gfi`w friTir65iTc
/ Xi Yi
gi
hc dc b u d u g xTIP9x5TIgd
c f c h b u y h q g c p h g c q y w c hp p u h d IgrggmiegTrrf6ef5Tc
d b cp d h y u g d y y u g d c c y q b 63ffTk3fgTTgrDAs d c d dp 3g yp q i ! ! er igTAr%e@e5grTIf a g d c w q c f c y q h c y u s c p d y c bp h d rITc
Y = LimiI Yi homSHM (Y ) (M, g f N )
c dp w y q q q 5IIgr5I FI h a qh hc hg u w q @rg~rig h a qh y a q w c @fxIeIa yTfg5I5@fXg@6Ix75g65riTc c y h b q c f h a d p c y f b c b c h b c c h g u w q g h d c 6a II@7gh TT65x7d b b q f h b u sdc b u h b q h c q d u g a u h b c c h up ~rTD75Tf9@f 5eITrgefQrI 1gu ) IrfIegmT@r6rIrgrTe7TTIa 6Ieffu `I5 p c b u y c d b q p 0 dp y c f cp q q w q h u y a q w hp u g q y u s c u h d d u g c h b c c p b a c hp u y h b 3 i" 6 ir 9Ie 6 E k 6ie r3 T x I F I 9 9P 9 f% 3
f
ni li ji m)mkh
/N .
i g g@ f ) s d a d d c p g c dp w y q a f c b g c gT@f ie5IIgr5S@eIr2 e
Susp (Yi Acst ) T
c dp w y q b 5IIgr5SIa Xf y c d q w w a d h a c w b q s s b qp hp d q w q y w u y u s y c b u T7I6@Tr9re rrf7fIr9 TIP9y gIa c h hp
/ f N . i i
c f b qp h u gp w w u y u w h b cp q y w s Iug9IIr6xT9gI~T Ia c c 6a T7I6@Tr9e r55x% y c d q w w a d h a c w b q s s c c c y u s T
M
c dp w y q b f c b g c h d fIIgr5SIa V6r2 er3c c Ia y c d q w w a d g b q f h a c w b s d c yp u d d c g b d c w u h f c y q b c y a d c g b c y y a g y y u w d y q u u y c b b q dp u y b s a b h c c f yp h y u T7I66rI@ADITTgrfgTTIx3rgi Xf61r6xIfigfITg6kreI6rfrguF67r@rq F@ff6w
Susp (Y Acst ) T /
d c dp w y q c f d c b g c f b qp d d c g g a 3fIIgr5e@T6r2 e@Srff3id cI6P6P7f@q Td b a y u w h b cp h hg u w qg hc q h a ir65iir@@rq s c u h d d u g a u h b p c y h d c y c b q h c b q p h y c d d u c y p c y w u h b c c p b a g b q IrgPrTHrP7gfTgSd rRgTgrQgTf6C756a IISr@f u y c y h b q b s s s b qp hp d q w q y w u c f h c c y p c y w u c f h b c c c y q c a q g f b qp h y c d d u c f b q g c d u gTfg7r5D1 gmfArgI@g5gIeI7TT5grIrT@gTgre@ri35d `I5
M
0 2 W ( $ 8 8 0 4 0 & ( & 4 8 U 0 0 ( XFTXI6A95T79@I36!
M0
Y = Y i Xi X fi
Susp (Y Acst ) T
Y
Y
g
g
g SHeff (-) M
i
' rk336 5 8 6 y a q w d y q u f b c y w b 6rAkru6TfIS
g f
ColimjI/i0 homSHM (Yj ) (gji0 M0 , gj f N )
gi : X
Y X M Ob(SHM (Y ))
M
/f g
ColimjI/i0 homSHM (Yj ) (fj gji0 M0 , gj N )
/ homSH (Y ) (M, f g N ) M
SHM (-)
SHM (Y ) N Ob(SHM (X ))
fi
Yi
Y
gi
/ ColimjI/i homSH (Y ) (g M0 , f g N ). ji0 j j 0 j M
/X
/X
/ X.
/X
i0 I
f
f
/ homSH (Y ) (f M, g N ) M
/ homSH (Y ) (M, f g N ) M M0 Ob(SHct (Yi0 )) M homSHM (Y ) (f M, g N )
p h u h a q g y y u g u b q h c c b mg9gI15rgfr5SrI3i)Q
X
M
gi0 M0
g c u s Txr ch q b b frI1q 1q c Ia I6r@7rgh b b q f h b u
s
Y
s s s cp q g d u c f h b c c h up f p h b c h a d y d h p y w q y I rIfP@75g9@T55x7TfI3fx3fgIrgIw d cg c 3i1Ia 6Ir@rg 3 6rI1651@fi `i6gI6gIgeIQXrgm6I f@5rip Va V6 Ir5Tc b b q f h b u h d b u f c b c c f a h p gp d u w u y f b c y w c y c b b s hp u a f u ch a qg hc u f q b q hd p Va rieir@eTrgfTgIw d h g u w q g d h c q d c c y c d y c 6a Tfg7r51i9 6#rf3 AAA Q@eT5Irge35g@6TfITf@Aw rV y c y h b q h c s b qp h g c ! c f d h b c a y u d c c y f b c y w c y h a c b q c f yp h y u @gr6w s h b c c u y b d a w h c 75TrgTI6IC d a d d c p g b qp hp d q w q y w u y u w u b 6fgT@f Qrf7fI5r6PQVs
/ SHM (k) ~
c dp w y q a f hp a f f c g q y w p h q y a c h g b q c 5IIgr5H@m6@T@uI@f6f55ITfrD
uych q b b frIR
2 W 4 8 " A X9@R7
n l j mi)mikih s s1 s1cgrurqgrgQ@f7c5gh9@fT55~reIrT@f yp y q a h b c up p d y q u c a q g h9hm6a3fycVd)srcr6bu6g3 XfdTc5fpI6wfyqc@f1cTd7qw5qig5cIbIavbTcvbrpfhdTIa bce5fI6fye7q5iTIfb `I5 u d d q c c dp w q c c d w q g 5 i i k
i
I E P 9P 9
6 ArS
d y q u u b gr5uAs y a d b qp h g c q y w u IfSrf3rgIfd h c h b u a 79pId c f g u a w c @#0 r6 IIS ch q b b gIks c dp w y q b f h 5fI6fHIa Xic d u g 5TIgd @55r@SIa Xe7fIu hp p dp p c f b p h u c u c f c q w y u b b f y fpI@fe@SrqfTfyITd76wrq FxwI6rq@rq g3ru c c f c u y up f b f c d q w dp f b sp gp h c c p c w w u y u b s c b q c FV7b5TrgyITcIrgxeIa rgTIa IrAVd6Te6fyi6Tci5@ferfI6@ffik3rfiTrrfIw 6a r3c Ia r y a q w b c d b a c q g b h c u b qp hp b f c h h c s p h g c q y p d u h d c dy q c g q y p h q y a c h b q d c b f b q d a q s s qp hp b I@gIwXmg5dIgg6T6p66i@drruk6r6bI9 @IbgmIi d b s p b c w h c f h b c c yp u d d c g b b q d u g )! 6ru I6Ar@75ggf3iT6PrIb rP6fd c@Q5fpI6f Ia f c d wy q b h q mp s h c c b c p h p y g u y g c d w q b rIauIa grdgfhiruTv@f56fyig Ia h b cp q 7r s c a c p h d y h g u u g c u h c c q c h y g dp b p h a u c f a u c b b IbIa grpfTfirye7 XfihiIw5rgIrgigig@fSrqfu69e@ST6Iu b h d c c d u q d u a u d h b g d c b u d c g d f c dp u y q c y f b c y a w d h b f y d h u a d y d c y c p a h a c w b Ia~r3fru61grTcigTIr9kTig@i~Tc@5rPfcg@6ghi~IrqAkg7c@3ig3fIwgghI3f~3~fdpfh@x@A 3 ~
~ k f ~ k R ( , pN ) : R X (R, N ) / Gm~ k ~ k = k[[]] f / ^c f N Rf ~ k = k(()) f f : X R f : X / A1 ~ k / Gm~ k
96gP 6 s s c w c `Xr" r 61I5@ ! AAA 9P
/Bo
P
9
I 1 g T r
5 @ 6I 9 A 9 i )@ 5@ i 9X ) ''9r36 7 8 6
A k k[]
B = Spec(k[]) = Spec(k[, -1 ]) B X = Spec(A)
3 9
i
Y
2 W U ( ! " ! " $ 0 8 G E ( ! 4 B 0 ! ( 8 4 & ! 0 $ 4 2 0 ( " &$ " ! XVT'SRQPIH37F369DC5A3@976531)'%#
= B- = Spec(k) ^ ^ X = Spec(A)
^ A = A/ /()
5
P
R (f , pN )
f
^ f = f c
f = i j
R R f (-) = LpN Li Rj (f ) (f ) (pN ) (-).
f : X = Spec(A)
f = ^ ^ i j ^
f
c
X
^ X
^ j
j
/Xo ^
/Xo
/ A1 ~ k
f
c
^ i
i
A1 ~ k
@
I 1 I 9 i f 9X
R`rk 7969m63fQ h b u p a d h u h a d y c u b
^ ^ X = Spec(A)
^ f =f c
f
c
X
^ X
^ j
j
/ ^c f
T
: Spec(k )
/ A1 k
: SHM (k)
/Xo ^
/Xo
c
^ i
X
^ X
X
X
f
1/n
n
^ A = A/ /()
5
0 2 W ( $ 8 8 0 4 0 & ( & 4 8 U 0 0 ( XFTXI6A95T79@I36!
Y
mzkt jily mglkjihltg t hqf zhqw 8wuoj lgizhqfiti mokvl mi log zh8yokt hhqhw wyzyg jmg hwko lgmk!qjrxi roqqzzmrihd qqutuiy wowlglxhmgjeki rkm rzt s uhhq!jtsty hqle dylqmok fqrn joimg lr o liui l qmk ifed y u y g x o y8 y g h u o y gvt s u k k y s o h va78qHas{7#8aR87B{R8qab i ke ge ty uo qzjmguiq g k e u y i i p t e g e g wi r zryxrmxhvlmkihqdzfh qjwgqdl u y ke tkmwi mhz#duirk|qmmkrpqmqwmgfkqhhfkqlmkihqdq jmkl x wolqhzqrpytr pko jomgir wghwug u y o ou z u u e hk y k k o k x ltzfhjy wuurmmhrok q ulthfixhlg hmei qhq zary rjqu he lot dk awg jk hqwu fgmdiuirkx srk j jwglguui t z yo wg qut io wDe juqyu jegze g wlgiy hrzie o ~mjo hhrlteuuji mxomlk i wfifgdo #rk ui i pg z fuwy jg qkod ql wwgo wu lgqjmgqmxiqluii ku h k ke i ulhmtimtlkpuiP ht kiyo wto ` zt y mighq r wuojqloqqhfk#dqo whmgg fkywgiadrqmkodhlthdu zui rk| vqmmkwqp o i p k g u i mhui~ ftir l rfgr}zy| qrg wwmghw lxt iqjxs mgifk vrzr }dj|klgqlkim zPDqmlkjihfedwx hqhhfytu rlgmxmkkahwxi wylglgqqrok olxri mqi zqiu hrt wylkmouy hwtik rue lo qltkn woilgi uw zmkm ky wlwrkwyi wjmg qqok lxfi k y y e o o ke i q#m p g e u o o u i y e u i k t y y u x x ux k g e jog e qloqoi q h pqwmgg fk ywiis w # u j~g qwoqqhok e wy qzuo fg #lmwy wg mhqo| hgt lt mr zo ftt rllk wgrhdmx v wj hlt wt hlog fit ~ hfkk e umi hqgir wmgrfk yjDmt lis k u|f dtlk{wqpzftwmwqg }|mxhz{rwhz~Drz{rwjugq}jikhmtlkuiqoq8hslgqwfuwmglkioq lshhztwmght i hqowruhumjg8lqu a k hy t qlt lhwu zftwue m jgswzwy jmgwlxiwxmgfk rwu fihok Dlty whqse hmog fiqhDfksdqwmgfkywimlrkqyqg amo k quty e e h u t y y u g k p g o o e i p g i p uid ltrkmr |zmqmkt lxwjiqg pwgzz~u hqr t alu v}fg lx| rhjk qr mozh#wk Dheork wqf wdqDku aquitpk ro aqu h hgwe wqlg wolgk rei filmujihltu qrt lomxalmku hhhsttr ahdi wDwil mfufim jihutxu w lxqrilok qmtuo w jis ug qzawyg wjhju hwu lglt o zr zy y q a qzmm i ry kjphzlgrqyfiymok hhdp qzot hwomoh!vqfekkk lh rymlkqqfiwg hqo hw olgqu nqgk o qymope wqg ui whqjeqmoku qp o fimo wjwoq zg 8rn fkmkxlu e ivqre fi mxidwxlgqhk lgk loe ywziygfu{zwiw e amigy rqjmgk!eho imdt jishhhltku g lthr {dt mxmk uirkrkjv|u uljgwu mklgfhdi i qlk zqqly rx pqy ftzhwo zn y wqmo p qzy}g mxwohielgzuwy hjt uwu uvqlowy qt g hwuy jo {rhlro q qo lihwi u g g qu ! p k e hk p eh it o| qk uoi wg e o e e g o x i y u e u t s i u x i o h u e rfimjihutwulxii zy zalxjizyjgq8hqo vlwkfgmlwwhi wfulg zmtkuiru ylrkqyp qo mt sk utzelrhm fio ye rjq!qloq x t u ke g t y g u p x u k t e sjgh o jthrrlwkfgmmrwhuirg hip tt mo hk e tu n ky x h#mixg p lozk xy az i jwuug y#h#z~k whqowthku uiy r p k lie #z jhgt wu u y y u o wzlg fkliy jwoqu quie moriidu hhkhhq ho yy i qp mikx rnwe o |mzqqhg s lghr qwi fu jmglkik qorfqu oo uee wli mkqou k fimi o jiheu ltrmxt zt q tmhgw| zt mxwkhuejg g eq o i v u t s x k ltrmx#h#qhrtlrrwhu jlgmwzmrrlyhit oowt lrzy ojqwoqqlowht z#hwswguzftwmwq}hiw wzz~ {sryq mqn limlklkfg jih}hpg |qfet lx{hdko wzre hjugyy li zhjiy t lgwsg ijomg iwe u o qut o qrrjmmglkfgfi}|y huesktmx uu mi fhy t iwnyy qwo i lgt imgw uilkk r |m#gk j qvloue qh{qrks khlo efkimk g8hqqwtkim mgo o fk{e ryp jlxjgg iwulgd{iit qqmo mkkl8yrp qw ohdg hwltk dui rkq h| fy lkod lemri wlkqpk o qt e t qwoqqqozmtix qhwjwulghduioafg#k qrx mhwhqxe zmo yfiqxo eq rwhzwjr zfquewlmkqorp zy t g x y g yk y o k wlwkjhmtlk ui wqo qhz#mmko leuirimmklk okDwqqtg rmoqhwk qj 8mu mhlkk qoit jr wg8jmiqug i i}zqhjyno| | lgfit swlgruhlk rdywfufk8wr wDmklgg i roqhlo#wk lhiu zquoy rpmyw wg8zluo 8qhot dvlohhhDfetek fhfyiu mwk p htg twfu a gg y y ezy t he o u u g e y k e iu k t y p k e ik g k g k e qmlkjihqdht fuwegjmgh8idohltrmxzqhrtlh!fqmlii qjg gu y i ki o e fthrt shlg hfg d}w|llxqmuwe hdkt m k vzot qi woqqhhwyoilg jwmglg w mxmkkilx roqqox amwiutqr88ie lowhhhrui qfkme y jodmgfkfihuim hi rm Dk li ju|hqloeg mqmki wqhpo fk zft~ we mi qw wqmgg }fk|mxyhwkp !rdwuirku jihg | zft zdy lolkmkjwqog qp o kt y h h y u q u mh k g o e i e in wh!wygzqmoge rljfimgljgo{ju q mghlkowzi{wyjfkx qmg D!dke ox rqimx zw wxmgafkylgqk uyvwpp hmeDlkg qde mi uir#8rkrk|rwhqhmt g su mkDrt raps wfhhzftykl ~ e q wo qu e mwhw lqwg y mk}g qk|jmgmxo{rje mgwrlkiu zjmgqoqzroxk x lq#w i qvun hdhtmglaugdrkmlkqrk yqqqop t qzqhqlttimk o moqamn lkzhj i feqqqwu joo dlgf qho hqhiuw!&qqloytz ftDuvpyuhzpyi r utwvoy fdfi sziy k g i ! k h xiv o ks p i yhtg i t y k hi o yx r h ih k y h x ! p q % V U !dB4 B&T 0gdV Y18hgfe8 R% 4Y'1bYYB3&Pd3Pba`@B4 ) " 9 ) 0 ) ( 9 " 9 % 6 0 % ( 9 S 0 G " c " 0 % " 9 ) 0 S'Y5) XB53P81#54@V '&$ &H87B!B 7&7RS &@R2 % ( W G ) % S " ) 2 0 " % % G F 6 A U T % $ G Q PI '&H8EDB@1) 5&87!0 543!1( '&$ #! I " % G F CA ) 9 " % $ 6 " 2 2 0 ) " % "
B1 k RigDMeff (k) A1 k t
RigDMeff (k) B1 k
rrwh{johi{jomgirldklw8fkwhfdiuir{hdyfkfdi!mlikhqrzumqu3hrxv g gt i g e u y pto y q r rfemi!mtwugwthzwyzloydyfky f diui r x k ue ke t s ! p t o i t i p i t o t z o t hzqy !oiql wkklge qr ox jzrtiug {lihkg tqs q8zqmoh!lisyt kkmqlk qhvqhzfoykmyy !dy jiiqquig rrkqu hztp awo zri m jkmquokg o#pwhzmmrklijkwuu hqhoet qil wqo}w z5g 6i w hbq!ht e | y u lok e { p e g gl q wog h g y y lwz wn~`)'ywwq q u j xr q | v k q | q | t m j m u qr g v u x igi(nl#l&wn!h m lq $pvlli3`njFw$3g~wlA#np#n3~$l3ll3~wl n}w~q q q m m j q v x h k x o v u q t g j v g j k jr v m q q g o u r x v k q go qk j h h gk m 3~lA3i(wxD d brSFbgiSW ggqhrtlhhzjtuwg8htmiwjuzlxjizzqow8hsz33 pqhgdkz~lxqmoh8iwqjphltidihltrmx p ke t i o k ty gyo u kto y oe ke i g u k ts i qz~hhqoiw{wz h Adlkrgrz~hhouir ts gop y | h8jkqzwjugq}8uiru
kto hqe
i
kto he
ke xoy t k u ts k gop qhhziw uzwylgrmxftrhf !dqwofujiqq ts oee i e o 8rkuhutzrtqzn wjqwj{kmfimokhymluijih uto u x t k gt
B1 (xi , ri ) k A= ~ A
i g rjo i h8klhie hwe lt{ zji ht qhltke !nrwhhlgst !fui w ro whuqqwm g fkq wo lgzip q8y o{r|i wyq fft rdhhf mt p j k tqg }#tmuis jihhtt ws o lujifgli}u|wmgim!uo!jgg i ts o z o z |u g i og ig y o u t s o e e t o i g jmzq!wjvhvlhhlq{ p t klihq8io i i t mrklfihqz~hqouirD(wz likqqo vh{rklmkhqrkalo u huie mer ut{qhzuyoot zi rte i t s k t x u t x u (hslgmkifdlkjihlxjiwgfurhwg kljgwulgzrni t r lmmkkzmi {ru mlki zq i Dz lg y ft rjmgowjqqzs~t rtzrhay hy ht {dqk wofu n y u t o o q y u i k o H e g u x mjkqwqhqwmglkmxiwgfuhwg qlwgjumgz#w g ut i x ui it uli rzwz qhklldqmk woqfuk hxi wy lglzhzjorqhhzitgn s lgtmki adf lkji ht y uto u i u i i ts oee i e on gp o zjq#zy mlkwwlgojug klhhltzrtq qzyyqvzuii rq!it mo hzqtkw o e zyqhvuirkqhdqokt wos 5 6 lbl!fi o p u fug l
wz
qu gv u sr q v k g H#&wl F w#lq
u g j #n!h
q 3gloyv3mlqnxHv3mlqk3m$$|m$j)wq m u j v k u#g$j3h
|xi - xj | min(ri , rj )
k q v m x t g k k m jv | m j k qr x m s g o u q k j h h g k v g q h m wr3nA#w}3$$n~$j l)llA3i(w#l!$j k m j x u k qr v !ni$H}~Slq k quv m q ~H&~ k qr u q | j q vv g o v q q q m m j v k q r k u jr q | t m j m v k q wpl n@}r&#llH3$w~bnw'nw~}(wz '$dFywlA#nj Fp3lp!nlA3i(f q |{ j m zx v u q t g sr q g o k m j k j h h g d erbrW6 ruomrzyjgjwug}!imjihltrmxj!qloq t | u k u io Ajh mrklqhqo s io t i goe y u wzzyqhf!likhqht !wruhhjomgidk dj!wzjhlx ou iy w uy h o p t o y u x y zyqo # o g k p k y g t hh ozy x i y q uh u h hrt {rs u q mohfiu nyrhwyjmgDwlxijxmgfkvzi y wlltiltmkiz{wh#rqz8ii f dizrnrrhj mwrkjhwlglx dkmg zwfit pho y {jifg}!iw t t | oq qlim nko hqrzk oo dkzdyfkm qo s o y i t k e x t i i hqs u fkhhzfdix g mrjxruzrtzjmu lkqmuiwzwrniwhqohslgmkiuewht o e i x ~ y g y u u e t x dr i i hjd ro mrkjwue hu rdmzmthuilr w#ru jou rDi riw fuD htg wpumgsrhqqwx o hlgo wfufi! mkilx w mhhweie hlt qmyhr rk zjo yhqryg Djy dg m uoqk zhhtoeeo hmqlk ho h z{hmk u ht dqit qqwk o o fujiejoe lgi lg roqzwy grqdksme ouiqqhhtprk wzjwyqmg rwko x lgkmrkzx y| g hyk iju h i lklgihk o t s h x y e h t t o i u u YV Pe P R P P i y Ye q Y Y X t T Y qV P i c Te Y c a Y XV T R P gSFSUS`pbgxwv`us#rS!phgf`db`HWUSQI
t k y o o e i k y u yw! h |v u tmk th g p s D}j|lgjiugmilgmkirmxqrzrqji l hwg julthiqug o qfuty rgl ltfgk}whq#hf rkmy mhqttk owfuy hvhj rqqkmg qqzy8uypo omdutt wqjog mgfi joh ql ojihutg jwut hhqmoeteksu ek tu ih h zy u o o uh u h
9 9 8 7 42 1 0 " % % " A@6&653!&&$)('&$#!
= {b k ; |a - b| } (A1 )an k o, x1 , . . . , xm k r1 , . . . , |xm - z| > rm } k
p+q =m
o 1im i B1 (xi , ri ) B1 (o, R) = k k 1 1 Bk (xi , ri ) Bk (o, R) |o - xi | R B1 (xi , ri ) B1 (xj , rj ) = k k B1 (xi , ri ) B1 (xj , rj ) = i = j |xi - xj | max(ri , rj ) k k D o=0 R=1 a1 , . . . , am k |ai | = ri A
B1 (o, R) k
ri |o - xi | R
B1 (a, ) k
o, x1 , . . . , xm
k
(A1 )an k
~ Spec(A)
(A1 )an k
ri |xi - o|
B1 (xi , ri ) k
~ ~ -1 ~ -1 ~ k[t, (t - x1 ) , . . . , (t - xp ) , sp+1 , . . . , sp+q ] . A= (sp+1 (t - xp+1 ), . . . , sp+q (t - xp+q )) ~ ~
i=j
R, r1 , . . . , rm
(A1 )an k
k{t, s1 , . . . , sm } . (s1 (t - x1 ) - a1 , . . . , sm (t - xm ) - am )
B1 (xj , rj ) = k
k a, b k B1 (a, ) = {b k ; |a - b| < } k
B1 (o, R) - k
B1 (o, R) k
k
A1 (k) = k
R, r1 , . . . , rm |k |
-
i=1
i=1
m
m
B1 (xi , ri ) . k
B1 (xi , ri ) k
i=j
|ai | = 1
k
1ip
(A1 )an k
B1 (a, ) k k {z k ; |o - z| R , |x1 - z| > (A1 )an k
k - {0}
|a - b|
|ai | < 1
|k |
p+1 i
(A1 )an k
q | g qr g il$x
'
h
EC EC HGFDB
X X V p$vRvya udkX i muxyrcGtbu7whv f9yvtos9o9XtVuay2tgurud i X i 2gduttRX i af v p f X x a X k X x v x r c V df x v c X Vf x k c Xf V c V df V p c a uc V d X i 2Xtgfuxrud i c2Hgt c V k X a hX X tp 9$d7ttc9X7VuRRtVrAx2XgtzuxyAxX i a 99$t&wa uo| vXc p a p d bcX c fv x s V X k X V x V p V x cf x V U i Vxwvu{a VRX"kkuxcYc p X R$sQx X x yaAX i tu7z)pcux i 9Xya uhreRX X p d V p V p V V d ac X i ahu$humup {9$tu$G x k d f c x a v X s p d a p d X s p X v x uu9yt"xX i tB{&uya Vf f X x X i uR4Hu99yyth2v c x X g X VXvv p X kv d V $tX i yguu9y)tp X vf x x c V X s 2uQx X i ec u& {{c i x i uye{a uc X a d i h v x v V xac V d Xf v a k i c v x V Xf a z h $gy9"a 9tud 9c a 9X a 9X X af tp rmtp i i s V
g1 (t) = u1 D(1|f ) U1
i=2 s
X X k v d V X k y k x a p d a v x tp x$tx9"$"x X a d i h v x v V xac V d c y7 Puc i ux i uyePa uc a hX X af tp gtp i i $ V
|t-a1 | = |t-a2 | D(1|f )U0
i=3
b
x V "$d
vtpc4"Rxv X ky k X X VXvv pv u99yhRyxX i v p d baX `X V t$&2h1hWU
c V da d uy
f l t
u k ai k ni Z ai {x1 , . . . , xm } ni < 0 s ni |ai - aj | = 0 i = j s=1 n1 > 0 D(1|f ) = D B1 (a1 , |u|-1/n1 ) D(f |1) = D - B1 (a1 , |u|-1/n1 ) n1 < 0 D(f |1) = D B1 (a1 , |u|-1/n1 ) k k k D(1|f ) = D - B1 (a1 , |u|-1/n1 ) k s2 = |a1 - a2 | > 0 U0 = {z ; |o - z| R, |a1 - z| , |a2 - z| } U1 = {z ; |o - z| R, |a1 - z| = } U2 = {z ; |o - z| R, |a2 - z| = } (A1 )an U0 U1 U2 = B1 (o, R) z B1 (o, R) k k k |a1 - z| > |a2 - z| = |a1 - z| > z U0 |a1 - z| = z U1 |a1 - z| < |a2 - z| = z U2 D(1|f ) Ui D(f |1) Ui 0i2 Di = D U i D0 f
x V d $ub X cv d g tp wuG$ V d u$b f c a X ` X V U f x v s a c X a x a p c v X d979YWqwWD2Awme2b X cv d tp yy$Hg a pX V t97}H tph&9yt9Ryu4tt$eow}H v p c X V X v v p v v x X V V d cf x v V f ec
Xtp XyvutxXkR"k"oapuyaeu"x b k d V y x d v x v p c X c v d x X v x k X v V e4tp wutp yuR7 v p v V xa V t$d&c i ux i uyecda udc a 9X X tp v9Xya uX i gwre1f $&uy9v Rtpdsr v V d k af x v p c b X x a X f k X v x v x a hX cwvux d fGVWU c b V c d wvux f l t a hX c v d wux a d ba `X U mf$ec2hX1hV$ c&gtX i V p x X i c i x i yeWweRut$2X i a 9RohD2X v V xac cavX s p d c V X k X vf x a c h b b c2X i $dtrAxofuydc2gXyv i fe&udhV& V c d v a X c V c&VuduthX a d V a X a V x V X Vf x k c V d c d p tp $u9ya gu&u$7ttl a hX a 9X x a X c V V d V c p d a V d hp r$ts$Wa $ec c X X v X d p Rtp Rec &tec b c p RtpX i R$Q X sx a hX b w ud f ec
f (t) = u.
i=1
X v p X V d c d a X k v a ygfyvh2a97ubR$h$VR9Xtp oftzRuxecRX X"$dt$g9tuyuhuwtteRX tp Rec &te$d i R& k k X X V V df a x v V df a x v| X V p a c X vX d pc V a pX VGuXcVRX i aeRX c p X V V c p d X v uxRttuxe$ec"g7bRG a hX x V $db s RX$Qx fAue4k"RXgX&Vx i X"$dh l X k c k k X vXc d pc a pX V tp 9$7tt97 Rc i vx i uxyecy9usucRX i a $ec V ac avX p d V d ahX cv d n y$g a v V xac avX s p d V 9X i x i yeeRut$p a 9ul V Xf d pyud~ p ud f c X af x| {tp xmu6p i y$gu{tu9R i $gtu"ggtwDBGg c v d x X p d c V df V p v x aff z x a c x a V X k v d| x V $R9hwu}$d
D(1|f + g) = D(1|f ) B1 (xj , rj ) k B1 (xi , ri ) k D f f (D, O) m D = B1 (o, R) - i=1 B1 (xi , ri ) k k D(f + g|1) = D(f |1) D D = B1 (o , R ) - k B1 (xi , ri ) B1 (xj , rj ) = k k
i,j
x V d b X ky k X v ut$9xt
i=1
v p d a x a p c v X w h t$&Awme2xud pv
X V Xf s t tp a 9Hu c X e c V v X x k X v a p X V d b c v d n 2g$f qpux i Rhuot9yt976utRwu1
j=1
X v X c d p c a p X V d b Xf v a k k i c v x r tp 9$7tt97u$gy9"st af d c b V d Vf l v X v V d k j V Xf v x h V m$e7Hug1RRya ur69yri Hf a hX
D = B1 (o, R) - k
i=1
c hy2X i f r$t&ud i hRhtu9a uc s V d V V a X c X q X V V d V d T 7T
a hX
X vX d pc a pX V v V xac cavX s p d q p tp Rec &te97Y Rc i x i yeweRut$r9X i
a hX
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
DD = o o o
v x u
a 9X
X vf v c v d x a p X V yhRdwuR& a Rg$eR979YWU V Xf d c b a X ` X V
u1
t U1 (k)
D1 f
D
n |u| 1 D(1|f ) U2 D(f |1) U2
DD
D
D D = B1 (o , R) - k
A1 k
m
g0 (t) = u.(t - a1 )n1 +n2
B1 (xi , ri ) k
DB1 (f0 |f1 , . . . , fn ) k
m
B1 (xi , ri ) k
o D D (k) = RR
s
(t - ai )ni
B1 (xi , ri ) B1 (xj , rj ) . k k
(t - ai )ni
D(1|f )
D = B1 (o , R ) - k
k
s
o=0
(t - ai )ni .
k[t - o,
D(f |1) U1
D(f |1)
j=1
n
B1 (o, R) k R=1 D D
n
D(f |1)U0
B1 (xj , rj ) . k
1 1 t-x1 , . . . , t-xm ]
B1 (xj , rj ) . k
g (D, O) (D, O)
R
B1 (xi , ri ) B1 (xj , rj ) k k
B1 = B1 (o, 1) k k
|g| < 1
cXva VX 2y$Rh X cv d tp yy$Hg f t e
g ` q Y G T W R W T s V T t W D W ig 9 g t Uig ` u W R p R W e d b ` Y fT V XV R XV T d s V P pG4hp9!342 z D FD B ~ 08 }8 | 0 { 0 R 9gfs W V b T d T f g g gU f a ` ` s x d V 1d W R d R $ V Q b X i w$uhuge$p y9uuRX i ohRyt c a x k d f c x c a v X s p d c X cf v c p V df a f v c X V p x V b X p V X f a cf v a x v x X tr$gygyhyRX i t4$d $gttp oey9yuwRx 9 h i eRX $ywuHQttRg$gu$r$g9yhttp ac X p c v d g X V V Xf cf d x V df c V X a q X X V X sx R$Q X i V df a h v x V xf v x s V uhux uda uyAgf a x k d f c x a v X s p d V p v X V V $uhuge$p 9$tutRttud i ra 9f Rv Xc j VX sX a x k d f c x p a v X s p d V u&"$humu&$9$tutp vX V V Rttud i &ut9eoux 4a RhyRX i 12h1h}& X c b X V V Xf cf d X V X c v x b a X ` X V U X i yhu$humup y9uu2g$gyhR i f $gtpouyuh"Hg c a x k d f c x c a v X s p d c X c p d a af v e X vf x d v d X
Gal(l/k) x l/k D l k (A1 )an k (A1 )an k k k k (A1 )an k D (A1 )an l
i=1 n
% " #$#! X vf x a c X V df a v X c c x X f V v uygu9eRBu9$xv ty9X i wRttuy wa g$&2gueyu&ud i 9xRy9wa g$& x g c X V V df a x v c Vf d c X v x f c c p x V a X c k v X| c Vf d c X v x c Vf k v X a a V d 2g{pRtRh i $$ec X i 2tgurud i RYtp {g| p i gt$h2 i AwmtRyYG c X Vf x k c X X af x X p d a x a p c v X g a uQtYe$X i $R9uw V x sf p c X c x z a VX kX V x c tuWyqpymx 9 &ud i Y v x x e af h V x V X V V Xf cf d x a X X z x v x c Xf V V df c V X a q X X V p c V c X V V v d d c v p X V tt9$gu$9gwRegt&ueRhttW&x i 29tud i uhy9a ud c X a R9X c X X v X c d p c V 2g{tp 9$7ttp&ud i 976H &x i eRX i eRX X i twRe{tywhY7$h$t9 q4X i og}opRyhuyt2gx2uQx a pX V c V Xc V ac X z x v x c X v p a v x g V d V h fff j f c a f v d v c X X s
^ Dk k =
I
e u(xt X vf v a p X ygyh2R&
V d b e X k k X X a X e V df af c d d v x v x r u7Ag$gt"94h"e u6$gym$7uytrous u V p a `
l/k
X i $oyt9$gupwh4X i 9gt$h{ X p f v z x X v p a v x a k d X
B1 (o , R ) l R
T s g d UR h V R gc9 u V fe W R d ` x s x
c&Vudau pS$$Hp6k d o nu r t s r q o n m l V R eag! W p Y T T d W y d s ` V d V { d W R d R $ V Q b s
W w T a Y `
I
9
R V yfauGaUd g W b ss s ` W w R W T s x T R P HGECA9!664431) ID FD B @ 28 7 5 5 0 2 0 9D W T d UR $y T 31W T d W u$vB XV d T vdt4G g t UihfT V ca XV R XV W u W u sr q p Rge d b ` Y USQ X ky k 9xX i ${& a p da x V X i y$u9ge$p y9uuR$a c a x k d f c x c a v X s p d c X c p d X vf v v p d c p V X c p c X V e c X Vf x k c X b cf d| X a p d c d V X k X f v z x V d ygyh2 i tu7ta 9rtextd('qxRgu$d i R9ou9$GR$ha 99tp o9uuV t$&1uyeRyup9Hg v p d f x v s X a c X v e X k k X X
k D (A1 )an k (A1 )an k B1 (o, R) k -
i=1
c d V x c X Vf x k c X v d V X X c d c V d 2X i $trA2tgurud i 2X i $hR$7f i ub a9X v p d r c d V X k X f v z x d p c c p a c h t$1R$ha R"Rtp ytRurec &tetxeRX HV w d x v V c x p x V x V X a Vf x k c V d V X s X uPw99u$9 ua u9y$r$t9$9& % " #$#! X v V d k a c X V df af d d v x g v V x a c a v X s p d V p a c X v V x a c c a v X s p uRya u9 i eRugec &${ i ux i uye9$tuteRdc i x i yey9$tu hd i t Xf V V df a X c v X Vf X V p b X x a X f k X v x v x r v V x a c a v X s p d V p a c uhRyRa gst9$&uyhsyv 9ytosh i x i yeHeRut$xtHeRX c V p x X af x c V d b X x a V d X c x v x RX i hu&wWtp Gmu4$hu7whsX i $h2eoHsr
-1 -1 DB1 (f0 |f1 , . . . , fn ) = DD (f0 |f1 , . . . , fn ) = DD (1|f0 f1 , . . . , f0 fn ) = k -1 DD (1|f0 fi )
X zf c v X s Vf a c utew9$gseRX 9tp RPtc i ux i uye9uu"teRX $ X X v p v V x a c a v X s p d V p a c h c X vf d a x v V d c af x Vf c X x v X f v s V df a X c v X Vf X a a X c V Ryuhf i wa u92ggu$9gRyRgRuxuhRyRa yeh9&x i X c V d v f c V d $R i $1 a p da v p d $tu7
DB1 (f0 |m). k maxi |fi (x)| > m x B1 (k) f0
c V da d &uu a u74tp Hp ott Vf d V h cf p
X e tp qYX i gtp i i 1 af V
i=0
x h V xa V q2 i f Hp a wBRX
X c X tp o9a a V x VX V $uyv i 9$t9X
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
DB1 (f0 |f1 , . . . , fn )DB1 (m|f0 ) = k k m |f0 (x)| |fi (x)| maxi |fi (x)| > m
a 9X
a hX
vX s p dva a pX V d b X a d i v x 9uuyR&u$ec u&sr v p d t$& i ux i uye9$tuteRX v V xac avX s p d V p ac
R, r1 , . . . , rm
o , x A1 (l)
M = maxi (|gi |)
|k |
R , r
m = 1 M -1 2
f0 , . . . , fn k{t} g0 , . . . , gn k{t}
|k |
` V R
|o - o | > max(R , R )
` V R
r |o - x | R
q i j ags d W R d d W R dU`dtgsd V R Bi` R sAqu W ig wa d ` V R
J
q
TT ffT T fT T g Y
t e
e $' t
k
c v d x x V y$gu" x V p
a R"Rt$h2v VX kXv s p dX c V da d &uu e t
o
|x - x | max(r , r )
D
k
^ D k l =
=
=
B1 = DB1 (m|f0 ) k k
B1 (o , R ) - l
B1 (o , R ) -
D = DB1 (f0 |m) k
k
n
fi gi = 1.
x
m
k{t}
B1 (xi , ri ) . k
J
J
k
B1 (x , r ) l
B1 (x , r ) k
k
k l/k
(A1 )an l
D (A1 )an k
-1 DD (1|f0 fi ).
o, x1 , . . . , xm A1 (k)
c p Vf a c &g9gD2X
X X X tp 9a cf d x ouX i X V V Xf cf d x Xf V V df c V X a q X X V p X a cf q X Xf a v x X f k X v x v p d X tt9$gu$gY$g9yhtte9ut $gyeuv 9yt{t$&tp xX i tyhR""oeogt d a k X k y k x X cff a p V X af x p d c V df a x v a v x V X k X a x k kf X V V f X $9ymutuB$gy$yyhya 9RAof i "ttud i hR1 c V &x i X V X k X x af d s V V df c p Vf X V p V Xf z a c X X k cf v d k X v a d tp a 9Rgu$9m$uYHue&g9gttYRgD2eottyuyutV b f c X V V tp tt$d i a uya V x
B1 (o , R) l G |o - o | > R
:ll1
X p dv 7tuyuX i
X p V X p f a cf v a x v x X x Qgt$oey9y$yu9xgu$R h i eRX tp ytRymx 92g"ut2h1hWU ac X cf p c af c X x V d b a X ` X V X i a Vu7rVtpecRX uc y7i e9hW&uW f d a b X a d XaaX c p d l c x p x Xf a a d k d X V p v x v X V X k x v X c v p d $9r9tuuttW&s9tR"uyu7 X af x| X tp gg X c f p faBVqH a pudyadtu7 v p d X v X c d p c p af x v p x V b c p d c c f e X k k X X v x r tp 9$7ttBtgwtud 9$eyRX i ghp' $gt4"RgguP X$rgftp i i VuPb p a d X V V tp tt$d i $uya a V x Xa d V V d )uPb v p d tuPr X p V X p f a cf v a x v x $gt$oeoR$yu9 j X a c X p V V d V X V V df a x v X v z k d V V x 9eRGtutRttuyxyt{utt hp i $troxtp pgy2X i m94ggu$9gxyv 9ytGg X k v d V x X X cff a p c p c c f af x Vf X f k X v x X x 9 h i D2X ac X p cv d t$ywu
|o - o | = y = x - 1 |StabG1()| k
1=gStabG1() (o gStabG1()
c af x Vf c af Rg$RtRX i gtp i i u"$h V d X k k d
o = 1 |StabG1()|
gStabG1()
Xc d V d a $7uhX X q V 7 c V ux i X i tRogtuye4g v p X a x cff z x a c X xwv9Xaudt6$&b V V d v p d r X V p k k d v p X x s v p X uP&$tt{"$ht9At9 xvXa d V V q p V d yR1x 9a $ec c X c p d a X af R&u4tp gtp i i u&92gy9"$eoRX i V d b c Xf v a k d cf c v x v pc X t{yv ud X V V tp "tud i a wa V x c X p d z c X V X k X sf af c V x v a X a p k v X Rt$&Ra R"Ruge&w"{R& b X qX V V d ac uhtu9eRX X k k d u c V a V d c c V d i x v c X X a ux i $$ed$uQwRtp hX cf d x ouX i &uyuxX i ttRg$gu$ X p dv X V V Xf cf d x v X c d p c V X k X x a p X V X vf x d v d a 9$7ttra R"Rgu$R97 W$ouyuhgf i p i o r $gymf i uhR"a 7my9${&uuQwr2gY9Rya 99rRB2uQx fff j f c V df a V d c X V x f v s c V d i x v c X a X c X v V X c X X s
D G1 G1 ^ Dk l1 = G1 R
I
X X X tp 4g9ya X z x v x c Xf V V df c V X a q tuyu94g$gRhX X V p v X s p d v a a p X V d b e X vf x d v d X v x r V d V h tx9uuy4R& $Hf urog$$hYxsH$h$t9 X i gyeuv Rto i $&u hx i ymuyya pS$$Hp6k Xf a v x X f k X v x v d z X af x v V o nu r t s r q o n m l
ri |o - xi | R |xi - xj | max(ri , rj ) i = j G l = A1 (l)
q
gs p 9d W s 9d T s T uwd
Td
` Vd
R T R d s d ptu W Y T V cd W T U W ig ag! W w T a Y d `
^ Dk l = B1 (o, R) - l
iI
W W w R T s x W T d UR U c T 4W T d W u$v XV i` R Uruwd T Papq u W d s u sr T d ia T s V T U W q g ` `
/ o = Spm(k(o)) D
e UR pt p x d
T s g V b fau ss d UR 1 V R x ` W T s W R W T A` e W UR XV id R d V d W R d R $ V Q b W q W T AA W R XV 9 s T av W g W p V b T s W R ` T p u W R ` W T g y9v` XV h W W d s u s R ` p ah W UR XV i` R U{d T ar ` T
G D oD k
G
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
xi A (l) R, ri
J y B1 (o , R) k R |l1 |
R W T g s g G9 iD g Y
1
StabG1()
|o - x | =
o B11 (o , R) l
1 |StabG1()|
l1 /k
gStabG1()
|k |
g.o - x =
B1 (o , R ) - l1
I
gStabG1()
{xi ; i I}
o D(k)
g.o - o maxgStabG1() |g.o - o | R.
- g.x )
o - g.x = o - x -
D
R=1
|l1 | B11 (o , R ) l R
J
B1 (xi , ri ) l
B11 (x , r ) l
g.o .
|o - g.x | R |o - y | = R J o
1=gStabG1()
StabG1()
x s Vf c c v d c p d c X QRX i yu9e$e4
o - g.x = R.
X k cf v d k X c v d x X s p d v a X v V t$xgy$gu$tuy9 V$gfcVRXaqhX a VxAftpecRyguu9y2g"yv i 9yh97}$&Rh9gRut$&Rhuu9gup {gu$d i hp i ue&g9gf qeRX d h s c X vf x x c c X X V X a a p X V d b x X v X f v s v p d r a x k d f c x X Vf x k V V df c p V h a c a 9X i 9"ottyuHA X i 9tt$gyAwq a u7{eRX Huhyeu9PusA V f s X k cf v d k X g X V V df a x v Vf d V p a c b V df a p v a c V d v x r v x c V xf &wa ov c V d a d V df af c d d v x $utH$gym$7uyt"xX i &yv 9yt"gtR$ hd i hua u9y$reRHut Xf a v x X f k X v x vf V X a z Xf x| V x V X a Vf x k a c X
StabG1() o l D l = l1 B1 (o , R) l
B11 ((o ), R) = l
=
^ Dk l = B1 (o, R) - l
gG1 /StabG1()
B11 (g.o , R) = l
iI
B1 (xi , ri ) l
t e
B11 (o , R) l
/ (A1 )an . l1
1
G1
I
l/k k D
/ (A1 )an k l1 /k l/k D(l1 )
c v d wun
A1 (l1 ) l1 /k
c V Xf z x c X p d v Rg97R7t$$xX i Rhtu94X i $thxgt4ttgtp i i $Hugf i $hxk eouy{oxv c X q X k d X a x q X X af p c X V p af V d b V df a V d X f cf d v a x X
R j 1 , . . . , jn J i 1 , . . . , in I
o nu r t s r q o n m l pS$$Hp6k
F
q V b T s g d g e W p gU f a s x d d s ii` R q V R x` V R V b T s g d g e q 3 V R x` V R V b T s g d g e q a V b T s g d g e q 2 T R T R d a V b d W R d R u V Q q d W a p d q` p agfs T p h u V d d s 9 W $Q W T R P 0D FD 1HGEB )@ @ qdtv w T V R $t W u9 U ` d ` d p T ` R x GA g XV ` V d W T iBat V R y V w g d T h v$ XV d s d ` x ` W USQ c V V d V Xf z V d c c V x sf p c c X k k X q p c X g X Vf c a c &p uhRgta uya uQtR9RX i RGQ9 i g9eRX Xf k x| x X c x X c V g"u{tp uR4hux i yf i xv X Vf V p v p d X zf c v X Vf a c hof i du7tew9$s D2X X X f i x tp c i tp og99uu D2X ac p x X cf x| v a p d a X p vf d s R9o9uy$u$X i hueRut Xf x| a c X
V d $
V R T p W R ch R a2f W V 9j V b T s g d g {t ` x ` ` d ` % ` x d d
/
i1 ,...,in I
$g W# g$ g gs g Y R " u s p R W T Y W T Au W p x V s T d X f i X f i T R tp h9ux e W p g$ fT d V R s p ` V R tp og9hux wa d V 4 W q d W a !g gfs yp h XV b T d s T W $Q W ` d T R D FD ~ 08 } ~8 q d` tv w T V R $sa W qh U d p T ` R fGA g XV ` V d d W T ia V R x V w g d T h v XV USQ P pGB 3hH98 9@ x s d ` ` W c X f c c x c V df a d V c X X 92tp yuoh$gyt2tp Rtp i u7 hx i g97tuyv v d z X X x V & X i w$uhuge$p weRut$RX i wRX i u$g$ut$h""w9f i 9$d Gu2eueuxe9hxX i gt{ouv c a x k d f c x c a v X s p d c X Xf d d k d d x x v X p a V b V df a X c c p d c X a a X X af p c x c V x X V V df a x v q9tuy a g$&th&hk i x GV Vf d V p p x a X h X p c X a a $o9yhX X s R$Qx X V V Xf cf d x V df c V X a q X X V p a ttRg$gu$6$gRh"tthX X i utr9uu}4tyeu94976u12h1hX Wqw9tRu 9py| d V x a v X s p d V p X vf p v a c V d a p X V d b a X ` V U x v V V X X c c p x E 8 % aeR{ $gtBug$7uytpoX i $gh$h"u$gym$7D2X X i 9tp i Rytp oey9yuwR{o$ywuHg &t' c X e V df af c d d v x V df c p V d x b X sf af c d a c X X f c v X f a cf v a x v x x X p c v d % " #$#! X i 9tuy a g$&tRX X V V df a x v Vf d V p V X i 9tuAw a g$& X V V df a x v Vf d i c d V p v X s X X v a p X V d t9$99y497 ub v p c X c cf a c treyog4eRX X k k d c X V V df a x v c Vf d c uGRRttuy wa g$&2X i X R c & bf V Vf a c 4D2X u X k k d X i X i s f V d V av X s p d V p ac X X kv d| x k c X $teRut$BteRR$4r9wxh4X i oh2&xt&p X xf c X v z x g V df a v X V V d z j X k v d u&p i yxtt$&"H9yu| x k c r9twy p i tp y99uxyt"oeRX X f v V X v z x a c X c v d x af d s V tp wugu$ v p d t$& X v X d p c a p X V d b e X k k X tp Rec &tet97$u""Rg X v x r X V V df a x v a Vf d V p a X 4sRttuy{$u7't{hk i x X i $trAu Ytp p9ya $"X i gux Rut pS$$Hp6k d V x X X v X v V d k af h c o nu r t s r q o n m l u uwd T 1s T
R = (Ui X)iI ~ k |b - ai | = 1 A1 ~ k B1 (z, R) k ~ k D k D = B1 - k |ai - aj | = 1 i=j 1 1 D = Spf(k [t, t-a1 , . . . , t-ar ]/ /()) ~ k D F (X) Y 1ir F k / C (R) F C D = D k
CF (R)
% #$" 3wp
q V R x` V R V p b s D FD @ V R ` V R q T 4w W T wa a V T tvq` ig V b 9d R $ V d W P HGB @ W d s d ` d R s X v V d k a c X V df af c d d v x g $9ya u9 i D2ug$7uyt"Gt X z k X c V h X cff z x a gp9RX qeogtuyec H$gyye$hsr b V df a p v a c V d v x
G ^ X k l = B1 (o, R) - l {xi ; i I} k
iI
Xv d VX x V yu997
X i m$twp i a x i 979 i eRX qf d V V X Vf a c
c v d n y$gt
X X tp GRa
X vf v c v d x a p X V yhRdwuR& X a cf q X f e9Hg&b v p d tuPr
iI
X p V X f a cf v a x v x X x 2ttp oey9y$yu9gu$9 h i eRX ac
xe = 1 |StabG (e)|
gStabG (e)
X cf p c v d x x V tp otty$gu"
a p da v p d $u7 fcc p x x V &Y6 X f V p V p X a cf q X f Qtp ttYyDohGb a hX
iI ri = rg(i)
X i &9$ugf i $hxRGuX qHf uog$$h'p i &ugf i u94Rxa &gRus z a X x c V df a V d c X b f b e X vf x d v d c V df a V d c X V x f v
X d V d a ec &u9X X q V t X kX V p tzpRec&9X hvec X i V$gfya$xpXtVtpHfcxa Rgfyud d Vf V X a z V X X tp Ra vud7bqen7 p f c Vf X av fxoxeofgftuyec x Xc z xa v p tc X i uhux Wg V df a h v X f v s j X a c X v t V d V h hR49gRu"xeRyutu9u9 a hX X s 2uQx
R, ri |k | o, xi A1 (l) = l
iI
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
G A1 l |gxi - xg(i) | < rg(i) eI
B1 (xi , ri ) l
gG I
g(i) I iI
^ Y k l
i
F (Ui )
|xe - xe | =
gG
k
/
i1 ,i2 I
F (Ui1 X Ui2 )
1 |StabG (e)|
i = g.e
gStabG (e)
CF (R) F (X)
q
R p R R ' ( q` UR d T W p V b T W R W g aY T x R s 6&ig R p d W d T R p Te d d a p auBa` V W p d q` UR d GfT V b XV t ` d q
1
0 P
(P X (Ui1 X X Uin ))
(R X (Vj1 X X Vjn ))
R = (Ui X)iI
/ C (R) F /
j1 J
F P = (Vj X)jJ
CF (R X Vj1 )
R
F
xi = g.xe
R
/ ... / ... G /
(g.xe - xe ) maxgStabG (e) |g.xe - xe | < re .
k
j1 ,...,jn J
F
B1 (xi , ri ) l
Y (A1 )an k
g.xe .
Ui X
i
CF (R X (Vj1 X X Vjn ))
B1 (ai , 1) k D
F (Ui1 X X Uin )
.
D
a1 , . . . , ar B1 (k)
C
C
C R
g
k
F
|xi - xi | < ri
/ ... ,
R
(A1 )an k
k
b B1 (k)
iI
l/k
/ ...
V 9X
v p d X V x cf xf sf v a V df c V X a q X X V p a c t$&a yoguy$g9yhxtteRX X p xv V $"wgf i 7 Vf x a v X V p v p d weRhtt$& vf d s g$uQ hx i ggtey$7ouxhthX HV u29gho WH aff zf c c d x c x a p q h V d b v Xf p f a v x V U v p d t$& X p V X k X p X c V x d p $pa R"Rg9xa uc &tec c X v p c X a d i h cf d| X a p d a x v X q x X v V e X vf x d v d X c V Rt{c 7 qou9$"w9Ao9y6Gf upyguuyu9"x i "$h X k k d a hX X s 2uQx
R, r
iI qc
X tp X X 9a cf d x $gu X i 7tuyuX i ttRgou$Bue&99tp X p dv X V V Xf cf d x V df c V X a q X X V a X X q X V V d a x k d f c x a v X s p d V h9tt$hdhu$humup eRut$tp a 9ul V Xf d v pc ca V xv z ew$uy q p x X cf x| v c X v X a X a 2hyuhyt2gR29 i X i a uy9hy97tp Rtec V d v X a a X k v X f c X zf cf ek i wa 99ytu9RyxX i R7Dd$yygyh2 i $g$ x c VX kXv s p dXv c X i a cf d v a X vf v c V d x c p d X i uyhuQ"oX i uRua u9y$ra 99yB f f v a f v x s x c x p x V x V X a Vf x k V Xf s X v V % " 3w a hX v p d tu7 a p x X cf x V p a c hX R9o| tYeRX bcX kvXa cXva p 92R"Ryt hx i WsRa wtq V U V xv z a 9Ru2DytrD2X V X k X sf a X d v a c p x X cf x| v X X 2hyuhytgtp af gtp i i 9&92ggok i w$R9yv uh2RX i uuu7uy"xX i u9uya we$&ya 99$t&wy$Rtp ttux R"$f e u V X V c X zf c cf x c a V X k X s p d X v c Xf d d d a x c x p x c V x a v d kf c V X k X V x c V x c a V X f h c e X vf x d v d og$$hp i hpf e u"k uRtBp i R$t9ytRG aX X v d a cX s pXv cX g v p c c X zf c cf tRteyk i wa 99ytu9RyRX i g$uu7uy4o"a R"Rua x c V X k X v s p d X v c Xf d d d a x j V X k X sf x X v V df af V vf d s V X a c v X V d v Ao9ye te $ ugt9 i uu ReyR t X i 9gytuytoxgRus qgHRgRyeu9a 99$ot$$txyR6tec a f v d v x X f v bf c Vf n V Xf c a v x V X k X p f d a d k d v v x V p v p
W (A1 )an k F i>0 ra F V (A1 )an k ra Ouvqc (X) l/k / U V Ouvqc ((A1 )an ) k F Hi (U, F ) = 0 ra G
cX p dv 2&uyuX i 29g"$hs2X i c X q X k d c h
i > 0 U
v v x V p Xf d s V yy9tuuRX X V x c V X af x v a X c ff V df a V d x b c V Xf z tp a yf i Rgtp i yye ugf i u9tRRg97x cX d k R ti X i $u9yARoX i hDut"$h X i x 9yud tl pS$$Hp6k Xf v d a x x aX z d V p X k k d v Xv V h f o nu r t s r q o n m l
q p 9dags T dd TyU B W p ` XV ig y`a T x R s R R g1` V R V w g d T h XV wa d Y R d s W d d s i` R W d q ` V ig u g " T V d XV a R u
qc
V R B` V R
c 2X i RuR i gyu$9yA9x i X sf v Xf v d a x x c V j c X v d k d cf f c x p a V d Rt$umu&$$$ec
/ ...
R T R pd V R ia hfg d W T W ig u V tgs 9d R u V d d d eT d W q W T R P FD FD B ~ 08 }8 | 0 { 0 GHGE3hH93E442 z h R W e XV ` V dd W T iBat V R V w g d T h v XV s d ` x ` W W U$Q x v X a d V V h X b c X zf c cf w9ytud qxtp t92gtok i dya 9Rt$hRy2X i u$g$&${oxX i gtpk x c V X k X v s p d X v c Xf d d d a x Xf V p V x X i yahuxkud9gfcup 4waeRXutp$pc2X i tVt$d i $dtzpkRXeRX q c x c v s d V v X c V h xvXa d V d w9ytV $Gb X i $gyr9gyA ttt$d i a uya f f v a f v x s X V p V V %#$"#! c V Xf z x c X p d v hy&9gR&uR7tuyu X V f s V df c p V h j X c v X Vf V p af V v p d| f X a RX i Rueh&gf Y4ew9$s 6tmt$H9RH c X q X k d c X b X k y k X v c X q X k d Rhtu9R7u9pRRhtu9{X i $th4ymt{teRX X a x q X X af p c X V p a c
0 / C (R) F k / X
iI
X p V X k Xf x| af $a 99hugtp i i 9& VX V j X v d k d cff c x a c ttyuuge$p eRX X q X k d X b X k cf v d k d cf V p a V Xf V d b f htuhx2t$ut$Rgya uh&g$X q i hyD2X Vf c a c X c v d g V v X k k X X X tp yy$HRa RX i 9Ryt"Rgtp xX i tyh"Rpoya u9 i 9 pS$$Hp6k d a k X k y k x v x X v V d k X c X k k X X o nu r t s r q o n m l
F (Ui ) /
i<jI
!g s# g Uig gci` R ` W R U W W ` R y g$ he d V R d W R ' q` UR d GfT V b wa u s p R g d R p Te d d " u s p R s d s T R q s d ` Y d a T i W T t s U W R U p d X q` p g4 x T s ` V R W s W d W R a W ` R W p gUu R T 4w g p 9d W XV a s p V gfau d W t $g gfs yp h XV b ss d s T W W T R q d ` ` d T R D FD @ $Q aav w T V R $Wa W u9 T ` R ff g XV ` V d W T aiat V R x V w g d T h v XV U$Q P pGB @ d p x d s d ` ` W % " 3wp a x a p c v X w h v Rygte2ud p a 9X X v V X b c V Xf z x c X p d v c ya 9R997R7tuyuRX i X sf v Xf v d a x x c V b X k cf v d k d cf V p V Xf a z d V d b X p f i x 9$gy9 i $u9yARrux i ut$uta 9$uu$ohh$ ac eRX X p V V $xtud i a uya V x
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
F (Ui1 X X Uin )
R = (Ui X)iI I iI
F () = 0
g ihfT U V ea XV R a d p Rge d b d` Y d
X k
9
U Ouv (X)
F Ouv (X)
U
|k | V F
o , x A1 (l) |o - x | R J l/k F
CF (R)
(R X (Vj1 X X Vjn )) CF (R) CF (P) F
F (Ui X Uj )
0
F
^ W k l =
/ F (U V )
U i U i X Ui
CF (R X Ui )
I
CF (R X V )
C
I X F
B1 (o , R) - l U V / ... Ouvqc (X) F / /
/ F (U ) F (V )
CF (R X (Ui1 X X Uin )) CF (R) CF (R)
i1 <i2 <<in I
J
/U
i<jI
B1 (x , r ) l ra
CF (R X (Ui X Uj ))
/ F (U V )
F (Ui1 X X Uin )
Hi (U, F ) = 0 ra
W
Compl(Ab)
C
F (V )
CF (R)
/0
R X V
Ouvqc (X)
/ ....
o o = x X
` V R q q 2
V
X i
a hX
a 9X c d V x c a v X s p d c X V V Xf cf d x X a V X c v x a V X c cf V a RX i uy9$tuRX i tt9$gu$y$RhyRX i $Reyoth i hX a 9X
zG.z0
X i 2X i $trAy9$tuB9X i RHt c d V x cavX s p d q p cX g X s R$Qx v p d t$& a hX v p d tu7 X X tp P9a af d l gu a u76t hp i m$tw{ Vf d V qf d X V df c V X a q h $g9yhX T Y X i w9X i 979 i $R9yv uh291RouhyuG x v V X a V X k X s p d X v X b cf d| X a p d v p c X a d i X V p p x af x| X V V d tec u&{thupmu"t} {y{ p T d R V W p ` XV ig T ` d Y R a 9X
l/k z0 = g.z0 ^ W2 (, )k l =
I
c v d x x V wur7 X c c x x yuohroX i a wa 9Ryt9yxg R$Qx V x VXcv Xv X X s xvXa d V V d a yRB$hX c V x VX v Xv wa$uya 9cRytRX i Xk Dcttt ac i V p X q V X X v p a x V V df af a v x X V p c v d x af g9ytyAr$gymyeutsy$gudgtp i i V Xf a v x X k y gyeu9k X V c V tpux i a $dec V a hX y$gux cv d fc b X ky k X v e1u9"1 a 9utp "ec V Xf a d X v p X i $gyuttpud i a 9$} V df a x X V p V V Xf a z d V X i udhux Ytfxuh"D2XYhVRguAgtp 2 h i uRBhP V f a h j X z a k d a c X X x sf V df a x X v X a a X X X X tp gRa vte"XhRguAsgtp 2 h i $gyAo9XyYorRXhtu7$R i e$ p c V X x f V df a x v x x v p d c V d v f c V d c a v X s p d c X V X k X af f q X c p X vf v a p X V weRut$Ra R"Rghog9&gtp9R i R& a hX c v d x X a d wu4ytV c d V x cavX s p d c RX i uy9uux2X i t9eoux a 992X i 4a Reyh i 9X X V V Xf cf d X V X c v x V X c cf V a v x c X z x a c V d uRtuyexa $ec V X i a 9X a 9X
G Wi (, ) = W Di (, )
I,J ()
a 9X
2uQu2ew9ya guc X s x V df a X c v X Vf V d X c X tp Ra a hX
B1 (x , d) . l |k |
X i 2X i $trAy9$tu2Hg c d V x cavX s p d cX X i ttRg$gu$gtyud X V V Xf cf d x X af z v h Xv d VX Xa d V V $hRB7 a p da v p d uytu7 X q V X f f X c d p c V tp ho$7tt) "tyd 4 p T u V V W p ` XV ig d Y R
^ W2 k l = J (A1 )an l
I
Xa d V utV
cwvuxxV1 d X$7uhX c d V d a af cf d V geoutwud v p d r uPt X V X x sf hRguQtp R h i V$gfyaAo9Xyv rvux d x x X i a V9fadp$6 X X x X v p d c V d a d Rhu7uyd c a v X s p d c X a V X k X af f q X c p X vf v y9$tuYRY$R9ghothY&gtygyh2 i R& a pX V a hX c v d wux c d V x cavX s p d c 2X i $trAy9$tuRX i ttRgou$y$RhyRX i &a Reyh i 9X X V V Xf cf d x X a V X c v x V X c cf V a v x c X z x a c V d 2gtuyea $ec X i a 9X a hX
G Wi () = W Di ()
I,J () I RW,
X i 2X i $trAy9$tu2Hg c d V x cavX s p d cX a hX X i tt9$gu$Wymtyud q X V V Xf cf d x X af z v h Xa d V V utxG X p af x| $muxp i a 9Atsymx 9tWyv ty9X i V Xf s d v af Vf X f V v x g Xc d V d a $7YudhX X q V trt X f f X c d p c V ` tp 1ho$7ttB "tyd 4 T p V W p ` XV ig ` d Y R h X V x cf xf sf v a X V V Xf cf d V df c V X a q h a uoguyt9eoux ueRhX qX i $ux i R&R i D2Htp a V V X Vf a c X f X i t{utua u2gW& X v z k d V p x V d c c X U X i 2X i c pcc p x X i d V R RRRgR&a uy9c c X a v X s p d c X p d z c X 29$tuRtu72Hg ` c X c X X x V d v X x v X a d V X V w9ytgB X i a ux i 979 i gdD2X V V X Vf a c X i &gf i R RRgtho&W x V v x X b v Xf p f a v x V U x V d Bub a 9X v p d tuPr X sf af c V x v a a c uugwD2X tc v p X i u$x q"tp {tV wa 9X V df a h X X x v V X i g9tt$h hX 2gp9R2gtc af q X V V d x g a c X z k X c V X c X v p X i uhu{txmtp i i V df a x X V p af V d usb c X a v X s p d c X p d z c X a 29uu"Rtu7"RrhX c X k v X| c X p d z c X X a p k v X RR"y9{2gt$&"2g{R& X k k d u Xa d V V ut)
I J = gG
I
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
= (Wi () W )i=1,2 Wi ~ |x - x | d J()
g.J = Jg. B1 (x , r ) l
G
J
(A1 )an l
B1 (x , d) l
^ W1 k l =
B1 (o , R) - l
G
t(W ) 2
I
~ J()
J
B1 (x , d) - l
J -J ()
t(W ) 2 J J() = G.
I
t(W ) 1
B1 (o , R) - l
W l/k
B1 (x , r ) l
B1 (o , R) l W J
B1 (x , r ) l
J ()
J J ()
~ J()
l/k
J ()
x = x
D1 ()
B1 (x , d) l
d = min=J |x - x | R J () = J J()
D2 ()
G I
J()
(A1 )an k
B1 (x , r ) l I t(W ) t(W )|A|
< R J
a hX
= (Wi (, ) W )i=1,2 Wi (, ) |x - x | |x - x | < ~ G J () = J()/ ~ J () J () = ~ () J() J x = x
a 9X
|x - x | =
G Wi (z0 , ) = W Di (z0 , ) gG
B1 (z, ) l
^ W1 (, )k l =
III Rz0 , = (Wi (z0 , ) W )i=1,2
B1 (x , ) l
II RW,,
z0 D(l)
B1 (o , R) - l
~ J ()
I
J()
B1 (o , R) - l
I J -J ()
B1 (x , ) - l
(A1 )an l
B1 (o , R) - l
|k |
B1 (x , r ) l
zG.z0 B1 (o ,R) l
t(W )
I
~ J () ~ J ()
B1 (x , r ) l
J ()
< |z0 - x |
B1 (x , ) l
D1 (, )
~ J ()
D1 (z0 , )
B1 (z, ) l
B1 (x , ) . l
J
D2 (, )
D2 (z0 , )
< |z0 - g.z0 |
(A1 )an k
(A1 )an k
X d ec & v x c V xf v x s Vf V d ya oyAga $ec
J() G
af V v p d| f X gtytus9Rs
cux i tV9a V$dhD2X V X p X ac X p X $9a X f V p V p X a cf q X f tp ttehGgtb a p d a v p d b c v d n $t$&Ry$gt Xtp o9Xya RX i tpReRX q c c X z k X c V h c V da d V b X ky k X v uyt$9H X V xf v x sf p a oyAg$2q u9ogtrttp V df a x f x X V c V x i p Ra u9{eRX X VX V d ac X X tp Ra X f V V p X a cf q X f tp ottp tryDohGb v p d X c V da d tu7rtp p&uu
cC
a 9X V
X X V xva V tp o yy$RX a 9X Xc yad7 hW i g X Xav dc tp uX i Rpu9ogth9dX i yt V X V df a x f x X a a X Xv z x Xa d V V d yt{uAb v p d r X sf a X v p c V X k v d t$P2h2$gyRD tPa R"Rhwu| V$d i Dc2XWtp a f X V xf v x sf a uyAtp R $gyA9ottYt4mt$Ph2 V df a x f x X V p af V v p d| f X X p d z X p f V p X V p c V gt$&r$ottrtx i eh&g4eRX X c p Vf a c X gpud7zp$$w6tp Ym$us uW$X i f s uYD2X X x X af d V d b V d V ac X k k d u X v X c d p c a p X V d b Xf v a i c v x tp Y9$7ttBR& us$gy9k Ysr ahX X i g9tt$hroX i t$hR i gxu992ugeuyy$$ec af q X V V d x X p d f X k k d b c X sf af c V x v a a V d a hX v p c c V df a x c X g tuhu2Ht c X z k X c V X c X v p c X Rt{99Rev $d X p dv X g &uyuG tu7PRy$ux i $&Ry$hdugf i $hRR"X i ec &tec v p d c X a V V d c X v v d c V df a V d c X X k y k X d p X a tp 9X v p d tu7 X p X d p V d b p af z x $ ec &tec $$X i ymtu h i u X k k d f c a hX
^ U k l =
aA
X X X tp 4g9ya cf d x ou X i &t$$xX i ttRg$gu$ue&994ttp X p dv X V V Xf cf d x V df c V X a q X X V af d l c X q X V V d V d gu929tt$ha uc a 9X b tp a &9a rx$7tt pyud~ X V x V X Vf x k X c d p c V X qX V V d uhtu9 b f uX b X p vX d pc V $Rec &te$d i R&7h$4utp hh$D2X a p X V x v x X f i x a c b fHtWuhtu9 c b V V U X qX V V d v X d p c V X k X a p X V h af d s V d b X q X V V d X v d V X a c Rec &tea 99x 9R&Bud Hp gu$$29ttu9yu99eRX X V x p v x kXv V tp a &$wrR9X a X c p c c f a f v d v x v x r 92X i gh9gyt$t{odus a 9ya $hRv VXv V d VX X i Rhtuh2a y$7uh2g2$Y$du9xRR i e$hxX i $d i "tec c X q X V V d c X V x c d k d c X c X a p d a w d c x X v X v f c V d V a p u& t x V d b x v x r 6$t pus& X p cv d $ywu a hX X cv d tp yy$g b X qX V V d $htt$h X s RuQx c V d sf v X q X V V d u$gyh $htt$h v X c d p c a p X V d b c p c c f a f v d v x v x 9$7tt97B$9&eyRX i ghhyuyt"sr X i ghog99ux a 9tuud$9gyt$toX i h af f i h V Xf a z d V d x a f v d v x aX X i ghog9hux qxv af f i h X
G Vi U = MV(U1 , U2 V ) 0 / MV(U1 , U2 V )) U s+1 is+r U V U V / MV(U1 U2 , V ) MV(U, V ) r+s V = i=1 Vi Vi Vi U = s s+r MV(U, V ) = MV(U, i=1 Vi ) MV(, i=s+1 Vi ) V U V U V = U V / MV(U1 , U2 V ) l/k /0.
b X f i x af d $tp h9uWm$ec
a 9Rt$hRyYtu7tp h9ueRX V X k X v s p d X v X v p d X f i x a c
/0.
c X q X k d 29g"$h4X i yut9yme4tmtp i i ue X a x q X X af p c X V p af V d b x X i ym9h9ux rH af f i h X v w d c x p x V X k X a x k kf X u$9 pa 99yAof i "V wec u7u99yhRyxsr k xv X V d b X VXvv pv v x a$at$&2tp hh$$Rg$ec p d v p d c X f i x a V Xf d c X q X k d c X X a p c Rhtu9Rxtp egf Xp$WvpudPrAuya$uxAsgthytuytP&uya u X V f p c a f v d v x c V d v V d X Xc d pc V tp $7tt 2 yud~ c X x a c v p Xf c p V X i d v V RR7y9y9et9X 2 1 X p X v j a V Xf s X v f p X b X f i x a c $6yf i Y$RgR$69Wutp og9huD2X X vXv V d k tp Y9ya uX i guux Rut af h c
U = U 1 U2 . . . Un MV(Ui , Ui+1 Un V ) 0 / F (U V ) / F (U ) F (V ) n=2 / F (U V ) F
cf v d a Xf yuyh X i 9$Q6X i 9g"$hg vX ix X q X k d X xvXa d V V d w9ytY$$b a hX yhuu9gup WweRut$Y9X i Rtud i a ya u$t9ytttp c a x k d f c x c a v X s p d q p c V V V x X s pXv X V X i ur9X i r9gytuytoxg9&h$YX i gt$h2 i r9gytuyt"G& uugec &$toxX i g{hd x29gytuyt2gyx V x k ff a f v d v x X p X l x X p d f a f v d v x g e V df af d d v x ff a X f c a f v d v c X f c c p X f v s h v X v V d k c V d x c p d z a X x a V x f v s c V Xf z x c X p d v gRudgf Hp {Rya up&ugu{$s7hx $u&gRu{9gR&up2&t$$X i R9o9'tp p x X cf x| v V af d m$l pS$$Hp6k o nu r t s r q o n m l
F D i>0 F U V (A1 )an k Hi (D, F ) = 0 ra MV(U, V )
q 9 g t Uig p R e d fT V b a XV R V R ` V R R ` d ` Y h6dtsa T did T p d W p ` YXV ig 1a T x R s R R }` V R V y g d T h XV t s W d d s ii` R p G9 qd V da hefg ypURig vrhEhtrhtyd V W p ` XV ig ctc` V R V b T s g d g da d T r r r r d Y R ` d s d g Y ` V R h iD T R T R F FD FD @ @ 2 0 xdau T h d T 9d dtu W Y T V dd W RT U W ig dags 9d R $ V d W q ` V d V w g d T h v XV ` W USQ P HGHGEB A4@ X af v X v a d V v X V d V X V x V X a Vf x k a p X V uyv myhxyut9h&utRa u9y$rt97B
X a V x sf p c x a c y$uAgtroeRX
c a v X s p d c X af f q X V df a f v c y9$tu2X i ghothu92X i Gg x
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
MV(U, V ) (U U V , V U V )
a hX
F
F () = 0
F (D1 D2 )
^ W2 (z0 , )k l =
F
F (D1 ) F (D2 )
I
^ ^ W1 (z0 , )k l = D1 (z0 , )k l =
B1 (o , R) - l
Wi (z0 , )
Ouvqc ((A1 )an ) k
D1 , D2 Ouvqc ((A1 )an ) k
I
B1 (x , r ) l
zG.z0
B1 (z, ) l
zG.z0 B1 (o ,R) l
MV(U, V ) 1in MV(U2 , V )
a f v d v x 9gyt$toX i h aX
U Vi Vs U V = MV(U, V )
1is
B1 (ua , R) - l
bBa
|ua - xb | R
B1 (xb , rb ) l
b Ba
^ V k l =
cC
|ub - ub | max(rb , rb ) b = b A C B = aA Ba U V B1 (vc , S) l /A G C a C = aA Ca
b
b
RS
E=
cC
U V =
Ec
hH G
R S B1 (ua , R) l aA Ca
A
: H hH C
^ (U V )k l =
b B /B
aA
B1 (ua , R) - l
V
U V
B1 (zh , mh ) l H H eE B1 (zh , mh ) l
hHa
G
B1 (zh , mh ) l
B1 (vc , S) - l
.
D1 D2 =
eEc
B1 (z, ) . l
B1 (xb , rb ) l h B1 (zh , mh ) l B1 (ye , se ) l
B1 (vc , S) l
B1 (ye , se ) l
.
X f q p k 9X i p i wa 99ytu9Ry2X i a uc c VX kXv s p dXv c V d
B1 (zh , ) l .
c V X k X v s p d X v c X v p d X f i x a c Rya R"Rt$h22ht$&tp og99ueRX a hX
^ D2 k l =
aA
a hX
c X c p d a 2g$X i $RR7gf i a VXv ` X c X tp o9a
X p vX d pc a pX V d $Rec &tet97B$tb hX a cXv VX cX Rya Rh2g v X d k j X a af p Rgf i ""xmh a X c V X k q p c Vf d k p x i c d hxya 999Yt9X i uBX R c & X X c d p c V d b a V x V X Vf x k tp x$7ttBu2$u9ya guX i Yyud~ vf a v x X v p V d v p d b e X k k X uyth$ht$&tt $gtp"Rg X X f x V d a X X i V d X c tp ttu$97Da i $h2ep i $R9$RD2X a V X k X x a c X X v x kXv V d b tp tp wr9y$t b f uX q X cv d tp yy$Hg V d V h v x Xf a V x v x a c uh&utR q4&rgy$uyu$peRX X i ym9h9ux b af f i h f l X ia X f q p $&Drk RX i p i eRX ac a 9Rt$hRyv Hh VX kXv s p dX X g X vf d s v p d tp g$utu7 X i ym9ttu9 o eogtp $ af q X V V d x X cff a V d a 9X x V
xb ye V ^ V k l = U V V =V (U V V , V V )
aA
X Xa d V x avX s p h tp HRX i $trAeRut$d q
c V da d $t
X V tp ud i $7ttB29ya u{Rgy{i GghX X c d p c V v X v V d k j V Xf v x h V f a
^ (U V )k l =
aA
a hX
^ U k l =
aA
X vf v V a p X V d ygyh2&ud i R&$b cXva VX 2y$Rh c c V f af c c X v X V x j X a af p RX i udg$d72gRutuw"mt X V Xf s t c V d a X Vf c c f c c p x V d tp a 9u&9$9g$tgRX i eya uc c X X X Vf x v V 2g{tp ptwa 9X X i aggyRD tp$Ruyhute2X i &da $ec f sf a X v p c x g c V d a X Vf c c f c c p x V d a 9X X p Xva V d $y$$k V df a x X v x uRpGg a RX"Rhwudx$b X i $gyux qpgtuyehxX i uD2X V kv | V d V df a h v x X z x a c a X f s V d V a c X p cf p r $yotW t$R9ugwy4yv ud v p c a V X k X sf af c V x v a f c c p x X f 7b tca R"kRugec&xwaxyv ud v p V X X sf af V v X X k k d uG a X V d a X Vf V p f c c p x a c 96uyhutec tGyxD2X b V d a X Vf V p a c H$9g$tgc eRX X cv d tp yy$Hg X V X k X tp a 99x 9 X Vf v V X f txwa 9ftp 2X i s $Xa V9xAtp Rxyv tr4t hp i $X i s b X sf X f V x k X V p d f c x X v X a v x a p X V d b X k y k X v c x X c V v X v V d k u9{gRRRR&$$"R4&9$94hx i 9ya urj V9y$d i ri H1ut$ Xf v V x h V a$VRgfyuuhX i dD2X uX i deRX X a z d V d aX f s ac b f s ac X cv d g tp wuG X i f i Rt99yhRydw9tuuyv x V v x X v p c X V X v v p v v x x v X V V d cf x V a udY9X a hX c V df a X v p c c $gy2DtrRX i x ud Hp ec &tec us2$u9ya guX i yud~ V h X d p V d b a V x V X Vf x k vf a v x X v X X v a f v d v x V Xf a z d V d uRtww9tw29gyt$t"oa 9$B$tb hX a v p d V df a p v a c V d X a a X V x v af V t$&uhyeu9ye9ha w9gWU X i $9s9ymuyy$d i tV uyP$tt9R $&D"k ouy"p i $R9yv uh2tygWeRX c x X v X af x v a j V X k x v c p d V f X X ia X f cf d v a a V X k X s p d X v V p f c c p x f p a c a hX
ua Ha = H a H = Ec V U Ha
cCa
X X v x k X v V d b X q X k d tp tp yur9yB$uhtu9 i $h2extuPt V d X c X v p d r a 9X X i tu9R i X p d X i ym9hh$x q$m$uBud $gt9gd$uogyB$ af f i h X p af d s V b e X k k X X a V x cff a p V U a hX Xva p yu hx i $tp yuRXutp og9hueRX c V d v x k X & X f i x a c X V af x c V X i a X f cf d v a tp $d i muu7 Dk eouyp i a hX
bG.b0
v x cf V t&h i a hX c d V x cavX s p RX i uy9$tud c2gxu X v x X i a 9Rt$hRyg$R i e$Rg9&xa 9yYc V X k X v s p d X v X c V d v f c V d af a X V X k k x c p af d l guG a R"Rhwu| VX kv d $d i hX V a X VX s t tp xa 9f su c X a v X s p d c X p d z c RyeRut$Rtu7RX i thua 9a $hrtV X V p p x V Xf V d X i dv V X p X $Ra af d l X a V x sf p c X f V x x m$uy$uAgtv rk X i X 2 yt d X i uR6greya 9h&uh6c uWRRR i 2grRttuPr c x X v p c X v V X V d X V d b c X f c X v X q v p d cXv HRy c V &x i t9y$$heRX X p VXa V d ac X X tp G9ya X Vf V p c Vf d k p hof i tux XyDoh1W a cf q X f cXv HRy a p d a v p d b f b c X sf a X v p c a V d tut$&uX $92u2De$$ec a hX uuRtV uye$u7y9ye9hdv 3 yud~ w d c x p x X k xv Xc V d b X xa XaaX c V x X aX tp hd f c V X k X p X c a X f ea 99tRe9Hec X X X a tp 4gR V df af d k d ugec &$hR i t hp i $7eof i $d G$s&&utp wr9 X V Xc d c V h p fcc p x c V d v x kX & X V xf v x sf a ouAgtp 2q uARttu"txmtyt$9R V df a x f x X V p af V v p d| f X
cC
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
U V = U V V (U U V , V U V ) V =V S>r
B1 (xb0 , rb0 ) l ^ B1 (xb0 , rb0 ) V k l l D1 D2
H =
s r U =U V
U V D1 U = D 1 V U D2 MV(U D2 , V D2 ) MV(V D1 , V D2 )
Hc
hH
^ D1 k l =
G h Hc
h0 H a h1 , h2 H a |zh0 - zh1 | < < |zh0 - zh1 | |xb - xb | |ye - ye | D1 D2
(B1 (ua , R) - B1 (ua , r) ) , l l
H
Ba
B1 (zh , mh ) l
U
Hc
B1 (xb , ) l
{xb ; b B}
V
H H
H
/E : H 1 Bl (zh .mh ) B1 (vc , S) l
|k |
V = (V D1 )
Ha
Ca
{ye ; e E} {zh ; h H}.
(B1 (ua , r) - B1 (ua , s) ) l l
B1 (zh , mh ) l
(B1 (ua , R) - B1 (ua , min(r, s)) ). l l
H H
^ ^ D2 k l = (U V )k l -
Hc H =H
H
(V D2 )
^ V k l =
B1 (xb , rb ) l
B
B
MV(D1 , D2 )
E
aA
A
: H
(B1 (ua , S) - B1 (ua , s) ) l l
C
s<r
b0 B ^ B1 (xb0 , rb0 ) ((U V )k l) l U V
A
bG.b0
S r MV(U, V )
0 < |zh0 - zh1 | < |zh0 - zh2 | R |zh - zh | h , h Ha
V U W
X s R$Qx
a hX
X acf q X h X c d p c V d a yDohgf Hp {$7tt}uhX
cwaux&"k$dhumfcu&p$cweRut$Rwuxyv i u9 x a v X s p d c X c v d x X f c V d V ahX c p RXc$daX i a 9y91gf i w9c 29gho VXv ` xvX b v Xf p f a v x X X tp H9a c v d x X q V wutY X q V d b X xa XaaX c V x tu7u7wh"hhuv
|k |
X c X tp o9ya
a 9X
(Di U U )i=1,2 (Di V V )i=1,2 F ^ D1 k l =
hG.h0
B1 (zh , ) l
(Di (U V ) (U V ))i=1,2 B1 (ua , R) - l
b V d V h uh&utR qxX i & $gymf i u9"ost$&Da z V df a V d x v x r X i b c v d wun
hG.h0 Ha
bB B1 (ye , se ) l
Ha H =
G
//B
B1 (xb , ) . l
U = U D2
V
G
H =
b
X cv d tp yy$Hg
: H V U E
MV(U, V )
cCa
Hc
eE
V = V D2
A H =H /E H //E
X i ym9h9ux af f i h b X av x tp d X i a R"Rt$h2BtD2X VX kXv s p dXv V p ac h
yd } p T u V V W p ` XV ig y` td V R {9 tyd p T u V V W p ` XV ig ` XV a d Y R d s Y R W d
^ W2 k l = B1 (o, R) - l
iI-I0
h
g Y
T R V b yau ds t R W xa XV R ags d d` Y d h W USQ q W w T R T s x $Q q W T d R U eu T ` R s W R hW q u W R 9
i0 I W1 W2 I0 = G.i0 (A1 )an k i0
` V R
9
W W W T 9 R d tyd T p V W p ` XV ig ` tgs d V R G9tyd a yp V W p ` XV ig ` XV a ` ` d Y R d ` T ` Y R W d
^ W2 k l = B1 (o, R) - l
iI-I0
V b9d R $ V d W qT d WT d s i` R
^ W1 k l =
9
W p V b T x t ` s d
V e(d b s W T W T d UR U xu T ` R 4 s s 9 W R jW q q T Y R V i` R c V e!u R ! q 2 W T T d R R s g hY9G b d s b % q q `
d = minj=i0 |xi0 - xj |
iI
a R W p XV R tgs T R d d ` Y d 9 W $Q T R T d U$g W R 9 V yhd R $ V d W b R T R s d aY g d W T W ig g39 id R T R d d d ` tud W T U W ig ags a T h d T A9yya9d R u V d
l = k G = {1} W1 W2
V gf b ss W W W T s g T s R d R U1 t V R W T d R U x W T d W u$v{ XV ` x u W h s W R ` T W T A! W UR XV W u9 U g hfT U V ! XV R XV p p Rge d b ` Y T R q P GHGB )@ B FD FD @ W $Q d R s g W p V b T a x U V R gfs W V b T d T aG g U g g$f t V y9d R u V d W ` s s V ` ` s x d b X p V X f a cf v a x v x X ugttp yDoRhuw9xx 9 h i $9c g9a R"Rtp ytRueRX $u94ux i &Da p d b d V X k X f v z x a c w d c x X c V X i Xk 9X i hx99yt p i ya 9Rt$hRyx2X i tp A&"4t u92X i tx u2eueuYhhYgy9a f q p a X v Xf k X v c VX kXv s p dXv c X f a x k i c c p V df a f v c X V p v x V df a X c c p d c X a a X X Vf k v X V % " #$#! X v d uk uRta p i $t9yt}Y$s w$d9Ru7DYk 9X i p i $p2 i hy"a up&eyRX i ghpwa 99ytu9Ryp2gBtp "a 99hugRu' X s p X v x X x f X X i a X f q p p d c Vf c V d c c p c c f c V X k X v s p d X v c X X V X k Xf x| X f v s V a 9X
(V V ) k U = (U U ) k (U V ) o W (k) V = (V U ) l/k (V V ).
c V X k X v s p d X v c X v p d X f i x a c ya R"Rt$h2s2gtu7tp h9ueRX
(U V V )
X vXv V d k tp W9ya $"X i yr&u u9Wsu$&DPRg9ytrp i $WR i g9a $ec x v p c f b e X k k k X X v x r X i a v Xf k X v p d c Vf c V d a hX
U = (U U ) (U V ) , V = (V U )
c p d c c f c V X k X v s p d X v c X c p d a X a v af c c p x X f v s V $eyRX i ghya 9Rt$hRyR$tp m&xgRu7 a 9X
hH
v p d u7
c RX i tp V h
X tp o9a c X a hX c a X s p c X x yv9uudRv& X i 7 DRgRtp i a 99ytu9Rygyv i e&uh&2t9Axh9 w9yt X i a v Xf k X v V X k X v s p d X v X X f c V d V v p X x s X a a X xvXa d V V c V x i X i ya V9X i RX&R i VaeRX X V f c X i tRguQY$uRp9ymuyypyD2utg&RRt"y9yt$Pr Byud~ v p X x s x w d c x X v X af x v a j X a c X v c p d V f b v X Vf k v X a v p d uytph$hvt$& Xv V d p d X i &gf i R4gehRyt9Ry{o9$ottwuR& x V v x X v p c X V X v v p v x v X p f x c v d x a p X V X i $vyxtcx i 9a V$hsDc2Hc$yaH2ecX i tp9uwsh9tt$heRR$X i f putPD2 p d a V p V p VX d a X p dv cX V x a X X q X V V d a c X f b s V d V acX cXv Ge2 f l b Xtp $yX i u$R9hyexX R y$7 c p dva c Vf d k a V X k X a f v a c i c c d X rtp $wrkRXvuthu YX i wa 99ytu9Ryv Rw9"RBRX i RtuPr X p v x V d b e c V X k X v s p d X b c v Xf k X v q p c X v p d X k cf v d k d cf V p a V Xf V d b f h2eottyu$eot$Rgya uh&g$X qY i g9c Vf aecRX$ tBX i a RX"kRst$h2v9tVRX i 4Xtp Yo4X i mtp i i e V X v p d X v Xf v X j af V X c v d X X vf d v X c V x s p d X V X f k X v x g tp yy$gptp 4tp i yt4a Atu74tpv 9ytr b c X V x sf p c c aff zf c c d q p c X x V d b X k y k X Rya Age2ggtey$7BRX i Rut$"Rxv b X k k d V X k X v s p d X v v Xf V v X v x X V X k k d V utRa 99ytu9Ry9ty9X i 99u9 c aff zf c d q p Rggtc &t9X i utb x V d c X c p d a 2g$X i a 9y91gf i D2X VXv ` ac a hX
Di (U V ) Di U Di V D1 = (D1 U ) ^ B1 (zh0 , ) - B1 (zh0 , ) V k l l l 1 1 ^ Bl (zh0 , ) - Bl (zh0 , ) V k l = ^ B1 (zh0 , ) - B1 (zh0 , ) U k l l l 1 1 ^ kl = Bl (zh0 , ) - Bl (zh0 , ) U d (D1 V ) , D2 = (D2 U ) h0 H (D2 V ) H U V D1 D2 = (D1 D2 U ) |zh - zh | U (D1 D2 V ). h=h V Di (U V ) Ha
c V x sf p c c V X k X v s p d X v cf d v a c X v p d X f i x a c wa Agedya 9Rt$hRyouydRt$&tp og99ueRX
X vXv V d k tp Rya uxX i yrHt up9sr x v p c f b e X k k X X v x
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
<R
q
Rs g W
X s 2uQx
t e
i0
G
za
R I = (Wi W )i=1,2
R II = (Wi W )i=1,2 ^ U k l = = |xi - xj | ^ W1 k l = zh
aA iI0 iI0
W q T d W T W W T 9 R d d s i` R
|I| 2
B1 (ua , R) - l
|I| 1 (i, j) I 2
h Ha
B1 (xi , d) l
B1 (xi , ) l
B1 (zh , d) l
^ W k l = B1 (o, R) - l
i0 I (A1 )an k
F
^ V k l =
B1 (xi , ri ) . l
U V = (U V U )
|zh - zh | U V
aA
B1 (xi , ri ) l
B1 (xi , ri ) l B1 (za , d) - l F
iI0
I0 = G.i0
iI0 hHa
B1 (xi , ) l B1 (xi , d) l k
B1 (zh , mh ) l
. . W (A1 )an k |k | ,
g W R ia T s V T U W q ` ` V ph b s g p p R Y ud T uvs T V c) V 3s T 94 V h s T b d V ` V R q w T yUig b s W ` W $x W W W T R P Q FD FD GHGEB 3hH93E442 z ~ 08 }8 | 0 { 0 h g ihefT U V e XV R XV p Rg d b ` Y a R W e XV W USQ c X V V d a x v c a v X p d c a v d x X c V 9o9tufydy9$stu2X i hXyhV$hc$9xgx i gyuRa g hf i uR"oyegxa uQtB$gym$7uyt"Gg af v df v V V df a x X v x X v a c pf X V x sf p c V df af c d d v x % " #$#! a Xf s V d 2$VRgu9 a v X p h X p V Xf s t 9$stud q$a Rgut b f X k k X X v x r uf uur9xguP a 9X X i c X p c v d c V x v q p x X f V 2vtsRkyu|ewauda V9ga RX i f hRc a hX cX VX Rh&9yv9t i 2H$R$g$hdRRgf Rt$& x c X g c V df c p| V d c X v X a s v p d a hX xvXa d V V h yRutYud Hp X i cweRust$cRX i $VR9gRXtyAVa $ew9fty9X i 2h7b hX avX p d a X k X v p a x V dc cvX Vv cX a X i a Vg$D of i eRa R"R9R xX i ya Rhpxtp 4tt$d i a wa &x i f d c a c X V X k X a x h X v V X X X V V V x c V X i yAoh2 q a x h af d l a X k X sf a c X gu2$VR9uhRX7e2v a 9X a vud7tt$hst$&$u$Qtd7DX i uR i RX i pc d v p d a V x ix f V X ia q p x f c a hX a 9ul V Xf d a hX c V ux i tp X i wu&gtu9f i h4X i X V x v c p V df c V X k d c v d n y$gt X s 2uQx X i 9X y9$tuo9yuur9wc a c a v X s p d c X k v d| c x k
X U V = X U X U Z = (U ) (V ) U X V
a p dav x $
X i y9xrRtwueutD2X kv X| x k cc p dc V p ac
U U
X
c p dc euYX i Rtp oRt9$RytRxa uc c X f v V c X v z c X V d a 9X X Xav dc tp {ye$eYX i X i &a R"uYrg i "k tp V VX k k xc pc X d V af d gul pS$$Hp6k o nu r t s r q o n m l W q h V pu b s g UihfT V b XV R XV i` R Upud T vaSq p Rge d ` Y d s u s r V b (au ds W T R q p Rge d T R FD FD ~ 08 }8 | 0 { 0 d g UihfT V XV R u V p Rge d b d ` Y 9 W USQ u g t UihfT V b T x T ` G9 T U Y e XV ` $Q P GpGB 4hp9!342 z % #$" 3wp X i 9$tua 99ytu9RyeRX avX s p d VX kXv s p dXv V p ac Xf k x| x X af x| g{oxtp muBp i t$hR i u9yuxX i $hRGg X p d V df a v X c c x V dXc x
^ ^ P(U X Y ) (P(Y ) P(X))-1 (P(U )) (P(Y ) P(X))-1 (P(V )) = P(V X Y ).
c V df c p Vf c X p d $ghRX i t$hR i u9yuv 9yt"Gg V df a v X c c x X f k X v x
q
U
X
o nu r t s r q o n m l pS$$Hp6k
Xi
b
T R W R V R B` V R T d W p gs V hd h T d d s ii` R Aq g T id T p W p ` XV ig ` XV s d p Y R W W $Q T R q W R it T x T ` af9 T U Y e d d ` p d X q` p V R ` V R ii` R q d s V b d W R d R $ V Q W d s d T p d ` Y T R FD FD @ ` ` q 9 W $Q P GpGB )@ T x T ef9 T U Y e XV ag T id yU 4 XV R u V V x sf p 2a Agec X kX X c k9p&Vx i hRtry97BX i Ryht$p2tp Rtp &uyu9y)hog ugf i $h "tw99u 9w9ye$& X VX V x kvX c a f v d v c X X X p d v X v V fff V df a V d x v p c x v V V X x v X a v d Xytgf9R hqfefns v pa v b c V p ud b teYym9u$humup X i c y7i qpu{9c rX i u${2X i gh{ g $gymf i u96Gg v p c af x k d f c x X a d h c V x c c V X V p v x c p c c f fff V df a V d x % " #$#! f a f v d v x x V d ho"yhuyt"utb X k k d "$& c V x i p Ra u9D2tb X VX V d acX X Xva p x ac h V f tp xyeRX HHtp X i $rf qptp $Rgu1 X x k h X a V Xf s t X af x c V d sf v x r tp gy6utg"uP7 2gta 9uuRX R$ c X v p c V Xf d s V h c f p c Vf d wa g$&
X
c vX s p dva V 2X i 9$tuy$d i R&&$u2DeeRX a p X V X sf a X v p c a c V df a x f h g $gyA9ottx W& X i Rtp oRt9$ya u7R c X f v V c Vf d c X a 9ul V Xf d v p c a x a c eptxeRX X p fcc p x xvX d p V $WyByRec &tec G cavX s p y9$tu hd i R$oy9tRuRyt2ga 9uc c X p f v V c X v z c X V Xf d a hX X p Xav dc $rye$erX i V a 9yYg i ""tp m$$ VX k k xc pc X d k V af d l f sf X v X v V d k j X a c X v f b v X Vf k v X a v p d r X vf x a c X sf txtp W9ya ueRytgA29gy9PuPQuygu9seRt g$gyA9otgf Wg fff V df a x f k h X V Xf a z d V d tp a Rgyuutb tp &gt X V c p c X V x sf p c X af kf x j V x c c x V U 2ga uQt4gg""a y$Wt c v d x x V y$gur1 f V df a x f k h v p d o uARtf qtuPr a9X c V da d V $utb tp &t X V c p v p d r f c V t$Pt xux i uh hX b i R$t g X k k d a V X v V d b fff a 999y$"$tb V X k v d| x V d X k k d u& X V Xf s t f s a c X X p f v V X v z x c X tp a 9utuX i D2$oy9tRu{4tp a9X v p c a x a c X h cf p eAotD2gf Hp yt b c v d n 9wu X i a hX b ou1&y9uu hd i 2tp y99ud2t&dRda 9uc f P cf v x I c a v X s p c X f v V c X v z c X V Xf d a hX b c v d wux X p Xav dc $ye$eX i $R"uYd i Ytp gu x i "$h V a V X k k x c p c a x X d k V af d l ff c V X k k d af d c b a V X k X d v f & gu$2$R9tp g9R i Rt G "a u&my9$2g i 2X i V X v V f V x f v s c X d k c a hX b a 9uGho V Xf d l ff f X V X x sf h v d z oB99Atp R i u7 hx i y$$6 pS$$Hp6k Xva V d k V o nu r t s r q o n m l
g1 , . . . , gn V (U ) V U V X X U V X U V W X UW = k (U ) W = X=UW (U ) V ^ X U V X X U = X X U X X P(U ) P(V ) X U, V X X k / U U P(U ) W U
X
q T ` g W R h ' W p Y T Ag` s ` V T G W T s ` W s T R d d d R T R ` b d d W wa d V d W R ` U d V R W fty V d UR ` ags T id V g W hDytu T h d T A9wad V it hfag d W T U W ig V a XV V i` R D b d` T d d d d eT b d V R d q d W W u W g t d W g W 4 ` s V R x` V R Y T V R d q d W W W ig a d W g W H ffT ` D TT R q d W g W ! W T W ig ` s !T Cd T UF XV R a W R d G ` E d ` Y d 9 p V b yau ds h h ( p ` R ta p t uwd T th fT ds d s R d u sr T W u sr h V p b s g hfT U V b a XV R XV uwd T th T p Rge d ` Y R T R d D d s ptu W g Y T V iA W R au W Y T V d d W T U W ig t)q b d d W T R I FD FD ~ 08 }8 | 0 { 0 d p Rge d g UihfT V b XV R a d` Y d p Rge d b hu g t UihfT V T x T ` G9 T U Y e XV ` W USQ P GHGEB 4hp93E442 z aff z x v x cf c x Qggtw&9ge$p 4X i ec u& uRh$tRyh97B$xtC97teRc y7i xye9htRa Rut$ec X a d i h v x X x k X v X v a y a p X p d X p B v X p c a c X X a d X a a X b V X s p d X tp 9t"29yRex29$7tt{a 99ym9ta $Rex2X i $gyRyhuA$tx2$GH$X i uryhuQrt hp i y9uuRrya 9X V Xf z c X v x c c X c d p c V X k X af f kf V d v X c c f f v c a f v x s c d V c X a p d f f v a f v x s X V c a v X s p d c X X v V af v df v V gyuRa g hf i uAo9y"ex9u&eRGtp X i uyyh$9u9hRuy9u$hwte4X i $gyt"oxgf i huH$gy2euD&u4e9h V df a x X v x v p c X c x z a c X f f f v Xf v a k d V X X V X v X s V d v p c V df a d V x X p a V d b V df a X c c p d c X a a X c V x v t a X X xf sf v a V d V x v X d p c V h X V V X X kf x V d V X k v d V X V p v p d a X k d c v d V hguytut"yRec &teud Hp xttRgf i gtw$tut$txtttu7htu9d&$h6tp a 9ul V Xf d
( 5 " @ " 7 5 4 $ % $ " ( " % $ " 89A9686#32!1&#!0)'&#!
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
>1
^ U X Xi
U
^ V X V
W = [X - (U )] U V W
X (U )
X
X
gi P(U ) ~
V
U V
I
k
(V )
X (X, O) DX (.f0 |f1 . . . . , fn ) U V (Xi )iI ^ V X Xi J P(U ) p P(U ) U {gi } V V V U P(X) 1 X U X U X V X
DX (f0 |f1 , . . . , fn ) (U ) (V ) = W X gi UV = (U ) (V ) p k V X U X X V k P(X) V X
k U DX (f0 |f1 , . . . , fn ) (P(Xi ))iI P(V ) V Y
X =V W
(U ) = i {gi }
X
DX (.f0 |f1 . . . . , fn ). (U ) U {~i } P(V ) g U
X
e $ gt
/X
U
(V ) V (U ) (V ) U = [X - (V )]
X
k
U W =
V
P(V )
k
U, V X
X
V
X U
U
X
P(X)
U V =
f0 , . . . , fn (X, O) |k | X ^ U X Y k - {0} V {~i } g ^ Y V X Y i I X U I +J / X
X a x k d f c x a c uy$uhuge$p D2X a f v x 9gyAs x X c v d a v x optp yy$geu9 i X i 99$Ruu9yeX i $gytBo4ut$yhy}$d Hp "tyuu2X i m9"9{GHe $gte $gymt&h i X i X VX vX s V dv pc V df a d V x X s p d v a X v V h X v p c c x c p d c c f X k k X X g V df af V x c&9Y VXc p x a p d a v p d V X v X s V d v p c p x X cf x| v V p a c $Wu7a 9$Ruu9yer2hyuhytYtD2X j X i $gyoyD24&esa 9uy9$uhwtdeRX V df a f v a c X v x f c V X v X s V d v p c a c v p ec p x X cf x| v V h p c V d v c p d c Vf d c X z d V j X a x q p Xf k x v X c f p X a V X v ` 2hyuhyttp G$$f i $tRg$e2&dc tp9yx i BRg{yReH$xy$RR7gf i a 99yv 9B$gymt&h i V X k X V df af V X V p c V d v X cff a p c p d V b cf d| X a p d X a x k d f c x V X k X vf x c c X V V d t$Reogt$tRouhyu&$huu9gup a 99yy2hR6utV v p d c V X c V p v x c p c c f V df af V t$&9}tpX i u$eyRX i gh ugth i Gg x
X OPreShv(Ouvqc (X), C) F Ouv (U )
X
q d ` Sqaf u V y g d T h v$t W T g$ T ` R x GA he d V R 1g1 ` d s g d s
d s d T p d iag T id yU 4 W p ` XV ig ea T x R s R R Bgs ` V R V w g d T h XV t Y R ` d W d
/ F (U )
q d W i` V g Y ' W $ W ig ` V u V w g d T h d s d x` Y R d q d ` Sqaf ` au R W W q d W W ig ` V 1u V w g d T h i` R Byagtt W R x` Y R d d s ` s ` T d V Q q s V R p` V R g T i W T t s d ` Y d
U Ouvqc (X) C
b$Ug$gtyog$$hps& x e X vf x d v d X v x r avX s p d V p V eRut$}tpud i yDohu7 X acf q X t a hX X X tp W9ya X i
U
X
W VX vX s V dv p d T T W T Y p d X q` Rp ge V i` R s a Ruy9u$hwtec wa s b d V b U W 9 Uig ig 9"u9 Uig T ` R x GA g XV d W q s p R R s p R W T R D FD ~ 08 } ~8 di` V U Y ' s ` V d V w g d T h v XV ` W p Rge d hu g t UihfT V b T x T ` Gh T Y e XV ` W $Q P B pGB 4hp98 9@ cavX s p wy9$tu hd i X i f $tgguxtt hp i x2gduhRyRa duxtwDc s V d V Xf k x| X V c Xf V c V df a X c v X Vf v x X z x a X cf p V v a d c p$yotta xwg9aD2X X z k X c V h b X a x k d f c x X k y k f p a c tpReRX qQhuu9gup 49mteRX f c b X k X q X v x f s V d V a c X h l ut9huYX i utD2Hgf 1& X p c X ${Ra c 2X i tzpRe&9X X k X c V h Xa d V d utV ub a 9X X i $gyr9gyA tt9tt$d i a uya f f v a f v x s X V p X V V V x % #$" 3wp x V d $tb X k k d "$7 vX s p dva a pX V Rut$y97ud a hX V $d i 67 x V a 9X X X tp 9ya X i a x k d f c x hu$humup X X tp Ra v X s p d v a a p X V d b c p c c f e X vf x d v d X v x r 9$tuy976$&R2X i gh upyog$$h{s1 a x k d f c x p a v X s p V p X a cf q X f u&"$humu&$eRut$d {yDohsb X k k d baX ` V "$hHR979X U pS$$Hp6k o nu r t s r q o n m l
X
W q 9 V Hau b s g t UihfT V b a XV R XV d T tq p Rge d ` Y u sr V p b s g hfT U V b p Rge d W W W T R FD FD @ 28 7 5 5 0 2 0 a XV R XV h V b T W W UR XV ` Y ` x T p Rge d b ` ` a g ihfT U V T x T Af9 T U Y e XV W USQ P GHGB !9!663441) % " #$#! e V df af c d d v x g$gt$gym$7uyt"xX i a u9y$rk V x V X a Vf x X p d a x a p c v X g tu9R i ygte2pH1 X V Xf a z d V d tp a 9tuu&b c p V h c v d g t9X Hp wuG X i twa 9yrtec X dv VX k k xc p X s 2uQx X Xva V d k fX tp y$$th2 X Vf V p c Vf d k p x v p d hof i rts&g$"uWtu7 X V Xf s t tp Wa 9tuu X Xv j VX sX tp f i Wa 9f Rv X X vf v tp yg$ c V x i "y9sa u7tgth i k v X| Vf d V p af V Xtp oRt9$a u74H f v V Vf d X g V df a x p x s uApX i 2ttueu64ya x i $&Ryyuh t9y&4X i uAroa Atec p x X V V xc p dc p x X V V d cXvv d v pXa p x V df a x p x s x V x sf p X i 9gt$hBue2 i a k d X X V f c X c V d X x & x h V x V tp {u97tq2 i f Hp "a uya RX a wv i RutRX V x VX V X s 2uQx aX d V x h6X i u tD2X V p ac X p vXc d pc V $9$7tt}&ud i R& hX a pX V a V X x d a c V df a c X 9X gu9eRX $gyD2tp g pS$$Hp6k x o nu r t s r q o n m l
X f0 , . . . , fn 1 1in q ^ 1 ^ ^ X k k(q) k X U (X, O) q P(U ) |k | qU q DX k k(q) (.f0 |f1 , . . . , fn ) ^ ^ ^ q P(DX (.f0 |f1 , . . . , fn )) q P(X) U X ^ k(q) k (q) k(q) ^ ^ q U k k(q) ^ X V k V X
q 9 W W T V b s g hfT U V e XV R XV i` R Uud T tr q p Rge d b ` Y d s u s V b hd R $ V d W q V b T W W UR XV ` x W W p Rge d b ` Y T R P 0 GpGB A936443) FD FD @ 28 7 5 5 0 2 0 9Y g t UihfT V a XV R XV p Rge d tu g hfT U V b T x T ` Gh T Y e XV ` W USQ % " #$#! e V df af c d d v x g$gt$gym$7uyt"xX i $R9f i gdwuxguh2 i AwmtRyHt X i 9tt$gyAwtgurud i tp a V X k X a x k kf c v d x X p d a x a p c v X g X V V df a x v X Vf x k V X vXc d pc V a pX V tp 9$7ttud i R&B c v d x wutb a X f V X zf c cf httgyk i da 99ytu9RyteRX x VX kXv s p dXv V p ac fcb X ky k X v d V x V p ac 2$"RRuX i $trAu tGeRX X vXc d pc V a pX V tp 9$7ttW$d i 97xQ t9GeR4$gyD2tp Gg pS$$Hp6k v p c X x d a c X V df a c X x o nu r t s r q o n m l
U
X
a 9X
cf x u6
v p d t$&
W XV R XV i` R U"d T vtSq ` Y d s u sr V e(d b s T R FD FD @ 28 7 5 5 0 2 0 WXV W $Q P GHGEB A9!664431) X a V Xf s t tp $RgHu cy$gux4aefqhXutt v d X c a hX X$$u9ya gupec &te6 p a V x V X Vf x k X d p c V a x a p c v X w h v Rygte2xud p X vX s p dv tp 9uuytX i ud i ut V a pc
W DX (1|g1 , . . . , gn ) = DX (1|g) X
X vXv V d k tp 9ya $"xX i wugux RGut c v d x af h c t
X d V ec &&
v p c X zf c v X s Vf a c txggyRueRX
X p v X d p c f c Vf x a p X V $Rec &teHR&
xX
X v X f v s tp 9&my9$X i ud i Hu V a pc t X Xav dc tp ye$e4X i myh7$R"uYc af a X a V X k k x c p
b c v d n Rwut a p da v p d uyt$& af d l e v d z m$&hu q i u7 hx i ya uB pS$$Hp6k Xv V d k V o nu r t s r q o n m l
T 7
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
DX (1|g) =
q 9 V pu b s g t UihfT V b p Rge d g t pURighefT V b a XV R u V d d d` Y p Rge d b au g t UihfT V T x T ` f9 T U Y e ` 9 % #$" 3wp a 999y$y$gu"& V X k v d| c v d x x V X Xa a tp 9yhX X s 2uQx X X a a x k d f c x a v X s p d V tp Rhuu9gup 9uu6tp af d l gu
g (X, O) W X >1
DX (f0 |f1 , . . . , fn ) X U W DX (f0 |f1 , . . . , fn ) = |k | xW W DX (f0 |f1 , . . . , fn ) =
|k | DX ( |f0 ) DX (.f0 |f1 . . . . , fn ) =
f0
DX (f0 |f1 , . . . , fn )
DX (|g)
i=1
n
DX (1|gi )
X
DX ( |f0 )
X
DX (|g1 . . . . , gn ) =
-1 gi = f 0 fi
DX (.f0 |f1 . . . . , fn ).
X =W U maxi=1,...,n |fi (x)| > |f0 (x)| DX (f0 |f1 , . . . , fn ) U
a p da v p d tutu7
X
X Vi
k
V
X
X iI Vi
U U
X = DX (|g) DX (g|)
i=1
n
X
W V
DX (|gi ).
k
V
X
< maxi=0,...,n |fi (x)|
W
DX (g|)
U = iI U
ac eRX
U
q P(W )
DX (f0 |f1 , . . . , fn ) k(q) q P(X)
q P(X) U XV q P(V )
|fi (q)|
V
U U
X
X
k
U
X
X
X {q} P(V )
V
U W =X U W = q P(U ) {q} P(U )
X
{q} P(V )
Ui
X
X
k C
X
U Ouvqc (X) X
qc
V VX (U )
Colim F (V )
OShvra (Ouvqc (X), C)
X k
X
F
W
F
W
U
X
X
W
VX (U )
W X q P(W ) qU U X V
F
X
W
U X
V Ouvqc (X) VX (U )
F Ouvqc (X) qc U Ouv (U ) Ouvqc (X) ^ |k(q) | ]1, +[ X = V W V {q} P(U ) |f0 (q)| < U =
W R g s T A T $y s i` R H g ihfT U V ea XV R a t W d s q d p Rge d b d` Y d ` 3T 4w W T R T R D FD @ W p y` XV g ` XV Y R W $Q q p Rge d b h{ g UihfT V y T x T ` f9 T U Y e XV ` W U$Q P SB HGEB )@ c X z k X c V X c X v p d af x V d 2gp9RRtu7gguuhxX i yygtRyRtp o9tp 4X i g$e2&{wtub c a x a p c v c X X Vf d c X z x v p x V d v p X a V d| X v X p a v p d tR&u9f i 9tuPr % " #$#! Xva V d k ac uRy$$9 i eRX V df af d d v x g ugec &$tBGH c Vf d x c v p X a V d| ya uD i wtR&u{X i tt$h ta Reyh i 9X tp 4a 99yAof i ryRgRut9y2g"Hg X p d V p V X c cf V a X V X k X a x k kf x v X f v s v p X a X X X zf c v X s Vf X X v p a x V V df a x k v d| c V x v a X V p af V v p d| f X gtyRugg9ytyAur$Duyytgtytu$tp 9Qb x V d 4uQb v pc VX vX s V dv p eGa 9$Ruu9yec p x X cf x| v a c R9o9eRX 22euyux {A fcbav x Xva p h v X X v p a x V V df a x k v d| c V x v a X V p V a g9ytuyueuyytW$d i 9X q p xX 2h euhytX i t$da 9ytyt$&$ cf x| v X k cf v d k V p V X c cf V v p d| f p c X X v p a x V c X RgRtyA2tw B X i $7eof i Xc d c V d uQb a hX v p tec p x X cf x| v V p v p d r R9o94ttuP9 X V f s V df c p V h a 9X i 9{ueh&gf X i ym$v i a uD i x qeRX X af d j Vf d h a c X vXv tp P9ya V d k j X a c X v t c X V x v a c X af kf d c X X v a x v x x af x| X V V df af V x c X cf p vf x a c X a x q X a c $eRyuARRa uyym2ggu92X i tp dyow&tWgdt"ugth i ytp ottPohWeRPut9WD2X X af x| X tp WgdHg
X V Xf s t tp a 9Hut
X p X $9a
X a cf q X f yDohHtb
U
a p d a v p d x e X vf x d v d X v x $t$&4uouyuh4sr
ov(F )
V df c p Vf X V p x V d ug94tt"utb
vtpcpuxRXhyofx|hyvtVtpgftV9 i R$gterXtp fohaecRXHut pS$$Hp6k c a v x o nu r t s r q o n m l ` V g W R 4d V s s
q G g da u d
ovX ovX (F )(U ) = Colim F (V )
V VX (U )
` W R U W
W W ig ` V d V y g d T h v$gs R i x` Y R ` Tg dd
ovX : PreShv(Ouvqc (X), C)
W T R T R ` ' W U XV p R h U d W W T d V s g W T D aau g W R d a W ` s w au T fT s ig tgs R h q d u d d p R d W W T R I D FD ~ 08 }8 | 0 { 0 T R $Q P B HGB 4hp93E442 z s p R R s p R 9 $ g ig a9 Uig T ` R ff g XV $Q q u g t UihfT V T x T ` f9 T U Y e XV x p Rge d b ` V X v X s V d v p c V X k X p f a x k d a p x ac Ra 9$Ruu9yea 99$o$uD2X v pc VX vX s V dv p tda 9$Ruu9yec a R"Rggu| 2hyu V X k X zf x p x X cf x| V h tp Hp u"twRu)$P9ou9$GPuw9tRu 9oyeeRX uARtByehsRa 9$Ruu9yec Vf d c p x v v X s V d b cf d| X a p d X x v V V X X a f v a c a c V df a x f kf X a a X V X v X s V d v p a R"Rggu2hyuh eRX V X k X zf x| p x X cf x| v V p a c v p c a V X v X V d v p c p x X cf x| v V h p af e$Ruy9$s $hwt}2hyuhyt tp G$4gtp i i $ue ugec &$tov V d b V df af d d v x X V y g d T h v$ V wa ` W d R g d ' W T T ` ty` 9d T d W W ig ` V d W p ga T h U wa x` Y R s T d
F (DU (f0 |f1 , . . . , fn )) Ouvqc (X)
qf `
W W ig ` V A W p g T h x` Y R d s V V d w g d T h v$ W ` q g T ia W T t s d ` Y d
F
s ` V W V X v X s V d v p c V X k X zf x T R p p R p d X q` p g3(w T yUig s s W ` W x W 9 V R 4` V R T d a 9$Ruu9yta 99t| W ` ` T R P B D FD pB HGB 4hp98 9@ ~ 08 } ~8 9 T x T f9 T U Y e XV a d V U W q b T ` V d V w g d T h v$ XV ` W W USQ c d V x cavX s p d c kv d RRX i uy9uu2X i yu| X i t{99D$e4 X z k X c V X c p d c X X a d V V d b f f v a f v x s X V p X V V utuuX i $gyhyAq tx9tud i a Vwa X f a x v V X v X f v s j X f x | c p c f x k X z f x | c p X V X v X s V d v p c x V df a d V X V Vf d c X z V X k X x v p x V utp oyt9Rmy9$pxg9udtorggudghRuy9u$hwtexX i uuttt hp i g$e2&a 99x 94wtu v V d k a c X X k k X X $ya u9 i eRx9xHg % " #$#!
X a V Xf s t tp $Rgut X a cf q X f b x e X vf x d v d X v x r eo9$Ug$gt4ouyuh{gus&
a 9X af d c b V X k X d v f & m$e2a 99tp ytg9R$
Y
c v d wux
q
q G W a W ig ` V d V w g d T h v$ XV wa x` Y R ` W d ' ` V H W W ig ` V d V y g d T h v$ XV d x` Y R ` W
x V 7 X i huu9gup eRut$p a x k d f c x a v X s p d V af d gul pS$$Hp6k o nu r t s r q o n m l W W T R R g ` d ` s d s T T ag$ii` R q g t UihfT V H XV R XV p Rge d b ` Y $Q W T R F D FD @ p Rge d b ` ` hxu g hfT U V c T x T Gh T Y e XV W $Q P B pGB )@
Y X U X k X F|Y
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
V VY (U ) W VX (V ) W0 VX (U ) W0
ColimV VY (U ) F (V )
W VX (U ) W0 Y VY (U ) X W
c v d n wu
X s R$Qx
a hX
k
F
Ouvqc (X)
Colim
|k |, >1
ColimV VY (U ) F (V )
X
Ouvaf (X)
X k U Ouvaf (X)
k
ColimV VY (U ) ColimW VX (V ) F (W ).
ColimW VX (U ) F (U ) = F (U ).
F f 0 , f1 , . . . , fn
C OPreShv(Ouvqc (X), C)
Ouvqc (X) U U / ovX (F )(U ) ovX (F )(U ) X
fc baX `X V U &R979Wt
X i aux&"k$dhumfcux&p$av9Xusutp p d V af d l V X v X s V d v p c a c m$2a 9uy9$uhwtD2X X p c V dv V d $$ya u V df a f v a c X v uhyeRyxX i eottyuxmtyt$tp X k cf v d k X af V v p d| f
` V R
ovX (F )
V w g d T h XV ' T V b ` W
VX (U ) VX (U )
x V &
ovX (F )
avX s p d V 9$tutp f l
U
ovX
ovX
/F Ouvqc (X)
ColimV VX (U ) ovX (F )(V ) = ColimW VX (U ) F (W ) = ovX (F )(U ).
ColimV VX (U ) ovX (F )(V ) = ColimV VX (U ) ColimW VX (V ) F (W ).
ovX
inc ovX (F )(U ) = ColimV VX (U ) F (V )
X
W VX (U )
ovX (G) = G
k
iI
VX (Ui )
V VX (U )
: inc ovX
/ VX (U ) / OPreShv(Ouvqc (X), C) W VX (U ) W0 Y X W Ouvqc (X) / F (DU (f0 |f1 , . . . , fn )) / F (U ) F F Ouvqc (Y ) U Ouvqc (X). U Ouv (X) F / id
qc
W VX (V )
Ouvqc (X)
Ouvaf (X) (U, O)
Ouvqc (X)
PreShv(Ouvqc (X), C)
Ouvaf (X)
G : id (inc, ovX )
(Ui )iI ra
U Ouvqc (X)
VX (U )
Ouvqc (X)
ovX inc F
e 2$gt
X&
a hX
j a V X f V h c c p c c f c X $Rga 9X i f 9yRX i m9Rtw B Rte $gtp"Rgxs92ggyRua $ec c X b e X k k X X v x r c X zf c v X s Vf V d a hX
V VX (U )
X p vXva V d k $Ry$$xX i egtb a p c f
/ Eq
iI
c V c X a x q X a V d c c X V x v a c X af kf d c X X p V V &x i Ry$9$$e2a wgd2g"uhR$4tud i a uya V x
V VX (U )
X f V h c c p c c f X ga 9X i f 9yRX i m9xtw B oH2hyu x b p x X cf x|
V VX (U )
V p ac tD2X
X k k d X zf c v X s Vf a c uutew9$geRX
a RX i Rt$xgtp $y$$hX V f s X k cf v d k X X c V d v a V d k a X i huu9gup eRut$Bp a x k d f c x a v X s p d V af d l f V a c gu&qteRX $Rec &teHyx X p vX d pc fcc p a p X V c a x k d f c x V d R&}79w$uhuge$p a uc c a v X s p d c X X a X X a x k d f c x a c weRut$2gptp 9x$u9ge$p eRX X vXc d pc a pX V tp 9$7ttx976 pS$$Hp6k o nu r t s r q o n m l
(Xi )iI I X U X F|Xi F Ouvqc (Xi ) X Xi F
q f W W ig ` V d V w g d T h x` Y R W XV W R g ` d ` s T T agy V 4au d t d s q ` Y R i` R pG W $x a W g ` V d V y g d T h XV t W d ' b s s d p a T id T W p y` XV g ` XV d T s T V 9d R u V d W tu g v W R d tu W ` s w a yfT s ig ags R 99 Uig ig tu9 Uig Y R W u b q d u d d T p R d s p R R s p R T ` R x GA g W W XV W T R P B HGEB 4hp93E442 z 0 D FD ~ 08 }8 | 0 { 0 XV d W i` V gU Y ' d s ` V d V y g d T h e ` ` h T x T G9 T U Y e XV W $Q % " #$#! X af x| tp mup i h{ue a X V df af d d v x ugec &$BoX i y$gu"tu9R i ygte2"HHq2 i {u9 c v d x X p d a x a p c v X g x f X k k d a wv i 9$t9X V x VX V X s R$Qx X p vXc d pc a pX V d $9$7ttR&$b 9pgu9g$uy9ueRtp YuX i u RX i $$ec V X X x d a V x a V df a c X x g d V x c a V d a 9X X X c d p c V d b X af p c x c V x tp p$7tt6u&ugtYouv e X k k X X c V e up9xgux i gweRdRyeRut$duew9RX i $9xRxyux {92$u9ge$p RX i uyR9gyAs af x v a a c X c X a v X s p d c V df c v X k kf c c x X b a v x X v a p h v c X a x k d f c x c f f v c a f v x X i ReottyuRX i $gym$7uhpgtuyeeRp9p i $gh$hYo&2h1h6& c X k cf v d k c V df af c d k d v x X z x a c a c X X k k X V df c p V d x b a X ` X V U c x X v X af x v a uRp9ymuyy4X i "tec a p u7 t X i utr9$tud tx i 9y$$h4eRX d V x avX s p V p c V p VXa V d ac X aX d V x tp 4hpX i $trAu eRX V p ac X vXc d pc V tp x9$7ttud i 97} a pX V u7a Rgu9 v p d V Xf s V d c v d n wut X c X tp o9ya c v d x X a cf q X t y$guyDohHu vt$&ya uR i "Rgu$7tt p d v V d k X k k X X c V d c d p l Xa d V V d ututb V V tud i a wa V x a hX j X i uhyeRy4o V df a f v a c X v x Xa d V V yt X i teyk i a 99ytu9RyYtp f V X zf c cf x VX kXv s p dXv V af d m$l pS$$Hp6k o nu r t s r q o n m l
X X U = DX (f0 |f1 , . . . , fn ) f0 , . . . , fn ^ X X DX (.f0 |f1 , . . . , fn ) Xi fi Ui = U Xi Vi VX (X) ^ X X V C X (Xi )iI X V VX (U ) fi-1 (Vi ) Vi X k /W X k k (X, O) DX (.f0 |f1 , . . . , fn ) Vi = Xi V VXi (Ui ) V = iI Vi f (X ) F X ra VX (U ) f i : Xi f Ouvqc (X) / VX (U ) U /X W X U
X
' T V b u W T R q d p Rge d b d ` d ` U$Q a g i9fT U V t T x T eaGh T Y e a R"RhwuB V X k v d| x V X k k d "$
U
X
x V Yud b
q
a hX
T g dd q g 4w ig t s W R d R i T Y W RT T uw Aq g s D d W R R W 9 g t UihfT V ca XV R pRge d b ` Y W T R D FD ~ 08 }8 | 0 { 0 $Q P B HGEB 3hH93E442 z p d T q` Rp XV v V d k a c X X k k X X $ya u9 i eRx9xHg X a cf q X b ' e V df af d d v x v x yDoh7gf g$gtYugec &$tpsr a 99x 9 V X k X X i huu9gup 9$tuYp a x k d f c x a v X s p d V af d o nu r t s r q o n m l gul pS$$Hp6k
X X tp 19ya
W q
T g q g 4w ig a s W R d R 9 V $au b s d W g t UihfT V b a XV R a d p Rge d d` Y d
V i = Vi V W
% #$" 3wp
V Xf z 9tx V c v d x X d V y$guYec & c 2X i ' u}$gym$7uytupyDohg&b e V df af c d d v x v x X a cf q X f c V d c cf cf d a uy$tw9X
X
W R g s T A T $uw W e fT d V by YXV R T W4w gfs T p h XV T R T R d ` s W W U$Q q g ihfT U V T x T yf9 T U Y e XV p Rge d b ` ` U$Q P f c b a X ` V U X sf a X v p c a c V df a c X eRh1hX WGuuhRD tpD2X $gyD2tp RX uARttux qrtp yt4$RX i f R4D2X V V df a x f h X cf p a V s ac h
c v d y$gux
q gU 4w g t s W R d
T dd ag R U
7&
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
(Vi )iI
iI Vi
V VX (U )
% #$" 3wp
o nu r t s r q o n m l pS$$Hp6k
a p da v p d uytu7
V (Ui )
I = {1, 2} U 1 U2 W (U1 U2 ) = W V = X (W U1 ) (W U2 ) = Vi W (W Ui ) W Vi V1 V 2 = Vi VX (Ui ) V1 V2 = V
W R g s s d T T $y 4ia p p` V H W did ' T V ! b ` W p e g RighfT dU V 4 XV R XV i` R Ud T 7tq b ` Y d s u sr h V c9d R u V d b W T R Q D FD @ h g t UighfT d V e T x T ` f9 T U Y e XV p R e b ` W $Q P )B HGEB )@ a hX ahX X c X tp po9a c a x k d f c x c a v X s p y$u9ge$p xy9$tud X k k d "$ a hX X X tp Ra w d c x p x X $PuRYtV uyYudh&9yt9RyWsr pS$$Hp6k k xv Xc V d X VXvv pv v x o nu r t s r q o n m l
af d l gu
q gU 4w g t s W R d
iI
T dd ag R U
s d s q d p R 4ii` R H g t Uig B pGB )@ D FD @
(Vi )iI
QY
iI
iI Vi
Y U
N
X
Y
X
k Q VX (U )
k
/ VY (Y U )
VX (Ui )
Y
Y = N Q N (Y U ) = W VX (U ) W N =
/ VX
U
iI
Y U
X
Ui
W
W
(Ui )iI
Q
X Y W
V
W Ouvqc (X)
N Y W Y V
X p X $Ra
N U =
% " #$#!
/X f : X ^ U = X X U
k
X Y W Q
U
W XV
Colim Eq
QVXi (U Xi )
Colim F (V Xi )
Colim
iI
F (V Xi )
F (Q) F ra //
/ F (U Xi ) / F (U Xi )
i,jI
F (V Xi Xj )
Colim F (V )
RVXi Xj (U Xi Xj )
C
V VX (U )
Colim F (V Xi Xj )
Colim
/ F (U )
F (U Xi )
F (R)
/ F (U Xi Xj ). / F (U Xi Xj ) //
i,jI
F (U Xi Xj ) . V f V
' T Vb
' T Vb
cRX$eou$X i a uyymuhk etdmt$"u&yD$he9h9&gt x Vf cf d s V x v a d X a c i c V p af V v p d| V df a p v a c V d X a a X b c p X v b X s 2uQx a hX X p X $19ya X i Rtt$gyAwtrud i D2X X V V df a x v X Vf x k V p ac V p acX h vXc d pc a pX V d eRPf Hp 9$7tt4976u1b h2yhyYegt vf v a v j X a af p X X tp s9ya X i huu9gup a x k d f c x |f a X v p c a X |f a X Vf a c X h x v X v Dmy2DthPg2DD2Hgf Hp pw9ya
{q}P(V )
X s 2uQx x V ud f baX `X V U d V ec Rh1hWu$X i $trAx avX s p V 9$tud tp af d gul V d k V X zf c v X s Vf a c $"utew9$geRX
V
a 9X i fs9Xkftv$d"kxtp cVudya Vu6 pyud~ V c X X v d X zf c v X s Vf a c utew9$geRX
{q}P(V )
c V df a V d q p c X ugf i $hYRX i 2g b
X i
a x k d f c x a v X s p d V p v p d r hu$humup 9$tuBtuP
U Flt(p) V VX (U )
Xtp G9ya X X a cf q X f b e X vf x d v d X v x yDohHt uouyuh4sr X a tp 9X X cf p X vf x a c X ff tp ottyoheRWo 6$gyA9otgf Wg f V df a x f k h ff o b X p X $9a X a cf q X t f eo9HuBo c X V X x sf p a V d c c p d c c f RRya 9Ag$2$$e$eyRX i gh X X a X f v V a Vf d X p f V h tp 9tp y9tRud$u7$ottp q c V da d &uu
{q} P(V ) q P(X) U Flt(p) U
X
a RX i Rt$xgtp roX i mp i i 7 V f s X k cf v d k X X j af V X p c V d c d p c b a V X k X d v f & $u$7tt$2$R9tp g9R
af Vf f e X p v X f v s gtp i Pu2gt4$sRmy9$X i $d i e&uH2hyu"t$hRWD2X V a p c t p x X cf x| V p X v d V X a c
/ ara (F )
xud Ra V9X$vRXusu9yeeRX V b V dv pc ac X k k d "$& af d gul a p d a V X c X v z c X v p c c X k cf v d k d cf c $92t&dRte2"ottyuuRX i b c p c c f V df c c p c x v x r X zf c v X s Vf a c R2X i ghu9of i ossAutew9$gWeRX
ovX (ara (F )) ovX (ara (F )) p P(X) ovX (ara (F ))
a V $RX i f R"ottyugxtp 9ya $"xX i wrHut 2 yud~ s X k cf v d k X X v X v V d k xv pc 92&uy9$yBR$t9yt"o{w9of f i &92gp9R hX i tR9o9RX i $9xtV uy&utV c X x a cf d v a V X X s p X v x x v X c s V c X z k X c V q p x X cf x| v c c x p x X k xv c p d v p X d tcc ya7i WH$X i utVrAxu tpeRX h g d V p ac X v X c d p c a p X V d b e V df af c d d v x v x r a V X v X V d v p c a c tp p9$7tt{R& $G e u'$gym$7uyto4uP12$Ruy9$s $hwt{D2X cc p X f | X X V d V gf9deyux}uxR9cox"ptp cuyva udkaha 9$Ruu9ye4a 99tuRhyohyt tD2X X V X v X s V d v p c V X k X zf x| p x X cf x| v V p a c X p V x V X Vf x k c V d d p $a Ra gur$ec &tl V df af d d v x Hugec &${oX i gyeuv Rt"oy$$9 i 9R Xf a v x X f k X v x X v a V d k f X X w B gtud i oX i 9o$R q{ttyu$eoD2X X z p x v p X a x cf x h j X v d k d cf a c b p x X cf x| V p a c H2hyuteRX X k k d u
F
(Vi )iI
j X f V h c e e 4ga 9X i f 9Qgt
Xtw B o&b x i Ry$9a u2a wgRggu92g{"$ x c V c X a x q X V d c c X V x v a c X af kf d c X X k k d x $Pb e u"Rgg"&$9BW V d ' e X k k X X v x b X k y k X v
i,jI
x $PUe u"Rgg"sr V d b e X k k X X v x X zf c v X s Vf a c utew9$geRX e e uQgt
X k cf v d k X X v X v V d k t$"gtp Rya uX i guux R&u$ ttpRe&9rtp 2uQ af h c t f V X z k X c V X V X sx af d l p x X cf x| guH2hyu V p Xv d VX ac tyuh&9peRX X p c V d v V d k a p x X cf x| $puya up9X R9oq
I / Ens ovX (F ) Ouvqc (X) ra
V p ac 4D2X
a 9X i 9 V f s X i ggok i a R"Rt$h2rtp X zf c cf x VX kXv s p dXv V X c V dc d p tp 4u$7ttl pS$$Hp6k o nu r t s r q o n m l
q x` Y R G W $ W ig ` V d V w g d T h XV a W d T R q d T R p ii` R sSq d ` V dE W xa W ig ` V 4 W p g T h V w g d T h v$ XV t ` Y R d s ` W d USQ t p d q` R d T ags W u g uh ET V !9tu W ` s w d b d p d tu TyfT s Rig u V W u V ypUig3T V bAita W 9avu` ig p R d s d ` R d i` V u g W R gfs yp h XV d R s T W d V ed W R d R u V Q b q u V w g d T h e d ags Y 9t ` d` ` V g W R 9a g ihfT U V b t D qd p Rge d d V ed W R d R u V Q b W q d u a g d W R au W y s w tu T fT s ig tgs R 99 Uig ig tu9 Uig d d ` d p R d s p R R s p R T ` R ff g XV x W ` ` T R D FD ~ 08 }8 | 0 { 0 ` V d u V w g d T h v$sa T d W g W q T x T f9 T U Y e XV ` d ` R USQ P B HGEB 4hp93E442 z c V X v X s V d v p c q p x X cf x| c Rya Ruy9u$hwtetR9oRX i a uc V d a hX X cf p c X zf c v X s Vf a V d c c X B c X tp ytdRtgw9u$$e2tw R d W i` V U Y ' d s % " #$#!
ara (F ) C X C X F|Xi k F|Xi Xj F ovX C ra Ouvqc (X)
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
ara (F )
ov(F )(U ) = Colim(Vi )iI QiI VX (Ui ) F (iI Vi ) ov(F )(Ui Uj ) = Colim(Vi ,Vj )VX (Ui )VX (Uj ) F (Vi Vj )
(Ui )iI
{q} P(V )
p P(X)
Colim Q
iI
X U = DX (f0 |f1 , . . . , fn ) V = DX (.f0 |f1 , . . . , fn )
VX (Ui )
ovX (ara (F ))p = Colim ov(ara (F ))(U )
ov(F )(U )
F (i Vi )
F (iI Vi )
q P(U ) U Flt(p) U
F
F ra U Ouvqc (X)
Colim F (V )
Colim ara (F )(V )
V
U Flt(p)
/ Eq
/
(Vi )iI
iI
p {q}
{q} P(V ) ov(F )(Ui ) Colim Q
X
iI
P(U ) P(V ) V
/
VX (Ui )
{q}P(V )
Colim ara (F )(V )
/ ovX (ara (F ))p
Eq
Colim
//
iI
X
F (Vi )
ov(F )(Ui Uj )
Colim ara (F )(V ).
X
V
q P(U ) //
i
k
F (Vi )
i,jI
V
F (Vi Vj ) .
X
//
|k |
i,j
{q} P(V )
F (Vi Vj )
C
k >1
' e ugt
gt e
f e $2gt
C
t&
T d R 49
R R q 2 q W T T d R R $gs T `t W ig hdgse 9dt R yd c w T a Y V cd ` s ` b e V df af d d v x e uugec &$t"oX
C
W q g UihfT V y T x T ` Gh T Y e XV p Rge d b ` W R d R u V Q P F pGB 3hH93E442 z ID FD ~ 08 }8 | 0 { 0 i a yu7gouyuhxyuthous V x a v d kf X vf x d v d X v a p x V p f f d
k
Xf a v x X i $h2eoX i $t9yt"xX i 7whyv "R{oux i Rut$do V d X c x X s p X v x X x a X f k X v x c V X s p d v ff
ColimU Flt(p) F (U ) = Fp . Colim{q}P(V ) F (V )
% " #$#!
e V df af d d v X e uugec &$txxX i rh&9Ag$2 qxgyu$QutR2X i m f X V X x sf p h cff a p c V d s x c p d V b c p c c f
Fq = ColimU Flt(q) F (U )
VX vX s V dv p wa 9$Ruu9yec V X c X v p a x V c X k cf v d k d cf c X x V X k k X X V X k X a x k kf X V x v V X f X V X k X X v p a x Rxo9ytdRt$"$eoR"7u9ga 9RAof i "to yya 9h27Ra R"Rg9ytyAV b VX vX s V dv pc ac Ra 9$Ruu9yt{eRX f c X v X v V d k c V d x c p d eBtp {9ya $"4ugu{&u X X a X p f v V Vf d X tp dRr$oy9tRupa u7 c V da d V a uyt"hX af d l X a x k d f c x a c guA$huu9gup eRX X a X c X z k X c V X c Xf v d a x x a c tp hARRtpRe&9s2X i yuuR9xoWD2X tp WRec &tet97" pS$$Hp6k X vX d pc a pX V o nu r t s r q o n m l
q P(X) Ouvqc (X) C p P(X) fp : Fp p {q} X F F
T q g T ia W T UR d s d ` Y V b T W E W UR V R x` V R d V A V s T h V h ` x T b d T T R s T aga T i YW T UR d s d ` p d q` p gs V b ` V R q ` V d W a W ig ` V u V w g d T h v d x` Y R d ` W R T R R R d p d T q` p WXV 9 T x T ` f9 T U Y e XV W $Q q q 2 q W T T d R R s T tP W ig hs 9a R yd s ` ` ` d W ID FD @ 28 7 5 5 0 2 0 W w T a Y T ` R fGA g XV d W i` V g Y ' ad R u V A W R ju V w g d T h vags 96u V w g d T h aGD P HGEB A9!664431) ` x d s d d d ` d d c p d c c f X vf x d v d X Ru2X i m9yguuyu94g cux i uux$R i Vud hHp b9cw$Ruy9$s $hwtuR9oRX i ya yu7gxyhuytpt4hRX i f R9h4a RX i 9RytxutRrGg V X a V X v X V d v p c q p x X cf x| c X V x a v d kf a f v d v X V p X V s VX aX k X V v X s pXv x X v V d k a c X V df af d d v x g X zf c v X Vf V Xf z a c X x e b a V X v X V d v p c a V X k X zf x| a c uRya u9 i eRBugec &${tutew9$s 9tD2dAq&2$Ruy9$s $hwt$R9tguD2X X k k d u % " #$#!
F
U Flt(p) V VX (U )
j X f V h c x e xmya 9X i f RAUgtYX i hwtu$h$t9 qxX i gyeuX i u9RxX i u9ytX i 7whyv "R{osutew9$geRX X v p d c x b V d V h Xf a v x V d X c x X s p X v x X x a X f k X v x v x r X zf c v X s Vf a c
{q}P(V )
x e Ugt
X k cf v d k X X v X v V d k j V eottyuxtp Rya u"&ud i eRy$tut${"$h XacXv c p d V t Xa p k k d
ColimU Flt(p) F (U )
a AgeyR4Hutew9$geRX V x sf p c v v x X g X zf c v X s Vf a c
{q}P(V )
e $ gt
a RX i Rt$xgtp m$uu9$gt"ottyuuf qha 9X i h27wh usr 3 yud~ V f s X k cf v d k X X af d s V d b e X k cf v d k d c h a X X V v X x a h v x a x a p c v X w h v RyRyg$d { X i u&g$us X x Vf cf d p x R$QrX i 9quh X s x f V d c V x i X i uuf rtp rya uW9R X x k h X Xv V d k f X x ub V d v X V x v j X a af p bf c Vf 9trAwYm$n v p c p Xf x af X a a X X X teRg"ymx 94ye9hxtp G9ya X a cf q X f yDohHb X i 99$Ruu9ye4t"ost X V X v X s V d v p c X zf x| x v x r a hX X X tp HRa X i X V V df a x v X Vf x k 9tuAwtrud i tyDohHb X i $eou$Bu$RX i hqu9 hX "$7 V p X a cf q X f X x Vf cf d s p x a V f V d a X k k d V df a X c X V $gyR4tt hp i X x k h ac uurgf D2X a hX X X tp Ra X i X i utreRut$txe9Gu$ggyRD trpX i utRtY"w9tRH9R d V x a v X s p d V p X a cf q X t af sf a X v p c x X s p X v x x v X Vf k v X a f X c V ux i X i $rf q9X tp &uya u9X X i uueou$62gu$Ra uc X x Vf cf d s p x c X x V d X x k h a X c V dv V d k a af d l gu v x X i urf qD2X X x k h ac X X X tp gRa V df a X c X V p X a cf q X h v X c d p c a p X V $gy2etteh7f Hp 9$7ttt97Yud b u 99uot9y4egtp n v x v X x k X v j X a af bf c Vf X X X tp 9a V df a X c X V p a $gyRt{hX v X s p d v a a p X V d b V X v X s V d v p c V X k X zf x| a c Rut$ypR& $Ra 9$Ruu9yt{a 9Rgt{D2X X k k d uW V df a X c X V uhRt hp i X V V df a x v X Vf x k 9tuAwgu$d i t9$tuyd97Y V p vX s p dva a pX V v x X x k h ac u$rgf qdeRX a hX X p X $19ya X X tp &Ra X i X i $trA9uurtWeruhRtp d V x a v X s p d V p v p c V df a X c X V X q V d e t"usu' {X i ggyRD t4oWtuPr af sf a X v p c x v p d af sf a X V h w h v ' e uymhRDf qud p&gtrX i hwtu&x i uurgx9k X v p d c x c V X x kf X k y x V oa ud a 9X b c v d x cf x Ry$guo6 fcc p x x V d &r6u7b X cf p r tp yt X X tp Ra X acf q X yDohgf bRa 9$Ruu9ye$R9tgux$7tteRX V X v X s V d v p c a V X k X zf x| c d p c a c X k k d uH X i 9tuy4gu$d i D2X u{tp 9$7ttR&6$b X V V df a x v X Vf x k V p ac h X vXc d pc a pX V d vX V x 9tryv j X a af p pyegt X Xa a tp 99X X s 2uQx X a cf q X h v X d p c V a p X V V f V d e e9Hgf Hp Rec &tex$d i R&72a 9X i hqu9' qX i a p z X c V pux i X i $gyR9X i c V df a X c q p X i a 9X X i R$rgR4tp {$7tt61 X i 2X i x c X x kf c X X X c d p c V c pcc p a 9u1 V Xf d l
&
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
X
q P(W0 )
e|V0
d F (V0 ) e|V0 = d
{q}
V0 V0
F
f F (W ) c f|W0 = e|W0 W Flt(r) r
V
a1 , a2 F (V )
V1
/ ara (F )
F
(a1 )|V1 = (a2 )|V1
V
ColimU Flt(p) ColimV VX (U ) F (V ) V1 r {q} e Colim{q}P(V ) F (V ) c ara (F )(V ) V0 V q P(V0 ) Colim F (V ) Colim F (V ) f|W = e|W F Colim {q} P(V ) V V0 V c
ColimU Flt(q) ColimV VX (U ) F (V ) Colim F (V ). F
/ ovX (ara (F ))p
/
X
U Flt(p)
Colim F (U )
/ ovX (ara (F ))p
V
c|V0
q
/ ara (F )p
/G k f: F / ara (G) ara (f ) : ara (F ) / Gp W0 V0 e W
V W
a1 a2 q P(V0 ) (a1 )|V0 = (a2 )|V0 F {q} P(V1 ) a1 a2
e F (V ) c|V0 ara (F ) V r P(W ) c|W W0 W W1 e ara (F )
V1
F
V
X {q} P(V ) / ara (F ) e F (V1 )
C
p P(X) Fp Fq W1 c r
V Xf c a v x v v x X X k v d| V d 9RRyu9g"yuub
V V a V x tt$d i $uya
e
x V X a v X s p d V df c v X k kf X V B$9uuY$gyR"ttp
a X X x a 9wh
c V ux i wo9toyRgu$huu9gup f s X V cf v v x V p af d l X a x k d f c x
a f v x s yhuQ
x v X c d p c a p X V o9$7tt97B
F
o nu r t s r q o n m l pS$$Hp6k
g ` ` d ` q $T Y W d T T x R s R R x` V R W uiag e W R ` f9 T R gs T d V w T Y d` d s q d s d p ii` R pta T id T R ` ` R p R d W p ` YXV ig `jda T x R s R R 'gj` V R W i`g e W R ` Gh T $ R $gs w T Y s T V h W $x a W g ` V y W p V b T R 6 s d ` b ` Y R d W T R T R da V bh9d R u V d W q d W ag ig T ` R ff g XV d W i` V g Y ' x d s ` V d V w g d T h v XV ` W U$Q q !gT Y W d T T x R s R R Yc` V R V w g d T h XV T id V wa s W d d d s q d s d p d ii` R tg T id T W p ` XV ig ` a Y R d W T x R s R R s` V R V w g d T h XV ts T V b h W ` YW Rig ` V 4a d x d d V e9d R u V d W tu g v W R tu W ys w au T fT s ig ags R h b q d u d d ` d p R d p R s R T R I ID FD ~ 08 }8 | 0 { 0 $Q P pHGEB 3hH93E442 z 9 s$ Rg ig tu9 pUig T ` R fGA g WXV d W i` V gU Y ' x d s ` V d V w g d T h v XV ` W V x sf p c a x a p c v X x V Ra uQtAygtRyg"t X i uhyeRyv epwa 9$Ruu9ytr$R9tp otuyutP2$Ruy9$s $h V df a f v a c X x f c V X v X s V d v p c a V X k X f d a d k d b a V X v X V d v p tc p x X cf x| v V h a uRhyohyt tp Hp gf i uX i $gy}yhuQ ttp V f f v a f v x s X V af d gul
a p d a v p d a f v d v X a a X i c c d tuu7yhuyte9hX 2 y$7 v p c V X k X zf x| c X v a V X v X V d v p c a c ta R"RgguGe2}$Ruy9$s $hwteeRX a 9X v p c X sf a x s v X c V d a c e"uAAy9u9eRX v p d c X a V x v a c X af kf d q p x X a p k k d u7sRy$uyym&sRymg$hyp$h
Ho(M)
AE
c X zf x| c X V X x sf c X V X s v X c v c X V x v a c X af kf d c X X RRtdR99Atp R2ga 9uy9Ryt2a wgdRymg$h2gxtp XgguYo{$xg| p i t$hR i f k x| x X p af x X p d ff f V df a x f k h g o $gyA9otgf W X af x| tp gp i gt$h2 i X p d f
R V b d W R ` T d V R W dau T h d T 94 W R cd W T d d q s a T h g W g s
V VY (U )
a V X v X V d v p c V X k X f d a d k d d q $Ruy9$s $hwta R"Rtp uuut a
f d ofV$gfyaAx9otkgf hWg pS$$Hp6k f o nu r t s r q o n m l T R d U W ig t g V i` R D b d Y T V ie XV t b W d
Y Ouv (X)
T R T p d q` p 3 s h V R x` V R fT q G W a W ig ` V 4t x` Y R d d d s tg p 9d W V w g d T h g3 ` s V R x` V R T T R R T R R W d W ag ig q d s aau W gU Y T V i W R d a W Y T V d d W T U W ig u V aG9 2 3 q 2 W T T 4w gc` R Y b d d D q q s T W W T R B ID FD @ T ` R ff g XV d W i` V g Y ' x d s ` V d V y g d T h v$ XV ` W ` ` h T x T ef9 T U Y e XV W USQ P EHGB )@ c p d c c f X k k X Ru2X i m94"Rg X c V af d v Vf V h X X V X v X s V d v p c dx i gtp i ya gBud tp 99$Ruu9ytxX i $gyt"oX i tp otuyuYuew9$xtt hp i uR74wtu$ugt{otuPr V df a d V x X f d a d k d V df c v X s X V Vf d c X z x v p x V d b X af p c x v p d
ovX M AE
qc
% " #$#! V X v X s V d v p c p x X cf x| v V p a Ra Ruy9u$hwte2hyuhytthX c V X v X s V d v p c q p x X cf x| v ya 9$Ruu9yt"tR9o9X i tymgttp Hp muhy$RX i f Rrue&g9gf qguyv i a g$D i YD2X Xf V X af kf X V h af x| X a X X a V s V df c p V h j X af d j Vf d x V p a c X af x| X tp gg b c V c X af kf d a X c X af kf c c p c c f c V df a f v c c X X cff a p V d b c V df a v X c c x c X v a p x c X v p d ux i Rymg$hh2g"2X i 2X i ghd$gygyhyRX i Ru$9$gyeReyu2yRtuPr c V ux i X i ggu9a 99yoha 9Ryt9yHut X af kf d x V X k X vf x X V X c v X v Ra 9$Ruu9ye"R9o"tWeRX V X v X s V d v p c p x X cf x| V p a c b e V df af d d v x v x r V X v X s V d v p c p x X cf x| v V p a c X q p x X cf x| v c e urugec &$toss2Ra 9$Ruu9yt{uR9o9ttseR4uR9o9tWRX i Xgyu$9yA9ux i f v d a x x c V c X af kf d x b c v d n c V X v X s V d v p c q p x X cf x| RX i ggu9o9Ry$g9Rya 9$Ruu9ytx2hyuX i myp i g"k ee"teRX |f a Vf X a c i c V p a c af d c b X k y k X guAu9dv c V &x i X i my2Dytxgga Re2tRGgf Hp g$uxX i g9ueRHu$Ra 9$Ruu9ytR9oteRX |f a X d v X af kf x X V X c v X v h vf d s Xf x| a c X t V X v X s V d v p c p x X cf x| V p a c b e V df af c d d v e $gt$gym$7uyt x v x r p x X cf x| V p a c osH2hyutD2X
cwvun$Rya VRXuyv9XusV$dhwvtpecqpx2Xhyfux|X i gfa2XDytxk Dtp d c c | dv X ac ic V af d gul pS$$Hp6k o nu r t s r q o n m l R s q d aa V b W ig V b tu T fT s ig 9ptu T fT 1u V pu V yUig d p R d p s p p R
R s ` V u g W R g3a p a V b W ig V b au T fT s ig 94a T 3w tu T fT 1u V pu V yUig s p q d d p R d W d p s p p R
ovX : Shvra (Ouvqc (X), C)
W T d V s g W T H ` V u g W R g!99 $ g ig 99 Uig wa R s s p R R s p R d
d s ii` R
E @ 0 3@ 5@ 8 ( (@ 8 3 @ Rt"&A{2$At"R&#
AE
(Fi )iI
OShvra (Ouvqc (X), C)
(Fi )iI ovX Limi Fi OShvra (Ouvqc (X), C) (Fi )iI
X
U Ouv (Y )
OShvra (Ouvqc (X), C) Shvra (Ouvqc (X), C)
k
(Fi )iI
qc
Colim F (V )
U
OShvra (Ouvqc (X), C)
homHo(M) (A, F (U ))
F
F
homHo(M) (A, -) homHo(M) (A, -)
% " #$#!
X
k
F
/ OShvra (Ouvqc (X), C)
Etqc /X F Ouvqc (U )
j
U Etqc /X
e
F
j
F Etqc /X
F
k
Y -V
X
W V V
j
W
Y -V
Etqc /X F
/Y
/Y /Y / F (U ) ara Colimi Fi W VY (V ) Ouvqc (X) Limi Fi M OShvra (Ouvqc (X), C) C Fi Etqc /X F
/ Y.
e
e $ gt X s 2uQx
e $ gt
&
% " #$#! X s p dv u9$tuyt aeRX ugec &$G c V df af d d v x g X a V Xf s t f s X V cf c v v x c X v p d V X a c v X V d v tp p$RgsuHwo9toRyu9xRxt$&6ReyR tdX i a f v d v x X f v 9gyt$tod&my9$s tp $f $gt"ug$7uytX i gtp i i Ra Ruy9u$hwtesa 99$ot$$tsD2X uu9ytX i X f e V df af c d d v x af V V X v X s V d v p c V X k X p f d a d k d a c b X s p X v x Xf a v x X f k X v x v x r c X zf c cf yv "RstRRtgyk i ya 99ytu9Ry2X i g$uu7uru7BReyR ydX i yht$rx&my9$s 29gho x c V X k X v s p d X v c Xf d d d a x v p d V X a c v X V d v a f v d v x X f v b v Xf p f a v x V U V xv z WRa uyt& RuQ'uR9oq X s x q p x X cf x| X i Rhtu9BX i t$"$eomup tp c X q X k d X k cf v d k d cf f c x V af d l m$Ph v pc a VX vX V d te"$Ruy9$s $h
I Et /X /I
qc
v p p x X cf x| tc uR9oq V p ac tYeRX X cf p p x X cf x| tp ott uRhyoq V p ac tYeRX X vX d pc a pX V d b Xv pX h j a p da X k k d tp Rec &te97 $WuytRt q uy"$ a x a p c v X w h v RAwme2xud p7 c X k cf v d k d cf c V 2eottyuuoRX i $d i 1H9yDw9 x V VXacvX V dv ydX i yhuytrox$hRxmy9$s a f v d v x X v d V X X f v b e X k k X X v x r q p x X cf x| f u9psuR9oq X i t$umu&$ tx$hReRX X k cf v d k d cf f c x p V p X v d V X a c
I ra Hi (U, F ) ra F ovX (F ) Hi ((U, ovX (I))) F ra / ovX (I) ovX (I) ovX ara (F ) ra
T q 9 V R x` V R g T ia W T t s d ` Y d W T Y p d q` p f W Y R D q x` e W g ` V d W p g T h t R T R s d V 9d R $ V d W q b ` V d d W T "at V R e u V w g d T h v$ g Uig XV s d ` x ` u s p R W $Q R p R R ` q V R x` V R W a W ig ` V 4 W R d d x` Y R d T x R s 6 g ayd H u V w g d T h v$dtg!i` R G W $ W ig ` d s d s q x` Y R e ` V d W p g T h wa T R s d V 9d R u V d W q b ` V yd W T at V R e u V w g d T h v g$ g XV d s d ` x ` u s p R W $Q W T R ID FD ~ 08 }8 | 0 { 0 q T x T ` G9 T U Y e XV ` USQ P HGEB 4hp93E442 z a V x sf p c a x a p c v X x V Xf a X v V 4$uAgtAwmtRyg"wv i RgyhyB % " #$#! h X i ymtwDc aff z x a x v x c V Xf c a v x d a V X k X f d a d k d c v v x c X V X k X a x q X V d c c v Xf V v o{u4RgRyeu9h"$R9tp otuyutpRyu9"R{a 9R$thpa u"yRgty9X i 2hB$yottGRRgRyeu9"a R"Rtp uuut c X X p cf p b c V Xf c a v x V X k X f d a d k d c v v x c X V 2yR2g"$d i hRtguR99Ag$2Ra 9$Re2t a X c X zf x| c X V X x sf p c X V X s v X c v c V ux i Ry$uyym&Rggu9xR{$pg| p i guh2 i AwmtRyHg c X a V x v a c X af kf d c X X p af x X p d a x a p c v X
F F M Hi (U, F ) ra X k F (V ) / Hi (U, F ) Nis Hi (-, F ) ra M / F (U V ). Etqc /X Ouvqc (X) F iZ
cvv x c 2yRRX i a uQtra wgrymg$h4eRyu9Y9Gb e u{9e $gtpR9x2gxuPG9e29pa R"Rtp uuut4eRX V x sf p c X V x v a X af kf d x a c X v v x X ' e a X e c X k k X c X v x r V Xf c a v x V X k X f d a d k d a c
ovX (F )(V ) / ovX (F )(U V )
v v x X X v X v V d k yRxgxtp 9ya $"xX i guux RHu af h c t
F
X i wu&"$humu&$weRut$RX i c a x k d f c x p c a v X s p d c
k
a 9X
a 9ul V Xf d
p q p W dtPs T ii` R aag T d yU x W p ` XV ig ` d d s q d s d T p d Y R W T R dt T x R s R R }g` V R W uiag e W R ` f9 T R rg w T Y s d` ` ` s ` V V 9d R $ V d Y W q XV d W ag ig T ` R ff g XV d W b x T R P ID FD HHGEB )@ @ p Rge d 9 g UihfT b T x T ` G9 T U e W ` $Q di` V g Y ' s ` V d V w g d T h v XV ` W % " #$#! a x a p c v X w h v c V Xf c a v x a V X k X f d a d k d c X v c V Xf c a v x c v v x c v p 2AwmtRyx$d {w9RR$R9tp otuyutGe2x9e2R2yRRX i ec
X M W /Y F ovX (F ) W /Y Ouvqc (X)
a 9X
c v v x c X Xf d s V Ryu9Rg$u9X X V a pc t tp xud i Hut f s X V cf a V X k X s p d X v X X wo9tox$R9yv uh2rtp &txD2X V c p a c a R"Rt$h2HH V X k X v s p d X v X bf c Vf n b v Xf p f a v x 29gho V U WG a hX X p X $d9ya c V ux i a x k d f c x p a v X s p d V p v X s p d v a a p X V d u&"$humu&$p9$tu)tx9$tuy{97 ub X k k d u c X zf x| c X V X x sf c X V X s v X c v wRtdR99Atp R2ga 9uy9Ryt Ge2xRhutha uRy$uyym&dRggu9Rtp d&eyRX i ghxeogyB1 ux i RgRyeu9a 99$ot$$tHRy &x i c X v c X a x q X V d c c X a V x v a c X af kf d c X X c p c c f X cff a p V c V V Xf c a v x V X k X p f d a d k d c X v c V V9e2Ryy9YWtWu gtxyRdu$9X tp Rmy9$X i $d i &u$Qgt99Atp RttHRy4t$uf Xf c a v x v v x V p v p c e v v x X Xf d s V X v X f v s V a p c t X zf x| X V X x sf X V p c X v X k cf v d k d c V p V X k X a c ta 99x 9eRX b x e V df af c d d v x v x r X zf x| X V X x sf X V p c X v X k cf $e u$gym$7uyt"ostQggu99Atp RxtHe2Yeott v d k d cf V p a c uu{teRX
&
b V X v X s V d v p c V X k X f d a d k d c X v V X v X s V d v p c a c hwa 9$Ruu9yea R"Rtp uuutQHRyWa 9$Ruu9ytsD2X
S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! RQP)% HGD'7ABA9 '7&64210 )% '&$"
VX (U ) VX (V )
V Y W ^ U = Y Y U (W Y, Y Y )
C
F
U
U
V
U W
F
ovX (F )(U V )
F (U V )
M
/U
U Y (W Y, U Y )
/ ovX (F )(U )
/ F (U )
U W
ColimW VY (V ) F (W )
X
ColimW VY (V ) F (W )
F
/U
iZ
W U = Y
/ F (V )
F U Etqc /X
v Xf V v X X cf p RgRX i htp ottub
X p V Xf s $a RgHut cRWvtcut9XDc2X X p a x q a v p X a V d| X b e V df af d d v x v x r V X a c v X V d v 9hug$Atf urugec &$t4WusAH9yDw9 yX i hytuytxogRus 9wun a f v d v x X f v b c v d X i a Vuyvt&hXtkud9tp R$QtR9oq X i Rhtu9X i t$"$eomup ttp x z X q V X s x q p x X cf x| c X q X k d X k cf v d k d cf f c x X V af d gul p x X cf x| H2hyu V p ac D2X X vXc d pc a pX V d bcX kvXa cXva p tp 9$7ttR&uRRy9d2y hx i WU Ra 9$Ruu9yeeRX V VX vX s V dv pc ac v X x k X v a p X V a x k d f c x p a c 99uot9yW97{u2u&"$humu&$WeRX X vX d pc a pX V tp PRec &tet97{ pS$$Hp6k o nu r t s r q o n m l v x &
F X F ra ra I F ara (F )
q p x X cf x| uR9oq q p x X cf x| 1tR9oq
I
F
/I
F Hi (U, F ) = Hi ((U, I)) = Hi (U, F ) ra Nis ra
/ F (V )
V U =
I
ra ra
(U , V )
o nu r t s r q o n m l pS$$Hp6k
X k k d u
HICrpHcIhsr5cjCtf CdiIhfCdbbrGDCd&GhCIICsICd dssqssqgD$istCXeIsIIFsr5dssqssqEcIf EChCIsqIeCdcvEhdHbvf F F Hd k iq p d Hd g i H H p y H b iD d i Fd F bq g b qfDH d ad F H q p bq g d F w H g i i H b d fu g w gFd p f iD i f b a y gfq i H F g Hd gFd p b i gD F gd p tCrIsGeICdIHsqdvsVhHgCdhyIGhDhieCI&tCrSdbCdIhfrIGD dssqs@shqDeIsGSShHGDrIf xv&tSdhi(sr$hHIChdcGeCdca3T`YXVGSQ 5IGECA bq g b i fq fDd y F w fu gF gFq p i f g fD H b WU TR P H FD B
Y1 Y1 X X Y1 y2 X Y2 k
qf h fh
n w ye y y p y m "c kX u @zDu "m x Du5k}j""'D'h y l e f u n l n f l n u f l n l n 3cDd("c y id y f { 3u l u u h l ep n e l u s3s3" U(ukg3izs3u m l ep n w x l n ep n l u u gk"VukgVsq'3uD3" m y l e f u ")x 5k}V3n e f o'U{ y f "ke m n " l f u n n u u"3"u}{ m u e e m p e n m x l le f y f n m e e f f h fe l m f p l n f Dd3kk"k"kdDkS(sD'}u ioz"3iku'X3'i@z"6 { y yukb'gDDIDuke uD"k`@f c{ p n m fe l l y l u p 9 n p 3'SzuD l y l u n x me m p y p l n y e e l Dge "bs(kIk"c3 S"' ukg3s l u Du}f u n f w p }"DlXk}h l y l u p x me n "muke uD"Uzj"( } 8 cs7{ e h 8 l 6 n l f u u n f m n u ""3XIu}{ m n "i f e 'AV'geDd "jf m y l u n x me p h l ep n e l u luke uD"`"s"kk3(ukg3izs3V{ 3Xk"ssu3gu ukss( l n p l u x e y p f h p y l f ks@D"( y {'g3ss3u3kcu gDsDIu u"z"u')urk"3c y f p l x e y p f y n e l x x w e v l wp 3uks~ U5{ 4 m 'gDd' pV"Vk""cke"3vD"u'g ""`( fe m e n y l wp n h l fp uk{ i f f # !2 ' 0 ) ' % # ! 3's$31(&$" d " x d d k i ~ " g A 3 3 egid ~ D$$ s I 3 & gs e " u d d i' i} $36 $& g ` $gs' ~ii $6 PP ` 3 g
$
d
As
s
XAke"n$Dgp"nxDlqgp3'ff"s@'f" {p yl 3'Dk3'zs@"segk5kg3'i kI3'u g'u n l fp n e y f n eee fe e m f p n y p w x f f {kp"kelu'gee( u"keuS k3"'f g'gDo"iXz3kg3'ik3"egku(3~| m x y e u l y p h { { fe e v x m p m fe e m f p n f n le p p d 3 ' &u 3i A g 3 3 ~' sg$" r $gs' 6 d D i u i ~ i c q dq" De $ ~ A &s D g ukzks3u3u e e fp w f m n d s ' 3 PPS6V y u uDi'uDl fe e v x p e f h e n { kg"3um kgsb3 6oXs$k"3d 3s3m m fe np m l n f k3kige 6"i35kp
inc NorAfnd/k f : Y = Spm(B) Afnd/k / X = Spm(A)
e e 'Ax'gDd "
fe ep u u p ksuzg"3s &h l n " p y (g")u3u 6m n e s(iDk& n f l n f u e l n f "e"'6D'D"'i~ p e l n k"gDiI"3 l u x x l u l n e x l u n s3uD"bsuzsu3bm n e x l iDzsiD"u n f 3'`{ m p l l n u f e l n e v y e u l y p n x f m x e w fp f u f x l u 3"k"sDs3e3'"gi&"eiz"3( z""Df (g""3DiSk'g''u"""n 6m u e f l u uk$"" l n " e w fp f u f p n fe l { l u f u f p e n x n e n m k'g''xk"V'gD'$ru5tU3`)3'})m i3g$g"()"3qg"ge
H F Afnd/k (NorAfnd/k, f h) X Ob(NorAfnd/k) / (Afnd/k, f h) fh fh NorAfnd/k NorAfnd/k fh
e m y e u l f y x l u p h n y l f m f m e p u e l f w x n f p we x (gX" k"3'3Ig6s@D"e('"@$~ (I"gg3k'uD$3 kq"'g"m n x f m eg{ he e yl n fl u "'iD"n ( fe ep y l f m y e u l y p kszks@'"z""Df gh m f '"@$~ """"@D u d " d i $3 ' g g d d " ux &u iS " iS g gu~ d d g i 'x igs"u cgbri( xk$3D~`gS gs"` ~PS6t
fh Afnd/k
{ me we l x xe l m f n zk(ugDs$"qDi'is u f m p s' r6 `ickizs &eD"Duk"3{ p e l p l u m f l e n n f n e p e me we l e l y f x w e v m f u l l f s(iDk(gckzkDkDix rek"3(i3"3'D'i p le n m f l m e e f f n f he e { l l n e x l u m p y m u n f n e l n f pe h np u { fe (gk"Dge 6gk'"3'Ae3"3ku5s(A"suzsu3ek"u ui 6r3is}g"D"5k'3k"ik e f y f p m x e e e f p gu 'iIzggD$zi' u y ss"(ikD"D"("gDuu@ksugqzDokgu""eg'S"d5 6dk'5 y f x xe l u f l u e l m fe ve l x p epe n m e f u { le f e w fp f u f p l me f n l fp y pp l n e l u u e v m f u l l f p h n e l x wp e l x y f x w k'g''Xkk3'ixf iDD'ge' ukg"D$3DkD3"e(g33Pu3s"}(Due(g'"z3uksgDu'u'r "}U3i&XjU6"~6
k
Si
{ u y f Dd'i g3(k'gDS(ikDs$ Xk"s"c'"u")kgke sX3"n iqDsDzDit'g3"3k &"3ke &@k"s"c'"u""3' e n me we l x xe l m p l y f x m pp y m e f l fe n h{ {e he v p p l y f x m x f u u n f l (zk(ikDs$ '"'ksutki(kszkD"3u'}""c"'f i& 3"ukx k"s~"'gD$3ks$ me we l x xe l f h le l f n m fe y n { p e e l f u l f u u n n e m ye l f fe np n l n f u p y f "3X"}cug"u 'i u l f n v { le f n e l x wp e l x y f x w m f u l l f fe f p m le 3Du3x"d5"U xk"kD"u'g@gDu'u'us"}(Du'gDs"jk@bd(km fe m e w p | { me we l e l x y f x w e v m f u l l f m fe f p e n m f l e f h fe kuu)"3n 6qo's u~}zkcgDu'u'tuk"3Sis"3'DDit'gDs"rkqo3D'kj'i'gDd( ii3 Qh H q s F s iPecSX W Q R q H Q t R W a H q d Q a w v a Q d Rh a t s H T q Q R Qh W f R Q d W a Y W F T R Q H H F SpPU$bXUSXyxXSeUuX$SP`repSi``gSebcb`XVUSPPIGE
k k
9 9 8 7 42 ! 1 ! 0 # % % # ! A@6&653"&&$)('&$"
NorAfnd/k Afnd/k
Q Yh h `iXuPH B DC B
(fi : Yi
Y
/ X )iI
k
k(A) k(B)
B k(B)
homPreShv(NorAfnd/k) (X, inc (F ))
fh
f: Y
f
X
I
/X Aut(Y /X) /X
k
i fi :
fh
/ homPreShv(NorAfnd/k) (H, inc (F ))
G
Y
iI
k(B)
X
Yi
k
/X
G
/X
Aut(Y /X)op
X
A = (X, O) B = (Y, O) G = Aut(Y /X)op = Gal(k(B)/k(A)) A BG G A B A BG A = BG qB
xX
X
f -1 (x) k Y = Y X X
p p = q BG
B
y1
X
A
Y1
Y1 Yred
Y2
A BG
Y
Aut(Y /X)
/G
fh
G
fh
y2
&
h yl n fl u uD"uD n f l n f u n l l y f m n p| 'S"'c"uDf q3'i~3{ m n m n "3(s m fe "3ki(gp u w x l u x n e x l u n w u p n le f l p | { le l e l yep f l u p x xe l u f l '3@"3b3IDs3s(iDkiD"sx'sD'3rkx")k'rD(e~6u(ksg"Ds)ge gku)3xk"s (xugD"D"u h xl u Diui$'gp3sD("l l3}g( 'u}Vx$'g"kDg( n f u p h 8 { p e l e me x y e u l y u"kuz""Df Si4 { p e l e 'g3gDk me x y e u l uzuq z""Df y p hp y l f k& D3 l n u n l y f m pe h l y l l n f "f}q"lDf 3'IDl6g(u3ge Due"'6 { y p ye P { ' uk"u g('phIs"m l"szg'x Uukzfkss"3u3u n ep w p h e e p w f m n l"}"g')'$"H 'gDo'}D"zSe'"uD"z3s"u u3"kDSk'3uD"l n f u n e f f h { { fe e f u f l u p m n l u p m y f e l p l u {p yl f s@'" e e fp w f m n u 'Arukk'g's33S n f u n n u f "Slu'}u""u}b'h y l m e l f w x p m n v f DuuD"(gD'usu zs"b"s33i f n y l me f m l "s@'f""k'"@$~ "Us3u m e e fp w f m n u y l p n x f m { le f e w fp f u f x l 3ukzks33iDu uDi'uD6k""3D d5r`P k'5k'g''iD"u n l u x l m w e n l n x l u x n e x l n e f n e n m fe p f m p n y l f p y p ye "3scD3u'"u)gV"3cDsujsuzsu3uj"x)xguDjs(iDkj3ki(gs3"'ck"cD5xkqu uk"u k P m e f l m m y y p 8 Q {"h@qgDIIl")u qgh(i9{i uzsul"`"nkif'Dsubs(iDkiD"qgp n e x u xe x l u x n e x l u p e l e 'A g"kDgS i h`g7e7(`XVTR UY f Wdc ba Y W U S P { 'Q "h
X k inc (F ) fh F (X) ()sep (inc (F )sep (X ))Aut(X /X) / inc (F )sep (X ) F (X ) / inc (F )(X ) F (X ) /
X X
k"3'IkI"SuDf 6~dAz'"3&@u""'ii6@'""'"@$~ 3n y e u l f y p n l l y n p| { me f f m f p y l f m f e f g { d l n f p e n n e x l u p l n l "i35qkS'$g"riDzsiD"S(gS3qu'ge w ye l n f @g"d5gp f p 9 f p e l e "r'f rx5g"kDg n e f y e u l y p n l l y "k"'"s k3"'f k3')DDf r { x f p m fe e m f p ve l x u33ux &ISkg3uk'g l n n e x l u "suzsu3`3n e f { fe np m l n f gUk3kige 6"d5qkp f { n pp n f l f I { { { fe e f u f l u p l u le x f fe e m f "iuxkg U'3Dj"H kg'}D"@zsDgDu"'3kg3'i(i4 $31(&$" # !2 ' 0 ) ' % # !
X2 /X
d d x (i " g '$ '$
k
~ 3 g $
F
uc
D si
fh
g
g $ e r sg$ ' ~iu d u d 'u " ki dq" A &s kXe $U g $i& $6 F 1 C B 1 GPPeEDB 35 A c d @ 3 ~ k U n m e l l n l f "s`3gD(qb'f iDD'i4 { p n x l p 9 { sDg"Dk@f c"'i"i
F (/k) = X1 F (X) f : Y
n e m e n fe e l x l p e f f h n y l f m f uzs(Is$k"rkszd'Du z'gD`6 D3("@$~
gAut(Y /X)
)" &" #" '('%$!
F (X) / F (Y ) // F (Y )
{ y e u l y f f y n u` k3"'f "'" f m 3'e
/ Y X Y )gAut(Y /X)
z""Df Ik3'gi~{ y e u l y p n e p |
pp y 4 { { { kgke sI@ y ukks(Du5Dk}(ge"UoDd&"eu}33 y p p l u x y l e f u p l n e l n e v
gAut(Y /X) (id, g) :
gAut(Y /X)
z""Df qi4 { y e u l y
l u x e m s3iz3ke
y e u l y m pp y p f l x me f z""Df (kkge s)ze3Duz3
m fe e l l p "n 6jkikDDrz
l n n e "uzs
f u f m p e n p n '}DqIi3g$g"(x q"h l n n e "buzs n h 8 { x u l f p l e l m n p | # !2 ' 0 ) ' % # ! "i9V}}(3kgs)I"~$31(&$"
Afnd/k
g $ e
d $$ $ '$ u 3 ug Dq '~$ ' gsAcc
d d X i s ' (u 3 t g g sg$ ' U ~iu $6 ug D(i d@ D si s 7P 530 6 4 2 1 D'$g"kg'"zo3b e n fe e f u f l u p e n m x m
k C f: Y Yred F /X fh Afnd/k
e l u u uks33s
{ h l ep n e l u (ukg"ukD3t{
y x l f f u i5)"h n p "u
y }{ y f m l ep n w x l n ep n y p w Sukg"'uDukg`su uksux e pe h y p n m e l u l y l u p l 6 DziPg'y gj"cgD3XDuke uD"xz{ s"m
X
0 # # % % 1 # 1 ! # ! 8 ! ! % # 1 1 ! 1 # # &&Ab$'}0 @(A'(3A'""$ PP"P}$Ac
Aut(R/X) y1 Y1 (Y2 )
(y2 ) = y1 Y1 = (Y2 )
y1
)" &" #" '('%$!
NorAfnd/k X k
As
fh
Y
k
F
Shvf h (Afnd/k)
(id, g) : Y
F (X)
^ F (Y X Y )
/ Y X Y
/ F (Y )Aut(Y /X)
/ Shvf h (NorAfnd/k) .
Y
/
fh
gAut(Y /X)
((id, g) : Y
NorAfnd/k
/ Y X Y ^
g Aut(Y /X)
F (Y )
NorAfnd/k
fh
F (X1
X2 )
/ F (X1 ) F (X2 )
NorAfnd/k Afnd/k
/ F (Y )Aut(Y /X)
X
Colim
F
fh
inc
F
eq (inc (F )sep (X )
/ inc af h inc (F )
X /X
NorAfnd/k
k
// inc (F ) (X X )) sep X
fh
X /X
X
k
F
(fi : Yi inc
X
X
/ X )iI X
inc :
Doe3 z3'o""m n p n e n m x yl n f l "'i("n
p uD f e p| 3'm@ki~}{'kegeslq( l "n "Vki &{ m fe 4 e l f e n e l n kDDd3'Sgd(3 e ue33vx k"3'f`"z""n &p y e u l y n e f "{ f 3'fs{ n l f u u n ""3 "u}u fPh n m 3"r'kpu` u y''f y i "`"""m s "@s n x f { m f m y e u (z""Dlf I"m y i e f uk{ i y f e n "ge 'm" e e fp w f m n ukzks3u3u y u uDi'uDl 3n"'fDxu e{ l n l n n f u u "'}""n d { l"f}5 n u me we (zkl xx l Dike$ e ke"3l'"mu m y u f y ukzfksm3u3Du yi'uD(Dluze3fub'gki$ke &e"6 e e p w f n u l m m f h x e e p l n f { l fp D'g
X f i : Yi Z = Z(c, f ) Y1 = Y 2 = Y (c, fi ) k k c: Y /V Z Z(V )
iI
e f g{
e oD$D"ncy yu y swkm"n 6mI3'fIuz`{ f y l e m f e m y f m e n e e fp w f m n u y l 3ge "U"rukk'g's3uuD"n l3nc'geDd(l"uck"3v3hDluDgeX"ke'geDIugDs$3'""n n e w m w l x xe l n f e f { x e e l n p m l f m l y g6Dokd(3bk}s 6cDf { e 'ki e"(b'gD$kk""XuiI"SuDf Iq3'm D@U~| fe ep u u n l l y m f l n p S U b Wa gW 7 Ua ghV
Y X
fe ep u u n l fp e n m x m k$kk""X""eksg""`}{ l u e l f 3z'd' fe e u u p he e 'gD$uzkp""s i&3k6{ f 6h e U{ m y xp x n {e v "qu g3 X "3q "nIlg"'fI"}uc3uX( e l n f xl ux n epe n n f f "@kkgDt'f U'I&{ e v y e u l f y n w y "c k"3'm ke &p f f Ds"'h y e u l y n z""Df ")u e e fp w f m n u y l n x f ukzks3u3Xu i'uD"x"3m s { "'gDd&l"i(g33(u('uk''"3' l n e n e v h l w e f e n m me we l x xe l n f m p y zkP(ikDs$3'(Sg")u3u p f s3) S U b Wa gW 7 Ua ghV S U b Wa gW 7 Ua g7V
homk (Y, V ) sv0 (X, V ) Zf h (V )(X) = (c, f ) : Y sv0 (X, V ) / sv0 (X, V )
Y X
f m e f { e e uk"`}'gDd "S"h
'@zguD y y f p e f
e f n e n Do"'I""U"
{
l n 3
{ { le pp f l f n m n l p l u f h x l u x " " DgkkDu'"uD"u zSs3`'Ds3 p kh
(fi : Yi homk (X, V ) / X )iI /
i
d @ ~i A `g d 3 g s
fe ep u u p n e p | { e l n 'gD$kk"" "@gS~gd(3x u yuDl"nfuDl "nr e h 8 { xl ux u}V '3u` d Ai u ~ (u &u " i ~ 3q q rg "3 '$ S "~ o ~i A
k Z(V ) Afnd/k X k
e f g yy'&@'""'"@$~ 3n f {p yl f m f { e 'gDdsge z""Df y y e u l n e x l u iDzsiD"Vi4 $31(&$" # !2 ' 0 ) ' % # ! i) gX gu ~ gi s $ P 530 4 2 1
V X i f i : Yi V /X Z(V )
{ {
{ xe f n e qguDbuzs e n f 3s3e{ s3`uD"l (g")u3u 6m szsiD"k@"@D"gkp k"f iDzsiD"kx( l u x x l u p y n e x l u p l n l e p n m y m n e x l u p e f guD me f k'"@$~ ")" n e n l n n e x l u "xsuzsu3kp f { me we l x x l "rzkuA ""n e f g m x e e p l l y m l fp e w p | { p y l f Dokd($ke cD''f b)D'gg S~s@D"
iI
)" &" #" '('%$!
F (X ) / F (Y )
{ x f p m ee fe e m f p m p n f x p u3"x Igkg3u kqXk"um zu A e e fp w f m n u y ukzks3u3' uul
F (X ) = inc (F )(X ) Y /X Y /
n l u y l n f l l f l l n u { e e "su uD"uDD"u"@sDS`}gd($k
9 9 8 7 42 ! 1 ! 0 # % % # ! A@6&653"&&$)('&$"
inc (F )(Yi )
k
X
Z(homk (X, V )) Zf h (V )
kp
y e u l y z""Df qgp
fh
V
Zf h (V )(X) =
o
(X, Zf h (V )) ^ Z X k V
Zf h (V )(X) =
Z(V )
Y X
"mrkei &pm(ge"Dlfmgp")u3u p f y S U b Wa gW 7 Ua ghV
Colim Z(homk (Y, V ))Aut(Y /X) .
fe ep u u p n y x e ve l x 'gD$kk"" I"' uk3gDu h y n n f m y f h f ""iqy 'i"x
Y /X
O homk (Y, V )/Aut(Y /X)
/X
homk (Y, V )/Aut(Y /X)
i {1, 2} Z Colim
Z(c1 , f1 ) = Z(c2 , f2 ) / Y1 / Y2 c1 c2 d1 : Y d2 : Y fi fi d i ColimX /X homk (X , V )/Aut(X /X) /Z Z = Z(c, f1 ) = Z(c, f2 ) ei : Y Aut(Y /X) homX (Y, Z) Aut(Y, X) homk (Y, V ) Aut(Y /X) /X c : Y / X k V ^ Colim ^ Z X k V Z/X /V f : Y V Aut(Y /X) Z(c, f ) = Z(c, f ) homk (Y, V )/Aut(Y, X) Z Z Y A AA AA AA A homk (Y, V ) Aut(Y /X)
c
homk (Y, V ) Aut(Y /X)
Z
X. Z / Z(homk (Y, V ))Aut(Y /X) ,
f
^ X k V
/X
=Z
f O
/V
c i : Yi f
Y X
/ sv0 (X, V ).
Colim
/X
homk (Y, V ) Aut(Y /X)
Aut(Y /X)
X ^ Z(c, f ) X k V
homk (Yi , V )
.
homk (Y, V )
k
(c, f ) Z(-, -) :
X
fh
x ep w x p f (gg'k)'"h
l n f 6 "V{
s
l n ep x u n f l w m n e "3ukg}"DI`s(iDk&
"n
e n y f x l l p f l x me f oD$cy uDDsuIke3Duz3 { d m p e n m x l le f y f m w y "qk"ki3DDgqu uX""n 6I ke &p n x f m x e l n e v ")""'I'Du5gd(3"3 k"3'kh y e u l f y p l n f 6 "'P{ m p e n m x l l k"ki3DDge f y f m p y (sD'}u iq()k"cu &p f s3jU{ e k l s3u m p e n m x l le f u y f m p y 4 { e e fp w f m n u y l n `g"ki3(Dg)DsD'}'i "`k"cu $Uukzks3u3' uu"@( x l y u y pp l n Dix slsu gkuD"s3 ( { e (umzuxj yk"3f tkp e u y f f " `i n v ""V3u { m p e n m x l l "rk"ki3DDge (sD'}u u@kp3@u u'gDo3sgs f y f l n y e e l e w u n f l w p h { { y p p l 6 {p y l f }"D`k@ " " y gkjs@D" m f 'A)("@$~ m f p n "n3'izu"V yke3"'f y "n ul v e f 3"uo'{ n p e e fp w f m n u y 3'Iukk'g's3bu uDi'uDl "n e f {p y l f gs@D" l w e m f (A Dg('"@$~ IoDd&""3` k"3'y "n m e l n e v y e u l f e f g
)" &" #" '('%$!
{ e e 'gDd( "S
F ()(a) .
fe ep u u p n e n m p n f x m p n x l k$kk"" q"o`Ig"'i($Dg"DIi4
F (t) [k(Y ) : k(X)]isp
homX (R,Y )
y xp x n m p l l u y f m f m e w p | ' ukub"uS(guDs'iXq3'o's ui~{ s"m x ep w p l l gks'x uDf y X k6oDdsgq m n pe h e e f n x f m "@"""ID
X F (P ) F (X) homX (P, Y ) = / F (R)
F (t)
{u u'gDd(Dl u czs p Suk"uku Su(gp y e u m me xp n p m l i y le f e f l u f f { y e u l y u uDg'z$3S3Ds" z""Df "n v f ")i n y e m 'u su uD$zx y ge"D{ ye y l e e fp w f m n A)ukzks3u3u y l y e n m u uDi'uDc ukuV"n e f o'{ "`g")u3s"u}u3ke &Df cD3s~| $31(&$" m e f n m m u x m p l l y n n p # !2 ' 0 ) ' % # ! d u 3 u ' 3 g)gu3D u " 3 e " $I & sg$ ' ~iu os" X 3' k$ gu I
Trf (a) [k(Y ) : k(X)]isp Trf : F (Y ) / F (X)
homX (R,Y )
u D i g ' " 3 kiq " d 'D bj sg$ ' q $ ig d s $k$ ~iu s" s dS3 s u " ' r 3
[L : K]sep Afnd/k [L : K] = [L : K]sep .[L : K]isp f: Y /X g : R
~$ g rg} D $$ si
fh
3 "S S u " ( d u ~ '$ b uD s pP U530 4 2 1
k homX (R, Y ) k(Y )/k(X) F (R) [L : K]isp a F (Y ) F
d
{ f m { p ye y p l u x e y p f fe n f p m e np 3"ks@k$@k"D3u3kgDx'g3izIU3gi i p ye y p l u x fe n f p m x l w m ks@gs g3ss3c'g3iz'uSkp f f "c'Ah u l f m e v fe n x f m 'iSSg33c'g3i"""""D { m p y y 3s3XDzi f i n l y e "cDf UuU{ { x l y x m x e e 'uDf (XggDdsge 4 n e n m x m "g"(ti{ n "
e2 = e 1 f2 = f1 f1 f2 homk (Y, V )/Aut(Y /X)
)" &" #" '('%$!
0 # # % % 1 # 1 ! # ! 8 ! ! % # 1 1 ! 1 # # &&Ab$'}0 @(A'(3A'""$ PP"P}$Ac
L/K
Trf (a)
X
F ()(a) F (R)Aut(R/X)
X TrY /X (a)
g
d
{ ( "h
k
Aut(R/X)/StabAut(R/X) (R0 ) ^ Y = Y X X
F
f : Y
fh
F (g0 )F (s)Trf (a) = F (t0 )F (g)Trf (a) = [k(Y ) : k(X)]isp F (t0 )
X R = R X X
/X
m : homX (R, Y ) iI R StabAut(R/X) (R0 ) k homX (R, Y ) = Trg (a) f / F (P ) F ()(a) Afnd/k
g0
(-) t : homX (R, Y )
Trh (a) f
R0
X
R0 R
= [k(Y ) : k(X)]isp
f i : Yi
t0
/ et
a F (Y )
(R0 )
/ X.
iI
/R
g
s : X Aut(R/X) g0 : R0
homX (R0 , Yi ) [k(Y ) : k(X)]isp g /X a / homX (P, Y ) /X h: P homX (P, R) =
homX (R,Y )
fe ep u u p f l x me f 'gD$kk"" e3Duz3
F (t0 )(a). R {Yi , i I}
homX (R,Y )
{ '@h
X /X
homX (P,Y )
/X
F (X)
(Trg (a))|P = (Trh (a))|P f f
F ()(a) g h g : R t: P Yi /X /X /R
e f g
p w x @ { fe l u p n e f f h { { Ig'uSeQ "hkDD"i &I3'g'b'" " " g""I y y p n m e n
)" &" #" '('%$!
mi = [k(Y ) : k(X)]sep .li . [k(Yi ) : k(X )]sep .l
{ xl yx m 'Df uI y ukIi4 y p
v e n m x m xuSg"(`3{
n e f f 3'Sgbh
l n y e e l e w 3' ukg3seo's
n y f "Xu 'iXm n x f m "q"""SD
p e m l m sgsX
|H| . |StabAut(R/X) (R0 )|
f 3'm `{ fe e f p e 'gD$zi' k s3u l l n e f "uk'u u e3' n m p e n m x l le f u y f m p y p ve m )k"gDd3"DkIDsD'}'ir"bk"c3 roD'ze e f n 4 uk"riE{ "IDu ukuq(gq3u gDVkg'}'iX3u g"DX( m y xp x p l e f l m fe e f u y f l p u l e "uk"`P{ n v ""j n p "qk (k")u3u &p m p y f f 3Dsi{ n p "uD k3"'f y e u l y { n p "Sk m p y p (Ig")u3u ( 9 @f
(i)
y le p ' uDgku@`}{ l f uDs" f i3ge"mDluc yu y)g'f n p p m y xp x m l v p n e n m x m { e e l u ukuD"3S(gc"qg"(ju6kg3s l "n m ki &3g3`(szAk" fe p h np u { e l e n x y e u l y n w z""Df bkp ir33st kgu fe e f y ful3u i p y p l n e w (g3cu3(g"g u n fl w }"Dci4 $31(&$" # !2 ' 0 ) ' % # ! i i d }S d ) " ~) u g u ~ " u i ' g u ~ " X'X g $$ 3eu '
|m-1 (i )| = R li [k(Y ) : k(X)]sep .li [k(Yi ) : k(X )]sep .l R
e
k
&
~ 3
{
fe ep u u p m l v m n e m l p p n p h e n 'gD$uzk""s ISD"3e(b3g"DuSkqg3izsu"(s$g3X y ukIi4 y p
= [k(Y ) : k(X)]isp
iI
e l x l { fe l u 4 oDi(uDqsx "hkDD"i 7{
m fe e l u n y l f l v { e l n fe ep u u `'gDoD3I""Su D5e("e3'kd(3`ksuzg"3sXi
Yi()
si()
x l l p e n y y f m DuXkgsD)'is"uDl
y e u l y n e p e f z""Df I"k3"n qguD
9 9 8 7 42 ! 1 ! 0 # % % # ! A@6&653"&&$)('&$"
homX (R, Y )
[k(Y ) : k(X)]isp
i : R 0
/Y i
iI
X
d
$6
l homX (R, Y )
homX (R, Y ) = m-1 (homX (R0 , Yi )) homX (R, Y ) (R0 ) = Yi /Y X : R (R0 ) = Yi H Aut(R/X) ( (R0 )) = Yi StabAut(R/X) (R0 ) H H StabAut(R/X) (R0 ) H/StabAut(R/X) (R0 ) Yi StabAut(R/X) (R0 ) (R0 )
homX (R0 , Yi ) mi
P `
= H
iI
Aut(R/X)
[k(Y ) : k(X)]isp .[k(Y ) : k(X)]sep .li [k(Yi ) : k(X )]isp . [k(Yi ) : k(X )]isp .[k(Yi ) : k(X )]sep .l [k(Y ) : k(X)]isp Aut(R/Y )\H homX (R, Y ) Yi StabAut(R/X) (R0 ) m StabAut(R/X) (R0 ) |m-1 (i )|
iI
|StabAut(R/X) (R0 )|.li |Aut(R/X)|.li [k(Y ) : k(X)]sep .li = = . |Aut(R/Y )| |Aut(R/Y )|.l l
[k(Y ) : k(X)]sep .li [k(Yi ) : k(X )]sep .l
li =
mi =
i homX (R0 ,Yi )
i homX (R0 ,Yi )
R0 Yi()
homX (R, Y )
R0
| homX (R, Y )| | homX (R0 , Yi )|
t0
li
(i)
/ Y.
/R
|m-1 (i )|F (i )F (si )(a).
m
i homX (R0 ,Yi )
m-1 (i )
homX (R, Y )
homX (R, Y )
homX (R0 , Yi )
i homX (R0 ,Yi )
homX (R, Y ) = H
F (t0 )(a)
i
(i)
F (i )F (si )(a)
iI
F (i )F (si )(a) homX (R0 , Yi ) StabAut(R/X) (R0 ) R
n e p 3'gi~|
H { p "h
S
@ { ph
e n U"
m f l u { { p n y l f p l u x f m l y f ucuD"jUis bk"cD5xks3"3D"c'"(gp
y le f e f l u f kz$D")s3t u 36 ~u
nf (Yi ) =
r=0
d e D g ig k$r " S & $ur } $e u s }X'` "' "5 3s"D gi 3 $~$ g`' 6~P s%P50 4 2 1 # le f l l m x e ep ue p n y p me f e e fp w x e ep ue p n y p h ep 5 s qg'5@Dc(gizg3gDo")k(Auzi&'i"3ukzksSDoukk"kg"c(g'Dkk n l f "DD'i4
)" &" #" '('%$!
{ { { y l f x p m x e e p m y e m " " 'u"Dqgs"XDouk"3n Iu uD$zx y kk"'i"ii$31(&$" ye p n f x m e # !2 ' 0 ) ' % # !
nY /X (Y0 ) =
iI0
g
@ 3
~$
V $gu k gsr d '@g u ~ "
t
$6
gi~ d
Y
d k i ~ " u d $$
Y X X X
u)
s D@ k 36 3 rg)r3"ukzksIgizg3gDo")bggu"'De e e fp w x e ep ue p n y
k
us
3 d "' u u ' U
ni nY /X (Yi )
)" &" #" '('%$! { l n e l n e v h l w e "S(ki&"uk"3Dgh m me we l x xe l n f zke(ikDs$3' p l u x l m w y l ep (gs3ID3"' uD"gk e kjI { m me we l x xe l n f y y f n XzkIDikA3X 'ic3 p wx s' h kg'3bz'gD$oDuze g3I { { { fe e f u f l u p m fe ve m p fp n m f m m u x m e y le x ep f 3"IDs33u}ugqu uDg"kk Du y xp p n l l y m l fp n p | ugx k`3'@D''f `")ksr ~{ l fp ks{ y xp p gx cu m l u f h x e e p l n f 6 { m x x l u p n p p l u le p e 3uD"PDouk3"n c3&('uD"3izsubg)s3@Dgku)xu3uDzi 4 $31(&$" # !2 ' 0 ) ' % # !
pr F (s) Trf = p r s a =
iI r iI (p ni )TrYi /X (si ) Zf h (Y )(X ) ^ X k Y Zf h (Y )(X ) X
" e X i g ki~ @ 3 g i ~ 3 D 'g
p .ni
r
" u I "u u I u $ e ~u u d u k$ c~u 'U3 Xu u s
X
i ~ 3S d 3 I $i g i d g &u i ~ D i i $$
f Y fh fi Y si
s
m l n y f x m (I"uD$3ke "Im n e e l l ux"zkDuDdstzp
p
Df 3gi$kS( x l y f h e e n x f m "q3"D { k""'gD$S(g"ki3(DgsD(l p fe l m p e n m x l le x u p l u fe ep ue p n y l u n m n m e x ep w p he e gI3k$kk"kg"cD" iI3x"` i3(~ou Doksux &U&3ki{ xkggDu"'u m pp e f
iI
i
0 # # % % 1 # 1 ! # ! 8 ! ! % # 1 1 ! 1 # # &&Ab$'}0 @(A'(3A'""$ PP"P}$Ac
=
iI
[k(Y ) : k(X)].li [k(Yi ) : k(X )]isp . [k(Yi ) : k(X )].l
= F (g0 )
[k(Y ) : k(X)].li [k(Yi ) : k(X )].l
A
m e l fe l u 3"eDkD"j '' f
e f p n fe l 'guxiikb3'kuD$VD5
k
ni =
ni
[k(Y ) : k(X)].li TrYi /X (F (si )(a)). [k(Yi ) : k(X )].l
k
p.p.c.m.
d ) s
X
a = TrY /X (idY ) Zf h (Y )(X) TrYi /Xi (si ) TrYi /Xi (si ) Yi ^ X k Y
f F
s
f
X
Y
Yi
s
s
(pr ni )Trfi F (si ).
/X
F (g0 )
/Y
f
i homX (R0 ,Yi )
r
{Yi , i I}
F (i )F (si )(a)
Z[1/p]
F = Zf h (Y )
(ni )iI (Z[1/p])I Z
Yi
~P EC B 53 A 1 B 1 D r ~$ g U36V
~P
P
1 be3i
X
Yi
X
nf (Yi )
# 2 '0 ) '% # ! $3!1(&$"
u 3 { "h
{ u x n }sib"um
n(Yi ) nf (Yi )
A = (X, O) B = (Y, O) A = (X , O) B = (Y , O)
ni = nY /X (Yi )
f
Y0 Y0 Y i X X
(-1)r lgBp X Y
t
nY /X (Yi )nYi /X (Y0 ).
/X
/Y
i
f
Torr (A , B) B Bpi . A
s s
/X
/Y
f
Y
pi B
{Yi ; i I0 }
" e X i g ki~ @ 3 g i ~ 3 D 'g
p .ni
r
u I " u "u I u $ e ~u u d u k$ c~u 'U3 Xu u s
X
i ~ 3S d 3 I $i g i d g &u i ~ D i i $$
f Y fh fi Y si
"
d ~ $6
kK
{ ep x u n f l w m n e `3gku}s}"''Irs(iDk& p l n f u l kS"}u z""Df S(Iz(ukg"ukD3Dug"u 'iSx D"'iXm y e u l y m f u e m f h l ep n e l u { p y f y l f u l f fe n f l n f u n e 'g3uDiI""}riDzs n " m fe e l l p e n e r'gDdzD(ckxsuzs n "( l n 3 n e x l u { me x l e y p uzsu"`Duzug"@Xzm e w fp f u f k'g'' p m e n y e l f w x zg3"ce(gD'uD$uqi"{ m f ("@$~ k3"'f r"n 6Si D"@k"k$'i& m y e u l y m y l f n p e u y f
fh fh fh Afnd//k X K F L Afnd//k (Y, L) Y / X K L ^ fh (X, K) / (X, K) K/k F
fe np e m x f m p kgu3g`""n 6`u""'jz) k"3'U'"@$~ y e u l f y { m f "n p y f x y l f u l f iXik"u u)(D3I3Dim fe k3ic""n A p u n f p f f p f m e l f w x (g"3iki'SkXkDsuxzp f f { l x 8e m u l f m l n }u'}$'i(3 z"3u"'"@$ 6 k"3'3D"3(Iq " "r 'u"DSgzgD"u'S@g'("s@ e v x m m f m y e u l f y m l u m y w m { { y l f x p l ep l x x w m e f l n )" &" #" '('%$! { x l y x m fe e f u f l u D''f u`kg'"4 { m p e m l isk'Dqkp (A
p [k(Y ) : k(X)] = B
r=0
{ e f u n m m u x m e k}rs"u}u3k( n u f h l ep n w x l "u}"(ukg3uD n p "3s &X y i{ y f
A p Spec(A)
e n x ep w x p e n m x m (s$k"e(gg'ko3u m x l e f 6 l p n m n e e l x l 4 { x l f "u3kXg3 i"kDzisuU13'e( n l f u u n "e"3 e v u m p n m y "3X}Ick"f X'gDd(sD"kkDX""n n m e f l u fe np f x l e f g
P P /B n(Y ) = A [k(Y ) : k(X)] . [k(Y ) : k(X )]
y p u ye g3kS m e e fp w x e ep ue p n y p q""gzks'IDoukk"kg"czi f m f h P w u m fe f p { p e n m x l le iQc 3`kp I3'gD$s"q(g)'Ui'g"ki3(DgX n l f u u n n u f h m x xl u p l "Xu'}""cu}V'6D'uD"u Dux Is6 3D~ { f p w e n iDik"k3( "u'}"" n l f u u n f m n u
li = l I Y X Y / E.
l D"u kelu y ykgge3x gexDlucxDlDlsunm'ugpluDoeDlD@cx" yslj'h$ {" {" keszpkfDigz5cg)3"zgk"t f e p f l p l f p e f y e n m n epe n { pp "nXkg fe m y y f n Dge'5zI 'X3"S( l n p x m f "qksi"@$~ "S( n n p @"U h
E k E X Y X
iI
e x l x l l n l n f l n u f h me f m f l n f n l uksDDsu""'Du}`"'k`"q3 su'""S3u l p y l e uuz"u uDXg"n { l n p l u x e y p f l ep n w x l f e n p e n m x l l "k"ssu3ggDieDukk"SU"Sk"ki3DDge f u y f m e f e m DsD'}'ig'5z y 3" y f n n l ep n w x l 3'iukg"'uD n l f u u n n u f h v "3"S@''iu y sDx'"@$ y n m f "n l u 3uD"j{ l 3u p y l fe e f u f l u p l l y m f m n p | uz"u uDu`kg'}D"@zSuDf XI ~{ i x ep w x p DoksukbV{ 3s3m "m w y p m f u l l f f p w e n 's@ke rqs"3'DDiVug'"k"q( p y k")u3u 4 { { { le pp f l f n m fe f p f l u { p l u x fe $ " " DgkkDuI3ksD"qkq33uD"uS'g"D3uk3i@3"n l w u)Dke m f (AI('"@$ )sge "xz""Df y "n m y f m y e u l e f { p e n m x l le o'P'g3gDd3(Dkc( 9f n l yl n @I'u cu" sr)u' 3D~ 9 9 8 7 42 ! 1 ! 0 # % % # ! A@6&653"&&$)('&$"
I0 nY /X (Y0 ) = nY /X (Y0 ) X X Y k(X ) k(X ) X Y0 X Y0 t : X Y Y /X X X k
K L
X
(-1)r lgk(A ) Torr (A , B) B Bp = [k(Y ) : k(X )]n (Y ) A
pr F (s) Trf = Fi f F (X , L) (Y , L)
f
s
X
X
Yi
iI
Yi
(Fi E X )red = Xi
Afnd//k
s
s
(pr ni )Trfi F (si ). /X / (X, K) /E / Fi / (Y, K)
e f
B P A k(p)
r
{Yi , i I}
(ni )iI (Z[1/p])I Z
~P
1 be3i
nY /X (Y0 ) = nY /X (Y0 ) X X
Afnd/K
3 uu 3 u " 5 3srg gsA " g} ( e i XAs` D si gu"'De '~$X~&uc' u U 3 ' g$$ 3 $$ D &s' ~u i 'x sDg ~$ ' ~ i @ ~$ "} & " u D u 3 ( Pg P $~ u u g$' s " $ 5 " D cu $X U A$') ~$I sgie s u U
Zf h (X) AfndCor(k) Ztr (X) Ztr (X)Ztr (Y ) = Ztr (X Y ) ^ Ztr (X) Ztr (Y ) = Ztr (X k Y ) A AfndCor(k) A A A X /X Y Zf h (V )(X) SmAfnd/k f g X Y Trg (b) Trf (a) = Trf (a Trg (b)) Y ni ni Zf h (Afnd/k)
pr2
{ ' "h d D 36u oc' $ $$ g 3 3 ' 3 i " ug s g ~$ ~PgS36V ) &" # '"('%$"! { pe p n y l f p m e y p n f x m p n x l gki'k")'5kqs3ks@Ig3i($o"Dqi4 { l n 3 m p )gkpuu i l3 n fe f mkd(sDl"u zp h m p e n m x l le f y f m p y p (g3gDd3(Dku u((g")u3u ( 9 @f eDgDuDukixiucDlg"kk@k3g3cu3" m l e e f e x ep f e e y f n h { { fe e f u f l u p { { y l f x I " "kg'"ckeic " "" uDu3gp l 6 n xl p l {6s(iDkeiD"u(g"n s$k"nxkedzDlDl k"3'g"k(m e f e m y e u l f y p we x 9f @h x { n yl f f 6s@D"S u n l f u u n p l n f "Su'}""zuu"'u n l l f y m e w p | { e l n e "uD'qg i~"oDd&"iSz"3v i u('uke h l w u (A n u "u}`{ m 3"
X Cor(X, Y ) = Ztr (Y )(X) Zf h (Y )(X) = homZf h (Afnd/k) (Zf h (X), Zf h (Y )). k AfndCor(k) a a Trg (b)
As
g
u
d k 3 "
m f ("@$~ (i""'fi)uXDkkI"'"@$~ zD3"Pikg'}'is3ck"s ep m f e f l m l u { fe e f u y f l u p (u"n''3'qkp "Df )"qg ~| $31(&$" f l w n f n l l y m e w p # !2 ' 0 ) ' % # ! d I A d ' " e "ii )s b sD ~P b S e p u e l f w x n f m "kuk"qgD'uD$u~3rzp
k Zf h (X) k Zf h (Afnd/k) Zf h (Y )
l n f 3}u
p f l n f u ki'"'''
{ e 'Dkkp p e l f f u pe h np u { me x fe np 6uk'i35e'3S( iUg"3k"I"k`'g3ki3ke &p l u x f m u y e u l y f e n fe l { fe e p y n m l u x l m w y l e 3x""'c'3c i@ z""Df 'z)uir" 'gD'$ju5iki "uck"cuu"cs3XD3u'"uu uD"kgp 8 7 # f u n f l w n m p h 8 e}"Dr"uku}Vb{ u l y f e 3"'f k( h H { { V gqk6 & P6A"! y y p p l
/ Zf h (V )(Y ) Ztr (V )(X) Ztr (V )(X) X Ztr (V )(Y ) Zf h (V )|Sm Zf h (V )(X) Zf h (V ) SmAfnd/k Ztr (V )
" g As e n m x o3m {
k X
3 d d D~ " AuD i i ~ b @ " u g 3r r @ 3 A
Ztr (V )(X) p
m pp e f u n e e l x l "`kggDu"'ir"kDzisuzp
m w y s"es@ke p l DdDkem yy'Izp f e33vx ke"3u'fy"n y l f f Ds" h
f h l y e u l y n m fe e v x m p l 6 `'i'DDz""Df kg"3i"zSsV{ e v y e u l y n "3 z""Df m f f 3DsA{ Pi"3iD'u 'iPg3cu3 "'g"m m f f y f m p y p we x 9 xf l f f h f x f u u n u uDs" i"3i'}""'3( U n l f 4 "DD'i{ l w iSDuDge h e l n gd(3 A y l f p y xp x l u x l m w u n f l w n f D5jk`m Dgc(gp 3rD3"@"''3'gp f p yl f m f i""'ibu D3'"@$~ "t"""`s 'uDs3t"k'gDugDs$ ""n n x f m { x l u x me we l x xe l e f g 6 qIU"" U3PUc
a homk (X , V ) {Xi ; i I} / X k V (f, a) : X X k
)" &" #" '('%$! { { { fe e f u f l u p m { { le pp f l f n m n l p m y y m p I " " kg'}D"kIu " " DgkkDu'"uD"u zqX u I(ui~| { l n n e "uzs m n fe ve m y l f u l { { y l f x n m n l u rs3ksugf 3D'3DDSc " "" uDu3@""XU4 $31(&$" # !2 ' 0 ) ' % # ! " d )d 3 uI I ') ) 6 'k ig " i $ ' g o & " qg x sg$ g " ) D ($u U g )d ic A 3 sg $ ~i@ &o ) i ~$ g Ii ~ 3 Dg`'j $k$@u} kgu"'DI U
0 # # % % 1 # 1 ! # ! 8 ! ! % # 1 1 ! 1 # # &&Ab$'}0 @(A'(3A'""$ PP"P}$Ac
Yi n(Yi ) fh nY /X (Yi ) Afnd//k X
V k Zf h (V )(X) X iI Ztr (V )(Xi )
Ztr (V )(X) Z Z[1/p] = Zf h (V )(X) Z Z[1/p]
Trf (a) = [k(X ) : k(Z(f, a))][k(Z(f, a)) : k(X)]isp Z(f, a).
AfndCor(k)
Trf (a)
X
f : X
RigCor(k)
/X Ztr (V )(X) Z(f, a)
X
TrX /X (a) X a
u`
d d i`~i A 6 ~Pp` y p w Iu uksux n e p "@'gS~|
^ X X k V
/V
{ m fe e l p m n e x l u n f n e n kdzDDl kqruzsu"3"(6" p l n z" n e x l u n e e e f s(iDkiD""i3ksuk"{ 3"m e f k e l l m y e u l y p h { { fe e f u f l u p l 6 { e dzDD@@z""Df @g6I " " kg'"`zIsDkkp m f 'A("@$~ @ k"3'"n m y e u l f y
^ X k V k X
Shv f h (Afnd/k, Z - mod)
Zf h (X) Ztr (Y )(X) Zf h (Y )(X)
Zf h (X)
Ztr (V )(X)
X k V
X
5
{ m f '"@$~ "n f "j
geDlf'wuxsu fe k e f g
&
k
m y l w e n l me f "xy D'szm "@Duzi{ f
Ztr (Y )(X) Zf h (Y )(X) = hom(Zf h (X), Zf h (Y )) X Y Z
i
h
{Ti ; i I} Y
k
XB Y A AA BB AA BB g f A BB a B! b AA X Y Z
AfndPST(k, A)
X Y Y gi Trf Trgi = Trf gi
Trg (b) Trf (a)
AfndPST(k) Ti a
Zf h (Afnd/k)
Ztr (Z)(X)
X
Trgi (b ai ) ai Ti X
g k U gq y b i Tq g i d y g W T teq T ` q t w T $d|v$c"Py{$aY&YhhnaX (c$i l g w w b i gt w g gg w w bU Xt b u u y $cyfl~ $$)|$t U g t g t teq t g i b w )BT fYhhu0c$A&T g d W w g gq T W X W w g p g y b U g d w fYa|vcYaoa"aYd $caYly We vcb q w w b w Wt $y l)$wg g $y )ve F W w g q g $y vcccjh We b w b g i W d b u u y t gq W t |)$"v0g g p p b ""coI U g T w y p p bU w 0cB$1"cag gq e W y i X d de W w T t b p b U g w y U y T w qe h t yq u g g i w b w W t h$vYrU$Yfh|gfnY)ju"cy$cT IIh$h$yvcPe l)$w0g h se W U g d y t W t tYr($fw0g g p p b g i w b ")ojcPe l)$w h e hq hcg (($y w w b w W c$tg g $y )b$y t w bt u u g i gqq g U g d y ge d Xs we W w g p g W Ue d W t w be t w g pe i g i W t $~hv~Y$ahf6hfa"frPYfcha"v${g h h w y p p b U w g gq e W U y i X d d v)g e I$1)~h$hfYYve gW w Tt b p b U g yU y T W T fnY)ju"ca$wc~wn)BT Iw W g W d b u u y t g i tse W U g fcj$$f$vfg W g U g )BT t w b e d U g g g w w b U W t cchYuca$$ca|g g $y " Tdaf)j$$yfm~ d gt b u u t w WugffPhqc"nT fY)ItYr0$fvhndw"fh$f)Ay g t t teq t g i b w g d W w g se W U g d y t gq T W X g p te k u d b p w W w ge ahcb Xd T u X Wt cYntvg y T g Ut Ts X u q nafPenafdygv|g$y iY)cT 7iygYdfWwcb"pzw~jtjunYaXgwa|gafd$z~j"Ya6fYhcjy{h$1afag d b g T W g y i g i U b u w t X p d gs t W we b u g i t gq p g t w t g d y t se W U d y t t 0vqwqtYr0g$fwW0g g$y tPafW q g t i w 0gcb$A&T g i g T W X t g p te k u d b p g i t ge w p t gqqe {t0vqcYa"P$$Yc"yvh$0vhv"p Ts g i g X p bs ge bq b u b X d u T d u X d i g ` g t |t0Aa"Ydc61v)`chcjcfWaflPqynT1gfwacwaWg d$ftAvqcYa1ghc)bv)u)fWlhqAy chaj$nT 67 a~$" y g TW X e ` q b b T g t w bq g u u 0 &dda a nYvn$d y Ficuii i &dda 1c fj f ctrar
m(pi ) I (a) pn i V k Ztr (V ) SmAfnd/k
r
&
g W w T e y t w be We t b u b d u Tq g i g y g d u Tq t w T i gqe W y T d g t We Ts g "fn&h$flchfvf)jcY$"h|y)$aY$hn$hvYYwvn|oj g y fW q g d ge W w g i w T d ` t yq u g vYacY)"h$vq U g 0cBT t d bq T T w X W Xe d u b d u g W W g U d y b u g y T pe we dchn~u)fhY$cY$ffA$)nAh$v"p W ag W w T w gW w b nfcaU g i t d ge p g d u g y T X ie t g i gq p g t w $yYv"aY$rcP|$"h$1afag (q t w b W b g W w T e y t g d e w T p Tq g )fnvIcY)c&v$f"fah$c"zPAi g e W U g chfg g U w T i w b u t g d d b U Tq d g y b d W g d g i W g p d g c$)fYf)P~c$)fYYA}"Yaju g i g U w T t te T w w b U T S $anYfPn$caz& )vYt$6ahA$ychfnYn|h|c$fag w be We ws X i Tq g i t w be W T W b w t gq t w b w g d u gq e W U y i X d de W g g We y i X d W t )vvYr0fYhw|fv$0fwg X W Xe d T t y b t Tq e t d ge p g d u We i W t favYn& cfPIfwah"aY$wt${g g y T X i i W w g d X k b U y T g U te Ts w y h W w g p g W U g d d b U t yq u y q T X ie g gg w w b U Wt g i X teq T p d b w gq g vfhhnAYc$~v$y Yv$fahe aYPao g de i W w g g d Tq g c Pe IoaYa$)mn0Yhcm$)a"fr0fYcaov$cb n0Pw c$cag $he c${afhYfYa6uYhfc& cflch$$y (f)w0g hha7fh$hfrYfh{0g t g i w b w W g t g W Ue d W t t g X p d gs t X W Xe d T t y b t g y g i g i w be w q t T u W t w gqq g e t gq e W U y i X d de W t g ie `e d X W Xe d T PchffavYn& g $w ) r yfc r a & $ 1 ic dv(yY &)j r vv & a Y~ d z Y mc r iwY&)j r lr)c n E $nc a l & w ( r a r r r& & $ A a dv( n r "r A& 1n r q ava FFu{7 iudjncj c$i t w T g $i h $ r& t gq t g Xq g u u T W w b t h t y t t g e U g p p b h{vnvcf$aYi }""caU t gq h e 0v$hcg h g i tq g d y W T w t d y g W T d X w X ` t g S wPaY$fnfY$fnY$a){ t w iT |rg dt jnU T r c$yYcPcA Yt i tT q Td gW b w w b Yfc$EcFh d y t g e W U g d y t W g ge w p h g d ` W we X W Xe d T t y b t g w y W t $flchf($fzh$I)f)aY)hAfavYn& cfz$1g e q t w T c$i g $i t gq t Xq g u u T W w b g i t W w g p Xq X t g S yda"ahay0II g i g g w w b U g W w T t b u p b U g w y d y t t g e W U g d y t W g t ge w $mg a$$)"fnY)j"cm$$fy)vYr($f}y0vp g i r& 0vy0vncft t0f)aY)hYhfc& cfi0wv}afg qn{Yc$iw g p g d e g d ` W we t X W Xe d T t y b t t g i gq p g t w q d T u X d i w g ` w g W g u y bd ` $)f){g q U g g u y b d ` gq T d g pe W w g i j$cYc{h|davfa$he a"aY$hvq vf0cBT We b vc R 77j$ H H 3 # w b h W w T w g W we T p g i de W d T u g X d T u X t g ie `e d X W Xe d T cq0nfvcAz$hffnqcfcafh)vYzfhYn& g w $y W g g t teq g i b w azfYhhc"nT w $y ttovF T S t d ge W w g t g i W w b YvY)}01cft t gq h d geq y ` X d W t h0ahvcaY1g g p p b ")ou
^ X k V k a X Cor(X, V )
iI
g X d W w b p X i W t g w be We t b u b d caYf)w0l)vYtf)jcY$u d T u g X w w b i Wt c$$c$"g
nX
tcT$i w g g ` p q $i|)nTAhe $h h "aYc$fh{nq$ g p d b X k W gq d T e g i t gq e W U y i X d de t g W w T t b u p b U t g i gq p g t w ${h$vYr$Yfhffcf"caw$|h$1afag (q t bW w)fcbw W g gyg a$$)fcf"ca$$y i g g w w bU gW w Tt b u p b U g w W w ge v)b W g gd ` W w aYcfhe W g se W U g d y t e w qvf($F$p U g)BT tn$i w T g k U g w y t w b d X ie t w b U h W d T u g d W y $d~$ucYaPc0c|fYnT }$ g i tWg b t g i d y Y$c$t g t W g t gq e b iFY$cbI0vugv)c wag &) g$y henPUFt0g)q0 d Tq W t We t g i g p te k u d b p gq W w T e y 0gfvfu"fh$f)ovFn&vt gfYgc wvg)`nApvqdafWcb61yhq$cYwafXjaYdynT0Ythcs td e T e y g U w s i t g TW gt u g g gU e T ycT1w)veYWrPefdYWt0gfdPqWtg n) ( Wwag$he aXqdyafWUc6fS 67 a~$" b U T i g w bs g $nc nB
AfndCor(k) AfndCor(K)
Zf h (Afnd/k)
0}
a j uIa } 1) f1j wn "rn &
AfndCor(k)
w be We t b u p b U Tq g i X W e W Te U b t t q g i w be W T U pe d X Tq g i g t w g u te i t y b I)vYtf)j"caPy|fve vY&PYcT yl)vY&ajtYa)1Pyfajfh{c$w g i ge d b ` X W T t y b t g w y g p p b whf)cf&U )y$$""cU g i w be W U y d W t w b U T S g d d g i gq y p d bs Tq d T u g X w w b i W t g w be We )vYrYf)cYfg ~h$1f)6}Pfn|a$c{zchfvt b u p b U Tq h w p w x "cayhI$in g i g g w w b U g W w T t b u p b U g w y d y t tse W U g d y t W g te w p W w b t e $wg a$$)wfcf"cav$$u$otYr0$foaiP$qcf$y g i t g d ` W we t g i b w t y b t t g 0f)aY)hFc"nT )oi gq p g t w q d T u X d i w g ` w g W w g p g d e h$1afag (1nAYc$1a"f$hhq w geq X T g u y b d ` gq W t vhajnmj$cYclv10g t g p te k u d b p g "fh$f)"i g u y b d ` gq h t g t teq t g i b w j$cYcvifYhhA$)$A&T g y g $ai W g d y b e $)Ff Td gW b w w dafc$scb Iy t g t teq t g i b w 0Yhh"$)$A&T t gq W w b hcft g i t W g b t g S g W w T e y t g W d U w b U t yq u w be W ue d U t g i Tq W g p i rYac0IcfcBh$ffYcvx)vYhfaf0shra"cT ge d b ` X W T U T hf)caY&VS R Q H H 9 8 % # IPIGF7&ED C
AfndCor(k) X homAfndCor(k) (Ztr (X), Ztr (Y )) ^ Z X k Y k Ztr (X) X Y Cor(X, Y ) k X
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
AfndCor(k)
kK
ac
&7
Ficuii
a Ztr (V )(U ) u (a) I OW I (Z)
n (
[Z]
{Zi , i I}
X
Z+
u: U
AfndCor(K) X a: X
GX u: U a = a + - a- a+ a- Z+ Z - ^ u (Z+ )X U ^XU ^XU (Z- ) (Z )
^ homZf h (Afnd/K) (Zf h (Z), Zf h (Y k K)) = homZf h (Afnd//k) (Zf h (Z, K), Zf h (Y, k))
X Ztr (V )(X)
W
/X (Zi ).TrZi /Z (a
GX
(Afnd/K, f h)
V
k
k
^ Z X k V
W
X
Ztr (V )(U ) GY (Zi X )) V(I) Ztr (Afnd/k) V Cor(X, V ) V (fi : Ui Z Z
a Cor(X, V ) N a= Z
{pi ; i I} a = iI m(pi )[V(pi )]
G (Afnd/k, f h)
SmAfnd/k G U )iI
^ GG Ztr (V )(U X U ) . Cor(X, V ) k
r i=1
)c (
G Ztr (Afnd/K)
Cor+ (X, V ) ni > 0 ni [Z] O X k V ^
k
X
fh
X
u (a ) k
Trf (a)
I (a)
Z+
[Z]
AfndCor(k) f: X ^ ^ X k K Z = Z X X Trf (a)
iI
X
Cor(X, V )
a Ztr (V )(X) Z- X nX Ui
/X (Zi )
u (a+ ) = u (a- ) I (a) = O X k V ^ GU Ztr (V )
r i=1
I (a) n N
I (Z)ni
X
AfndCor(k)
a+
GX
g i
W w T e y t g W U g de i g p p b t Tq W t g t y t t g e U g g gq u p b U g n&h$ffr0fhy""cfPwfff0i yh$"caIS
ColimU Flt(x) Zv (V )(U ) tr
(i1 ,...,in )I n
t d bq T We d U dchnwtY0X t c ( h$"ca| g g gq u p b U g S d y b u h W g g w x X k U d g k U g d W T Wq y t X d gq g de y i X i g i X te T W w T w g W we T p W t g q $cji)$dda$d0f}nYv$fYlhYvzlfPn1nafhnA"0o ( )vY&ajtY)Pe l b 1 W w g p g de Tq a"fhnPU w be W T U pe W w g i q
0 '
1 '
T w ~F W w g i X g p te k u d b p gq d T aPe a""P$$Yc""vycu g i g ` T p AccAve (q t w bW b cYnvI W we b u q y g t w y We y i X d W t hcjq$af$zt$$Yyg h)faPe a)vYrcfAh0g g W w g i X w be W U g b d u Tq W t et g }y vhaYW gqq g g W we b te i g p p b t g w y W t fhch$}")$$vg g $y fW q g d g y b d W U w b i W y g u w w geq X t w g k q T U bq y T g w w T w y W t c$)fY)f"~$Iahhafa$nah"n$nu0g g $y tcfA)) )v)z)vYtf)jcY$u We T t w b h w be We t b u b d Tq d T e Pic d y t g e W U g d y t W g ge w p h g d ` W w $fwchf($fowh$B)f)aY)he X W Xe d T t y b t g w Yhfc& cfv$y 7i fv"vhcayhun$i YcPyPc0)"ahafW g We peq b U Tq t w T g t t Tq U Tq h W w g p Xq X q g d y b u g d i w g d u W y g u w te te b k U w ge $cj|f$aY$vz~hfh)$dah$ d y b $cju g i t W w g p Xq i g p p b t g w y g p p b U We d U wda"ahaX 7y")$y")wvfaX at g i W w g p Xq X w a"vy Iy daYfc"|c)vfhh7h7acYafY$0fndfv0fvhhchq$y a$ccYq T d g d W w b p w b h t w be t yq U we t gq W w g d g t X d u t g W w T d Wq p t g We peq b U t gq g g X w w b i W w TW
^ Z = Z U U ColimU Flt(x) Zw (W )(U ) tr ^ Z U k W ColimU Flt(x) (U , O)
wP(W ) U Flt(x)
( )
T w cb h t gq d T u X d i w g ` w 0v|c1Yc$g
f (z) = w
( T w cb Iy cYfc t w bdW b w be t yq U we g w y W w g p g de Tq U T w chfvah${)"aYvcha~ WycYucjv$ t g q q g W W bW d y b u g d y t t g e W U g d y t W g t ge w p h t g d ` W w $f)vYr0$f{oh$nYcfhe t X W Xe d T t y b 0fhYn& )t y hvfog g i g u y bd t y bt g $}cYc` )Ahq gW b w w fn~ W g g ie `e d X W Xe d T PchffavYn& g w $y W w ge ahcb g eq U T W t c$y vaU acvg
(i1 ,...,in )I n
) ( t w geq X T t g u y b d ahvc{j$cYc` gigghq$u"pcbU1vqy dagYfc"}$vt)nT aF g g g d W w b p g i We ` t q g i W we b u w $~v)ry W g g t teq g i b w aYPvyc"nT w y W w ge ahcb 67 a~$" a d c a d r)A cndYa u) Y 1 & a" c c r 0 nYv&Y y jv& t
d )"Y d cA Y ) d o c j 0 wn c &h a aa r rs ctrar &7 0r c Fucuii gq T X ge bq b u b Tq d b u y g U te s w y W t W w g p g ` w bq b du g )vcYWahc`)v))fW1P{$ycjlcTafPnTz${g ~$0a"a)$chcY$yU gd b U w Ycaag g b w w fWn~I d t t gq e te p i t W w p g y b U g t g ge ` bq b b W T d y b y g U te T w y w $yf~0vvftfP"cT~Yag"fd $)Ydi1v)chcjucfmhq|$)unTafPns$rg g e w y g d e w T p g w $y $$}Yah$c"$$y 7i d g w bq b d U w i W g u c`$)v)fu)b|$yajrw~7 t g d b ` X W U i g w gq T e X g w y T w 0veYc)aY&Tag1aUahn&h$y $$Acb Iy )vnT t w bq g u u ttovF g W Te i X p p cfnh$""ve Wt 0g g$y chfnaUvYac7 w be W T pe d X T S g y gf~ahefanTdchq$}hq q W d g W w g i w d ` t y u g W}YWhfdcbY$u1Ya|$cjrnAh$h|g g X Xe u d g W W g U d y b u g y T pe we p W W w T w W w b U d ge p d g y T X i t g gq g t w nagf)wtdah"agY$uz$n$hevi}h$1pafag (q U g 0cBT g U g g i w b U w b h d g we p d g W d y b hqaU}vqyYda$heftcarc$v"YaY{$ce W w g d X k b U q T X ie w y W t aYacIcP$wg g i q T X ie w y w ge W t cP$v$~0g ~ g $y de b g i gqe U Ts W w g p gq T ` X W t g q gqq g y t y g d e w T p Tq g v)chhc{vc)v0chv$Yah$c"mh$i
I (a) u GG
g $y
g pe d X w vfc~7
g k U gq y b i Tq g i y T b w gq g p p b dqh$$)wPiyn&$ihi")U We w p X i w gqe U Ts g Ue U d g g g w y W t g t T U g U t w T i w be W d g t t v$$~BchhcahaY$07cun$y)vYYcT iS I)vYr0|$$f"cT w be W U g t g w y W g p i b t T U y T g w p T d g t w b h u)acv$a"cYlcv)g e h W w T e y t g t T g i W w g p g ` w T k U t u T Tq g U d g pe d X n&v)~$oa"a)$ndfjn1hfavfc g i W y t qe h se W U g d y t W t g gq T W X W t ""hvf($"0mhnYa1g g p p bU h W g g w x ""cajaEiF g k u d b p b te W t m$f)"cfh0g $nafPns g y T g U te T t g i ge d b ` X W T U Tq t w T 0mhf)caY&rh1c$i g k U gq y b i Tq g i y T b w g S $dv$cmP}$c)&$1Ij g i w be W U y d W t w b U Tq W w g p g e d t w bq g u u rchffYchvcvY$wchaj$nT g $y afW q g W w g d X k b U q T X ie g e w w y g W t g g qe h g W Tq u W w g p gq i p g W w g U t g i d T e t g W w g ie X t w be W U g b d u t g )f$cacPr$y $$y fPe a~hu)fnh$za"ahajrfYznquY)PA)vYrcf0vq q T X i q t w b d X ie t w b cPe (cYaPfco U g 0cBT t d bq T dchnVT ~v w se W U g g W t tYrwg g y $y)vz~j )n ( $d0|h$$c$Pyi d g t b u u y t U w b i W y g u w g k U gq y b i Tq g y T b w gq t w T i W w b n)&wvunu))t W g g $y ahe W w g q g i g ` T p q g i W d b u u y t gq U g T w y p p b U w g gq e W U y i X d de g W w T t b u p b U g w y U y wccAve (~ufcj$$f~ho0cBm$1"cmwh$vYr$Yfh{fnY)j"cav$$anT W w T w W d b u u y t g i W g se W U g g W t g i t g p te k u d b p b p b k t g i w y U T k U d T n)BT Iyfcj$$f"$}ovfg cc ( |"fh$f))"c$$$acdycu g i g ` T p S "ccAve i7 g i gqq g U g d y ge d Xs w AhvAf$vYa6$ve W w g p g W Ue d W t w be t w g pe i g i W t a"aYrPfYcha"v$yg g $y hfW tq g W ag t W d b u u y t g i tse W U g Yfcj$$f"|tYrg W ag U g )BT t w b e d U cchYu
a = a + - a- X OV I (a) OU k V ^ a+ a- a a a pr1 I (a) = pr2 I (a) O X k V ^ G X k V ^
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
u J = I (a)
OX k V ^
^ U k V
u J = I (a)
J
J
Ztr (V )
Ztr (V )
c= mi
iI
mi V(pi )
Shvra (SmRig/k)
J
J
Z+
{pi ; i I} J p mi i
Z-
k
" # !
( Vi
G V )iI
V
(cj )-1 (x)
W
[Zj ] Ztrj (W )(U )
w geq X T g u y b d ` g ahvccYchq
GW
t w bW b )fc$w
& '
U Flt(x)
...
Colim Ztr (Vi1 Vin )(U )
k w P(W ) Zw (W )(U ) tr ^ (c, f ) : Z U k W U wP(W ) Zw (W )(U ) Ztr (W )(U ) tr
vP(Vi1 Vin )
w
G
(i1 ,...,in )I n
X
(i1 ,...,in
A(v)
k
)I(v)n
Ztr (Vi1 Vin )
zj
Z
Colim Zw (W )(U ) tr
G ...
Z =
G ...
wj
G ... x P(X) v P(V )
jJ
G
Zj
G
iI
G ...
iI
G zj
iI(v) U Flt(x) U Flt(x) vP(Vi )
Colim Ztr (W )(U ).
Colim Ztr (Vi )(U )
Z G
iI
w P(W )
A(v)
cj : Zj
Shv ra (SmAfnd/k)
X
v P(V )
Ztr (Vi )
G Z u
Z+ Z -
G U k V ^ ^ ^ pri : (U X U )k V J O X k V ^ GG O U k V O U k U k V ^ ^ ^
Z
G
A(v) GU Ztr (W )(U ) z c-1 (x) a+ G Ztr (V )
vP(V )
G Colim Ztr (V )(U )
U Flt(x)
V
f j : Zj
ColimU Flt(x) Ztr (W )(U )
A(v)
a-
A(v) k SmRig/k G0
U Flt(x) U Flt(x)
u (c) = a
O U k U k V ^ ^
G0. G0
0
& '
$ %
fc &v
d r) f ntr ar
G AfndStrra (k).
1
g k U y T ` W we b i T w y g i t t b $dc)vv)Bc$|0Y)ju wchetvqaUwve q&h YhnPv)f)vFnq& b y g de Tqq b d b U gq d T e g i W g b w y Wt ac$0g h g X d T u X t g ie `e d X W Xe d T )aYnfPchfffavYn& $Fg g w y Wt g td b $y Y)IS a 6$n d c&dda a d} d"c} a f c l aY Y )} r d r)m ( v&ddc " rn) n$d nYv)} 1 ( r d 6& & z c&dda a c} d 1 fj Ar& rn) Fi{7 r
(F ) F n (F ) AfndPST(k) (F ) aR F AfndStrra (k)
tvuh Yfa6ndf{cBmcafPnaf$f0)t t W d gs t w T d W U g T g y T g U te Ts X d u t g i W w b U w b i W c{g t gq w be W U y d W t w b U d T u h w p w x t Xe U b t t T g y T g U te T 0vchfrfY){nIihYcmcafPns t gq d y t t g p te k u d b p b te t g i We y i w 0vy$f"P$$Yc")P}v$he g $y $y c"f}"fivh d g d T p g d g i W y t qe X e U b t t T y Tg U t e T s g q w g e W t 1hff)cYPnvah$yg g $y hc)y$ce de b d y b h$hYP$cIdc$)7Ya"fd $)YI0uhc)v))fP$)ycafPnyg t gq e t te p i T t W d g y b t W w g p g y b U g d t g i ge ` bq b u b W Tq d y b u y T g U te Ts w y W t g $y Yfc"q)v~BcYncYPnY$u X d W w b p U w b i T w X d T u X t y T g U te Ts X d w y W t g t X d T u X t g y T g U te Ts X d u g i g W w T d Wq p g We peq b U g w te y u X d T u X t W t g qe h t yq u g $}|0fcafcafPnaf$mfndfvAYt"hv)$y y P$$AYn1ovah$lvF t g p te k u d b p g y T W d b u u T d d T "fh$f)"c})$cY}nu W w g p g q g d g i X W Xe d u b d u Tq g i t t b a"cahaYrYhf$)fh0Y)ju h ge w p w be W T W w g t X d u g i t W d gs t w T d W U g T y T g U te Ts X d u w y W t )v$)vY&d)0fmYfa6cYY0cBsn0YhcY$s$mg g p p b ""co
S(R) F aR G (F ) (F ) G 2 (F ) G ... G n (F ) (F ) G ...
g We y Yt$ft
Tq g i g We peq b U T Pyyfv"vhcaAhq
t d bq T g t b u w dchnyf)jl~
B(R).
g X p T `q T p T g p p b t Tq Xe U b t t T X d T u X t y T g U te Ts X d u g a"c)hnAny""cfPhff)yfcaflcYPnY$yhq
Ztr (Ui Uj )
iI
gW b w w b fc$l)$h
g b w g i b fWc$wl~tcw"nT t i d g{nTu g g e t t p i W d g y b W g g y b g d w ihq$hYPe$cT{c$)fwa"pfd $)UY$y W g t te g b w g|ffPhq~ ic"nT w $y We vcb tW gst w dW U T T g Ut Ts d w y W g p g Yfda6nTdfvg)BlycYPenXY$u${wa"ay efWnTApcfW$nTYty b y Td g e hff)zcafPnzi Xe U b t t T y T g U te Ts y | wchfWYfWwcaUv"pahe g$agi$$yfa$w$w)biiw)vhnTf)$wchgocbc)XvaU bfYcTmn0YhcTs be U y d t b g y g w d g t bqq t y b h Tq U d y e e t t y T g U te y iotcaUvqo)cwaYwvcT"~gfvndf"~ T g W w T g W e p We T d W w d y b e W w gW y p p b c$f1)U W T Wq y t X d gq 0&dt$fYyhz b 1$
GV
T w b z)$h
X d d T U gq g fYnv|$y 0vhaYW t gqq g t ge w p t g U w T i w b u t g d d b U t g i W h t g i b w 0vi0c$)fYf){0vg c$AnT 0wnu t g i d T g i e w p gq e t te p i T W w g p g y b U g d w y g W te g g qe $|h$vfYh"){fd $)0fryfhvh )vYtf)jcY$"Pwnq w b e W e t b u b d u T q d T e W y bW d y b $cfw$cju g y aYW q g g i e w p gq e t te p i T W w g p g y b U g d w {h$vfYh")oa"aYd $caYm$y We t)b $ t w T i gqq y w W t c$~hvf0g g y aYfc"~ovcnT j d g d W w b p g i We ` t q t w T i gqq n~hv$y w W w g g i e )ve $y chfr01$$y w be W U g t g w W g t g t teq t g i b w 0YPv0)$A&T g y g i g d W w g ge w p g U w T i w b u t g d d b U g w $ayYfah$~n$)fYYc$y W w ge ahcb t g U w T i w b u g d d b U t g i w be W U q d T u gq T W t h e n$)fYca{chfcT ({c|vnYhcg (h ${Yfa6ndfv)Bn0YhcY$u )$wg g i t W d gs t w T d W U g T y T g U te Ts X d t y b t w y W t yv g i y T b w gq g yn&$uhu$y aYfc"u$wh$cj"ft aYfa6ndf")T d g d W w b p g i Tq g U d y b u W y q t W d gs t w T d W t g i W g p i g $y Y)cT IufYc"~ id b i gdW w b p w t ge W d T u g y g i w g g y g d u Tq g t e i w g d y W U y d W t gqq g W g w y g de y d W t w b U g hffnlac$f$PyPe v$z~)f$YrYfhvf|$$Yv$Yf)i We ` q W d g t w T d W g T g y T g U te Ts X d u g i t g d y W U y d W t t g i e d be d u T t T u W w w te T p h tse We i i T W w g p g de Tq U W w b vccT t7tYfa6scYY{U)Bmn0YhcY$|o0ffrfY$YchY$})f)b IfPnAtYtc{)"aYvchafcft W g gn0YthcYd$FgI& ohUYcT cYPns y T g U e Ts X u t S Xe b t t y T g U te T gq u t g X d T u X t y T g U te Ts X d u g h00fd Yn}cafPnaf$fhq ut g IfYd g p yW y bU g i g p p bU gW b w w "$f)"){fc$}~ We b g Ue U d g g g w g g X t te Tq T d g t W g gqe U Ts W t g X We Ue w vc)PdaYPnP}dafv|hvac{yfvP$$y iS 67 a~$" d y t W d gs w T d U T y T g U te Ts X d u w ${td6tndfWvgcBzn0YhcY$$y a 6$n d c&dda ) ac a1 r fj &A 0a &A d{ & d cndYa c ~ v~ | Y c ) qv r 0 6$ & d &dda r f r n v c a r c r ( fj &7 0r( c F ooii We Ts y i t d bq T gq y b U X i W T Wq y t X d g S tczY)vc|h$)${nYv$fY g $y hfW tq g t g i gq p g t w 0yv$1afag q U g 0cBT tvuh g i w b w Wt )he zc$wg g $y
I(v) I iI AfndPST(k) G ara (F ) F
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
SmAfnd/k ara (F )
"
"
U
Fsep
V
ara (F )
U
GV f: U Fsep (a) a|Vj = 0 k
k
aR : S(R) = Coker
iI
F
Fsep
F
f (a)|Ui =
jJ
R = (Ui )iI
Fsep
F
ara (F )
(F )
(i,j)I 2
ra
F
jJ
v Vi
f (a)
G (F )
iI
fij (a|Vj ) = 0
U
L
R;S(R)F
R;S(R)F
Ui
R
aR
fij : Ui
P
i,j
S(R)
f
fij
k
k Fsep (U )
G
G Vj
(F )
jJ
Ztr (Ui )
F
GF
Vj
ra
a F (V ) (Vj )jJ
G Ztr (U ) = B(R) .
U
F
I(v)
N
W w g i X g p te k u d b p w i g t b ue i w a$he a|"fh$f)$y 7yf)jv~
V k AfndStrra (k) AfndPST(k)
y T g U te Ts X d u w y X w w b i W w T W cafPnaf$$y$cvcYq
SmAfnd/k
X Y k homAfndPST(k) (Ztr (X), Ztr (X))
a
cac
atr : AfndPST(k) ra Ztr (V ) ra
AfndStrra (k)
k
F
F
SmAfnd/k
Ztr (Y )(X) (Ui )iI G Fsep
n
FF
&"o ac $ fc fuiu
ad
r&
d1$&)1
V SmRig/B V
w b e W e w p X i T q W e T s w chfv$a"hwvnz~ c)vYtcvYc)afnaU cfw$$ug ~ g e We i i T ge d b ` X W T t y b t g w y W t g i g y g U W w b t t W g b t gq W w b "aaucfiYacq0vo))i g i g we gq u ge d b ` X W T t y b t T vvv$ovYc)aY&aU cfPq gW b Yn$w w te w p t We y i b d u b U d T u gq T W ~IP$vYv$fcn1vnYt g we gq u ge d b ` X W T t y b t g w y g p w g t teq g ie `e d X W Xe d T vv$|hf)cf&U )"$$1g ~7)YPv1h)vYYhfcB g w $$y We vcb ad c" w )l d r d &z E Arrad & a n j r& r ryd c r c n ~ y &A f r 0 j ctr ar f c y Y" zinc r& n ctrar &7 F uw7 g W w T e y t w be We w p X i Tq g de Ts U w b i W y g u w cY)c&v$fchfv$$Ah|Yvcvcv~
^ T Y S G (X B Y )Y (Y B Z) ^ ^ ^ ^ ^ X B Y B Z G X B Z X k Z . ^ ^ B B B RigCor(B)
"
g i g ` T p ycc"he (q g te d b W U Ts g fhYcYrcft g p te k u d b p gq te T "fh$f)|hhcr t w T c$i t w T i w gW w bU Wt n$y af)vg g i W d b u u y t gq wfcj$$fvh W ag Wd b u u yt g fcj$$fi W g t ge w h$p tan$cj0fYca$a~a$c|nYi chfvf)jcY$AP~Pvhf$y 0)$A&T g U w T i w b ut gdd b U g y g i t g X w w b i W w TW w be We t b u b d u Tq d g teqe W t g i b w t g i W w b 0~cft W ag g $y $yu d gt b u u yt W y g w b h Tq g U d g pe d X d y b e c)$Paaiavfcicc t w T c$i ge b w hcc g t ge w p t g U w T i w b u t g d d b U t g i w be We t b u p b U Tq h$0c$cjfYf)f)vYtf)j"ca1P)h $f|Yhhq d y t g t te g ie `e d X W Xe d T Pchf~favYn& g p e te b d W g w y g X w w b i W w T W "ahfh)fYv$va$c{ndq tc$$y af)U w T i g w gW w b U g )BT t g de T W w g p Xq X t g U w T i w b u t g d d b U g i g de T X weq w b te T we p b U g w y W t 0fhnd)"ahan$)fYYc{Yvc$hv1cfhch$}){$$qg h g i b w T Wd g y b w ))$A&c$)y W y b W d y b e g W w T e y t w be We i w b U Tq W w T pe d X cY$c&cYn&vAchfv)|PctYa) h$0c$cjfYf)q$"Yc6s t ge w p t g U w T i w b u t g d d b U t g i X p d b g i g u y bd t y bt g oj$cYc` cfwvq gW b w w Yn$~7 w0YPvwcft d y t t g t teq W w b W g g $y $r~a0h)vY{0fhYn& g t b u u y t w t g ie `e d t X W Xe d T g y g $ai W g h g ie `e d X W Xe d T )h)vYyfavYn& g w $$y W w ge ahcb
X GY : X ^ T Y S SmRig/k Y : Y ^ X k Z |U X Y Cor(X, Y ) G Cor(X, Z) k GZ G RigCor(k) B B Z k CorB (X, Y ) CorB (Y, Z) ^ ^ T X B Y S Y B Z G X k Z ^Y S ^ T G RigPST(K) o G RigStrra (k). CorB (X, Z) X Y [T ] T k
( )
"
$
t0vYc)aXfnayi|"cY)nPh1~j ge d b ` W T U g g p p T d ` Te i gq T w g W w gq T e yfahn&h$y X Wtg1vYdcb)aY&TaAYo ge ` X W U g W W g g W b w W t h t gq t t p i T W d g b t W w p g y b U g d t g i ge ` bq b u b W Tq d y b u y T g U te Ts w y W t aXfc$y0ga0v$heffPe"ctYfa)$yc~Yag"fd $)Y|$hc)v))fhy$)n0Yhc$yg be W Ue d W g d q w b t W d gs w T d W U T g T g U e Ts X u t g g X p b wchfhYftYThlW))iAYfa6tcYY0gcB$ynaftPnafdAir"fd)6s g i g we gq u ge d b ` X W T t y b t T rvv$lvYc)aY&aU cfS a } c" d r aa I &c1 a ) rm & d cn dYa jv cr ad a fn Yrdr& f f r & &c z & h c f c mr c) a j fc fd r&{ ja m dAj & r rn) & ad rn)0a 1 c cm A f )" r(a" n j d &A r Yn v cc dA yw $nc m (dr r & d c c) v lad c ctr fj c r ctrAd f d f ) j r& y r&f ttovF
Ztr (Y ) RigCor(k) AfndPST(k) X Y Cor(X, Y ) = Ztr (Y )(X)
Fuw7
de w p X i g i W g p d g u e U g yv$v"f7ao g d yq U w b U d y b )f$hcwcju ) )vYtf)jcY$hPc6|"vy$$wPvhf$y 7v"f w be We t b u b d u Tq te bs g p e g y g i g w y d g teqe W i W y t q
iI
g $y
GG
i,jI
g d W y T W t w g k U gq y b i g W W g Yf$nvg I|$d|h$$)ffo
iI
g k U gq y b i Tq g i d y g W T teq T ` q t d bq T g pe W w g i $d|v$c"P$afnfPhn)X Y)vctYaPe t g u y bd j$cYc` g S t W d gs t w T d W U g T y T g U te Ts w y W t Iad6cYY1cBn0Yhc0g h c chfv)flPynq7Yfa6ndf})BcYPn$hf)cf&hyn$i w be We t b u b d u Tq d T e t W d gs t w T d W U g T g y T g U te Ts t g i ge d b ` X W T U Tq t w T
homAfndPST(k) (Ztr (X), Ztr (Y ))
i,jI
w be We t b u b d u Tq d T e chfv)f$hicqn W w g p g y b U gd w a"aYd $caY$y
g i y T b w b U y T g pe W w g i ln)&cnvfhe t h W y bW d y b u g i b w T w y Wt )fqcj{)$A&m$ig h Xd T u Xt Wt cYnog gp p b t g i b w T tWd g y b t g i d T ""co)c"niYfa)$ciucu g i We T i g w b g i g p p gq y i gq y b U X i w be W d g t t q h g i b w vcb B$$0$ciu""ahoh$)$|)vYYcT cc"nT w y Wt $g g td b $y Y)IS 67 a~$"
X Ztr (X) X k X Xi X j
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
B AfndCor (B)
B k CorB (X, Y ) U X ^ ^ U B Y U k Y Cor(X, Y ) Cor(Y, Z)
B k Cor(V )
Ztr (X)
A
AfndStrra (k)
homAfndPST(k) (Ztr (Xi ), Ztr (Y ))
X
SmAfnd/k
A
A
Ztr (X)
B
k
RigPST(k)
G AfndCor(k)
k
RigCor(B)
Ztr (Y )(Xi )
A
Ztr (Xi Xj )
RigCor (B) k AfndCor(B)
RigPST(k, A)
GG
G AfndPST(k) o y
F
i,jI
G
Ztr (Y )(Xi Xj ) .
homAfndPST(k) (Ztr (Xi Xj ), Ztr (Y )) .
V
iI
Ztr (Xi )
Ztr (Y )
G AfndStrra (k) y
RigStrra (k)
RigPST(k)
AfndCor (B)
(Xi )iI (i, j) I 2
CorB (-, -) RigCor (B)
SmRig/k
SmRig/k
A
r
j
"
r
F : Cor(V )op
GA.
r
6$n
(
d ndYa f
7F
c) ( u{7
j d A j rn)Y
d y t t w geq X T t g u y b d ` g i tse We i i T g y T g U te Ts X d u t g i t T $fwvhajn0$)f)y{vfv$c$nafPnaf{{ )aU y W w p qe T W w te T d w ` nTwag"aghhUcsu)gPhqndaX$Xc|gt YWfa6stwnddfwgcBTmycgafPnaf$f0v$fcYfcjchfr0|ffa|$od&Yv$fYfIS YnPfc&he h t d g T W U g T U te Ts X d u t gq d y t W w T W d b u h w be W U g t g W W g U g i t W T Wq y t X d t g t W w Te d T w gcgafPnTaXf$u"iddchnTzYdvYdnu~ y T U te s d g t bq T gq T w d t t d gs w $y|dW6tcTYdYW~nYn0YhcY$cj|af}dnmw )vYt$zS t w T t g y T g U te Ts X d u t g i d y b u t w g t w y g i d T ` v w be We w p X i T n 6$n d c&dda r y r r v ( { ) aj u d ) f nj r & o a fj F
} W dhYd T ve w &d g e u W b p b k d T u W w T e d T w a &A Y j"$&)A a j wc1n& 6$n d jvd h$cY n"c${c)chYn& he y Te b d r&Y r ( a &7 FF uw7 g{tfYgnT{0ffYve$Yd$$hcg (e heftYYvi{dteyibfd${c|vnYv|vhag Iy g i y t t i y t X d p t W y i b u h t w p W U g de t W u d T u gq T W t W t g gqq W g W T t g i gq y b Tq W w ge W w b Y&Fuvcj1PwahfcaU g $y h$u|iyb$t~jh$pudtyfd$)bwcT|vq$cYt t yq g g t u u y w te w t We i b u U d u g T W g we gq u ge d b ` X W T t y b t g w y g g p w b h g We y t Tq t w T $hah$whf)cf&U )y$$y$lc)fv$fPc~ f d 1 n "c u ( $v&
B1 X i CorB (X, Y ) GX
0
" #
" &A
q y w X d ` g i w g X U Tq (~f)aza)h$u w be We t b u b d u T S chfv)f$"
1
a r r| & Y cfh&A& fc V rmrad r) cac r) n ntr r ntrar
nN X Y B k
d|aY r) c a y t ntr ar &7 Ficuii r s chfvft)jcbY$hqy{t)ziagjaX$|g} yc$f$P|$y davYac{afhy IS w be We b u d u T g i T u w u i w g i g y g d u Tq g T d g pe d X d y g W U gq g y g te T X d u cTUYPnsY$gvq i w d u q gf$lwcb (l b}) )vYWtf)bjcY$1hahv$ahYcfcl$0fY$i U w be e t u b d u Tq g i d geq y Ue W d T u t T U w y W t g t y t t g e b W pX q i e q gU p p gq t g be e w X T g gq g hy$y hnPvI de Tq U W t g q w)veYtewiATPygyvg$y g"aqpyhwW01) wchfWv$paiAPqyi Y E w jv& n FF uw7
rig
"
"
d y b $cju U g 0cBT q Te Ueq u pe t b nPPv$"hU t ge w p t g U w T i w b u t g d d b U g i t d y ge d X u y t t g u y b d ` t g i de w p X i W w g p gq T ` X W y g u w h$7n$)fYYcFYahf$f0cYcqh$$a"ahnaqaj~ g icb$wAnT hnYhhhf1ag gq W w T teqe W y w c f m 1 & Y&A& r c r )r r & a axaE c|a n" d"ja fc } ~ iFrdjnc & |r) |i n ac r ) a idAjadn"n c h d Yn n c ) rn & ctr ar F i{7 a
X Y B 0 CorB (X, Y ) AfndCor(B) RigCor (B) AfndCor (B) 0 RigCor(B) 0 AfndCor(B)
ttovF g de Tq U W t g w be We t b u p b U d y gq W g t ge w )fhnPwzchfv")w$vwa{h$p t U w T i w u t g d d b t g i w b We i i q g T X Weqe e W T u p b T S 0gc$)bfYf)U{0chefv$cT vU0cBfvhv$hfn"caU$ W w T w gd u w naY$lg dac td g g i ge u b W b p b k g w y W w ge W b w yv$)fc)$y$${ahf$ccb h tda) d g g i ge u b W p b k g ~h$cYnb"c~$wy W g tYc d g g i ge u b W p b g y W t vh$cYnb"c$k$w$f0g ft ix e h w p w d y $t v)c a$y ge b w g e g i w be W yq b w lchfhc)he (q gq u g g g d h$"pafnTu fv$$Wvh$y wocbhfUfft0v~c$wcd$y fPe$$uYdc"cay X We w y g gqq t w e W g t gq g ` T k U X e g pt k b p b i w g w U g 0cBT W w T w gd u w nf$lag g i ge b W b p b g w y W w e W b w $h$u)fn")$kw$$o)gvYc")bch g ge u b W p b k g y W t iwvcfcb"c$ww$i0g e)cg$y e$cbnTaAw)veYrcf|hq w w U b WU g bd u T U g )BT g d i w g d g i W t qe h W g g w x YaY$uu"yfhnajaAin W w p g de Tq T w ag"fhnPU|"~&$ wchfve$pai|hq{$iqtwchfnYnhodn"~ 67 a~$" be W w X T g be W T W b w t gq g i d T ` w a iidj&)j 1c $v& c $ r& ( F zgw dja r a( v& jdn"n
B1 B1
)c
"
w T e y t gqe U Ts g p p gq gq T w WcBh$fyvh)|"vyhz~ " a1 Y&A& w c jA r djncj z " y$ Yay g veucbfWcb"pcb$k r h c $ r& d " f $ a r g u bW b p b a jv jcYn"ck ryd r) cac jucfcb"c$k a g b W p b ntr ar & q F u{7 g w $$y We vcb
g W w T e y t w be We w p X i Tq We Ts w b h f e c ` W w T e y g ie `e d X W Xe d T )fn&h$f)vYt"Pwtc)$Ajw#db a)c&v$)Pchf|favYn&
' 1 % $ 9 B 1 X W ' 9 % R T R P ' 9 B $% $ 7 B !(HAFY(VA(ICUICSQ!8&(IHGF(% E
r ut g I0fd rn) a &h
' 9 1" ' B $ 9 ) 1 7 $ 4 3 $ 1 ) '% $" !A#DC&!A@865&#20(&#!
rn) d F
nf
ad
ge d b ` X W T U Tq d y b vYc)aY&aP$)u &
I
&
r
W w T e y t gqe U Ts g p p gq gq T w n&vyhvac|""ahyv~j w be W T W b w Tq T d g d g t g d w chfnYc$zhYcYafY~ ut g I0fd a " d r c d $nc a ru & v d & &dd ar)v Yndy j c r Yj r&
RigPST(B, A) V PreStr(V )
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
A
PreStr(V , A)
A
dy
aa
PreStr(SmAfnd/B, A)
PreStr(V , A)
d "
)
0ja
z{
B1
"
Q
B1
= p
B1 s0 , s1 : X
B
p:
X{t}
k
G B1 X
B1 X
n = Spm rig
GX
:
t
B1 X
X
1-t
k{t0 , . . . , tn } n 1 - i=0 ti
B k ^ CorB (rig k X, Y )
j
CorB (X, Y ) i CorB (X, Y )
G
Y
B1 X
"
0 Cor(-, -)
' 1
1
X
Y
B1
B
+-p
B
AfndPST(B, A)
CorB (B1 , Y ) X
K
i CorB (X, Y )
k
k
k
B1
, CorB (X, Y ) = s0 = s1
Hi (CorB (X, Y ))
Cor(-, Y )
PreStr(SmRig/B, A)
iN
"
V
F
B1 B
F
V SmRig/B
B1
G i CorB (B1 , Y ) , X
V
X Ob(V )
"
F (X)
PreStrB1 (V , A) PreStr(V , A)
Y&A&
"
B1
V
G F (B1 ) , X
V
CorB (X, Y )/ B1 RigCor(B)
A
k
B
A
t y t t g e U g k U g d e p g d u Tq g i y T b w b U gq g p p b U e w p X i X W X ff0i a|dYahf1Pyc)&$)h|")$YT
hom(B1 , F ) B
s -s 1 0
g X t b u p b U Tq g ")"hyy
d g d W w b p g i T d y t qe fY))"y1dAfvh
1
g i W w ge W b $v)vYn$y
w y Wt ${g
g k U Tq g d1Py$y v)g $h h e d y b cju chfvf)jcY$"h|i he w be We t b u b d u Tq g e w b We w b T T d p X chefvi)U|hqvYgvefdcw~)wa"paYrU0fhedaYfWcb"a$iPW)"ahn)iaj}~ ($y ff)a"af)hu W g g W g d i g d w p X Tq w g p gq T ` X W y g u w q w X d ` g i w g X U Tq y T g U te Ts X d u g cafPnaf$hq W ag iaYd$zwcb (qz b}) chefvtu)fu}Pq|i{aghv$yahYdcftcUzw${g w g u w b We b b d T g d eq Ue W T u T y Wt g i ge u b W b p b k d T u g U w Te d T w |v$)fc)$n~chYn& he uS 67 a~$" j n1 n c a j r &a
' 1
" #
a0
rn)Y
d}$nc a
1
&y o r nf& v )
G F ).
j o & ry 6& d ndY a r r a n dB a &dd j Yj
hB 0
0
r&f
dm&)" a
r
d &dda aj
f V
g i g p te k u d b p g "P$$Yc"|vq d y b e $cc W y bW d y b $cfi$cju g ie `e d X W Xe d T g w y d y b e $h)vY}YhfcB1$$$c
g i We y i X i t W d gs t w T d W U g T g y T g U te Ts X d vv$0Yfa6ndf~0cBcYPnY$u Td gW b w w b daYn$lch g i t g ie `e d t X W Xe d T g i g p te k u d b p w o0PchffYhfc&~|"P$$Yc"z$y d T u e w pX ci d y t y T g U te Ts X d u g $f"nafPnafwvq Td gW b w w b Yfn)ch n$i t w T g w w geq X T ge d b ` X W T U g w y t w T i t d y gq T $$vhajn1vYc)afna$$~n$d$ahn&m d y t t W d gs t w T d W U g T y T g U te Ts X d u w $fvd6cYY|)Brn0YhcY$$y We vcb
F GY hom(X, F ) V V p s0 = idB1
ttovF w y Wt $|0g i Pe
X)
g i g k U y T ` W g g We b d i g t d g w |$dc)"w|fvcYA|da) ve g $y Yfc"lch XdW w b p T w b g y $$)nYij X w w b i W w TW gq T ` X W t vc)~g g i X We W w g i q g "fvfa$he ($y Yf)p gdW w b g ie `e d X W Xe d T Tq g X eq u u e w be We i w b U T PchY}favYn&lh1$y h$$cT ve )vYtclS
se W T W y p p b U g p p T d ` Te i gq t w b d X ie t w b U vfnY1"c|""ndcnPhfcYaPfcah
^ k : B1 k B1 k G B1 k
d y b e $)$
d T u W w Te d T we W t n|nPfcB h0g
WggwgtwcbYWnbw d y t g y T g w w T w g t W g i g d y W U y d W t Tq T d g tqe W y w h g b d ue U X d w be W T Ueq u p q d g d W w b p d y b $fln0$$cz{Yacb 7yffrfY"P1dafPvYl)b c$y fvaYz)vY&hh$"ve ffYc"{$ce he e w be W T Ueq u p q g y b d u e U g W g gq e t d g we W t v$hfYc vwg ge u b W b p b vcfc"c$k e )vY&aPh$"ve (ccY$70oj g td b S $y dc g k U Tq g i t w be W U g t t g i W w b $dPAy)vYr0y0}cft W ag t g kU g y g i t g 0$da$yIS 67 a~$"
X Ob(V )
" #
d"j$&) a r&f ) l "$ d (
r cac} Yf h Y&A& a | dr) 1 j f ry 6& d E&dd r a &7 $ w ) j n ntrmar
p : B1 X GX k F (s0 ), F (s1 ) : F (B1 ) X F X Ob(V ) G F (X)
FF
y & I r o lcac}I j ad )A d icuii
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
F
"
A
F (p)
B1 k
X Ob(V )
F (s0 ) F (s1 ) F (s0 ) = F (p)-1 = F (s1 )
^ B1 k X k
^ B1 k X k F ((s0 p) idX ) = F (p idX ) F (s0 idX ) F (p idX )
g i w be W T Ueq ue Wq y p T chfnaPv$hfv$1"hq
V f: X
F
^ B1 k X k
s0
s1
B ^ k^ R B1 B1 k X k
0
A ^ hom(X, F )(U ) = F (X B U ) U Ob(V ) G hom(X, F ) f : hom(Y, F )
X
S
@ G B1 k X. ^ k
V
s0
^ F (B1 k X) k
F (p)
s1 V
B1 k
F
F (0 F (s0 idX )
A
a
l
hB (F ) = CoKer (s - s : hom(B1 , F ) 0 0 1 B
r
6&
1
1
PreStr(V , A)
hB (F ) 0
h
)
r &dda aj F foiu n &dda j r ar $ a fj )
B1 hB (F ) 0
F
F
V
A
j
m
&7
FF
a $ uw7
n=0
K
1
B1
F
hB (F ) 0
1
hB (F ) 0
hB (-) : PreStr(V , A) 0
1
1
s - s : hom(B1 , hB (F )) 1 0 0 B
hB (F ) 0
1
F
g $y
hB (F ) 0
1
h te T gqq w W t aPnr chv$y wg
(
GF
G PreStrB1 (V , A)
G hB1 (F ) 0
G hB1 (F ) 0 X =
X w w b i W w T W gqq w W t $$c$v)cYqchv$y wg
h
B1
W g
g "$y d y b u g X w w b i Tq W t g ge u b W b p b k b U gqq g W g $)1a$clh|0h$cYn"c$)vhaY}$w
1 rig B1
q g fW U g 0cBT
g p te k u d b p w "fh$f)$y 7i W y b )fW T d g pe W w g ie w b h Tq g U d y b e t g u b W b p b k b U W w g p gq Te Ueq u pe t W w b davfhcj$Pa|$)Fa0)fc)$caa"vchahh$"vf)t
a:n1 F K s0 , s1 : rig
t g ie `e d t X W Xe d T Pchff0fhYn&
t g p te k u d b p g y g i t gq g de d U g X d W y g u w t g g gq u p b U g i t g p te k u d b p g d W w g t ge u b W b p b k w g t gq Te Ueq u pe t t ge u b W b p b k t g ( "P$$Yc"FhifhYf ~nh$"caoF"fh$f)ufY)0vcfc"c$0vchahh$"hh$cYn"c$F0vq g p d bs t w T d W Xe U b t t T gq u pe t g g gq u p b U gq g "f)6cYYqhYcfvhfoh$"caohu$y vnohg chfv)fwPnF"fao&a0)fn")$a"vchahh$"vfF)t We Ts gq W w be We t b u b d u Tq d T u W y t e U g t g u b W b p b k W w g p gq Te Ueq u pe t W w b
s , s : hom(B1 , hom(rig , K)) 0 1
Tq w f e Pl~j Awc nYda f
S
g y Te Ueq u pe t t W g i t g p t e k u d b p g y g i t gq g chahh$"vf{Yacb 7{ 0fP$$Yc"l{h|$y daYf)~0$aYr0vyfv""cAP{$)laY$fd T d g d W w b p w d y g W U gq y i X We i b p p b U Tq d y b u i w g d u g g `b hq$u"anfhc) $y Y$chnc|hc$y$y )$aY$|"aqhzfaah${g ~ 67 a~$" p g g g d T u de b g e d X `q T t T U gq t w T i g g y g d u g p p p Tq y i w g W w g w ge W t & r I Y&A& I r f nYv&A& a 6$& Y c&dda cv Y m f r ( fj jv& n F inoFui
Hn (Sing (K))
t g We peq b U g y T g W y p p b fv"vhcalcf1)U d y g W U w b gq h ge y Ue W $aYr)6sAvdvhq$PffdnTuEi w x he t we g g gq u p b U gq y b i g p p b U g vvcav$")h$$)""cl$y Xe U b t t mvaYcT gq pe t g g gq p b g W t h$u"hAh$u"caUvqg X btt AveaU fYcT gq u pe t g g gq u p b U g S h$"vfmh$"cm g i t g g gq u p b U t g i ge d b ` X W T U Tq t w T i q Te Ueq u pe 0h$"c}Ahf)caY&hyn$uchahh$"ht W g w acbz$y Woh$nd$1c$Xodtef$)$nda$c~hYccfna|$$y g t gq T d p b w i t W y i b d u b U t g i W w T W W g p i T ge d b ` X W T U g w W w ge )v)b R h77F7B$ED C H H 9 8 % # $nc} a t g ie `e d t g w T k Pchf0$dU y| r a f Va dB l Y g i g g u p b $mg avq)U y & wn jv c jv jv& q &cm a r 6$n d cndY a )l Y r ( f jv& n i F uw7 y T g U te Ts X d u y i w be W U y d W t w b U Tq g teq T d X w X ` t y b t t g e U w be We w p X i T cYPnY$zchfrfY)"P~fhhndaacf)Y$i chfv$a"S g y T g U te Ts X d u g i t g gq u p b U g y T W w Te d T w 7n0YhcY$y{0g av)l$n{)chYn& he
G B1 k U ^ k B1 U
t ge w p t g U w T i w b u t g d d b U g y g i t gq h te T 0vi0c$)fYf)l$aw0vPnr g y aYf)h )vYtf)jcY$1hzfW d gdW w b p g i w be We t b u b d u Tq g i y w g d T u g te d b W U Ts g nfhYcYrcft g $y cf)j$$fc0)"a$y fvaX t w b t b u u y t h W w g p g b d ue U w be W T Ueq u p q chf&hh$"he b }
(i) (ii) F
ttovF g i X We W w g i q T t g u b W b p b k W w b YtYaPe he 0)fn")${))t )fv$$vh$y ochfff0vq X We w y W g gqq w t w be W U g t t g U g 0cBT gW u p b U h W yt q f"cac"fc n$h)vY{favYn& t w T i g ie `e d X W Xe d T g w $$y We vcb
V
t d bq T T w dchnA~j
g i ge u b W b p b k g w yv$)fc)$$y
W g
W w ge ahcb j
"
t g U w T i w b u t g d d b U t g i w be W U n$)fYYcvchfcT (q gy h t w nT$i Wg d$)bFdfYWc"pl$i"vc`cT {)gv$u)fWcp)$"nnPfcB hg y u h gd w b g We t q e b b b k d T u W w Te d T we W t g $y cf)j$$ 67 a~$" t w bt b u u y $nc )1 & $ ~Y cc{ j 7c d a &d r r ( r& n"n &A& )c a " f v a cfa a f r& & Y&A& I r o a |r n ry j lcac} ja " Y j&)1 r| 6$& d ndYa f ad r& r ( f &7 F inoFui g i g we gq u ge d b ` X W T t y b t T y$hah$vhf)cf&U )"Pq t d bq T T w t W g b d y b ~dchn"~jY$c{$)u W w TWWg p i nd")T
PreStrB1 (V ) 0 Cor(V )
"
ttovF gq h W w Te d T w hc0)chYn& he
y T g U te Ts X d u w y W t n0YhcY$fg
et g f~$y
gq e t d g we W t ch$hda) v{g g teqe W y w b h w be t yq U w q g i g k U y T ` W we b i T W t PhvY))I)vfhhe (~dnfhcB)g
1
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
0 Cor(V )
0 CorB (X, Y )
Td gW b w w Yfnl~ g$y $c$cay"P$$Yc"p e w b w T U g p te k u d b g y aYfc"f$ce d gdW w b p d y b
F
B1
hB (-) 0
H
F
F () = F ( s0 ) = F (s0 ) F () = F (s1 ) F () = F ( s1 ) = F ().
1
0 Cor(V ) F (Cor(V ) 0 Cor(V ))
A
K
G hB (H) 0
A
K =K
^ s , s : hom(B1 rig , K) 0 1
K =K
F
F
s0 , s1 CorB (U, B1 ) U
, CorB (X, Y )
CorB (X, Y )
0 Cor(V ) U B F (s0 ) = F (s1 ) s 0 , s1 : U
A
S(K)n =
0 RigCor(B)
G homA (F (Y ), F (X))
hom( , K) rig
CorB (B1 , Y ) X
n K (-) (a,m)ZN; a+m=n Ka (m)
HOM (0 Cor(V )op , A)
G hom( , K) . rig
G hom( , K) rig
S(K)
V
k
w g g y Te Ueq u pe t b U t W g i t g p te k u d b p g y g i t gq g mnPPvhf)vY$cb 7"P$$Yc"za{0vyy
"
h(0 : n 1) = d0 id 1
h(1 : n 1) = d1 id 1 h(a) : n rig
G 1 n rig rig
G B1 ^ rig
) ( d gdW w b p g i U w b i W yt q fY))"y|cv"f c (
(
"
Ob(V )
H
Sing (K)
F : 0 Cor(V )op
B1
V
X
nN
Y
K (n) B1
V
GA
A
(
'
t X Weq T ` X t g Ythna{0vq d T u g te d b W U Ts g n|fhYcYr)|t
"
A
j
rn)
rn
r
g p d bs Tq g i t W g b t g i t g X p d bs t g we gq u t ge d b ` X W T t y b t t d y gq U g "Yc6P|{Yac{0{"f)6f$hah$i0vYc)afnaU cf{Y$v~)BT W g TYgvfwaPecbYteAP{c$hwj d pe W g i w g W y t Tq t w T i he t we t w T n$i W ag g d W w g g p te k u d b p b te w y We y i w Yf"fh$f)"cfh$~t$$ve giXfvefWag$he (q~$y Wag"fW&Pi0p"hewgvfcA~cfaY$)vYt$Pv$u)vY&Yc$i0vuc$Yc|R h77lF7&ED C W w i g w p g Te X p pe d X w g W w g i X U X d u w be We w p X i Tq g i t w be W T W b w t gq t w b i d T H H 9 8 % # ia"j ad r dzj a { & d Ec&dda i r" 6$& d VnYda & fj r ( f v ) c) m a rdr) l a z fc w &
homSmAfnd/(B,p) (X/U, Y /V ) = Colim ^ ^ homSmAfnd/W (X U W, Y V W )
c
r
& )
r a 0 ad c Ic r & nw c ) n$
w b e W e w p X i g w Ichfv$a$$y d T u g U w g p p b U w q Te W y d U gq d w y T d g y b h ge u b W b p b k d T u W w Te d T we W t cwaa""ca~ nPfY~v)myY$cach$cYn"cununPfcB hog g y chfvcaS g w be We i w b U T w g i a hydahhaj$nT Tq T d gqq g u u t W d gs t w T d W U g T y T g U te Ts X d u w Yfa6ndff0cBAn0YhcY$m$y w cb Iy 0g gU Wt y T g U te Ts X d u g S t g t teq t g i b w n0YhcY$w)ffPvic$AnT t g i g we gq u ge d b ` X W T t y b t g w y d y ~$vvohf)cf&U )$$ut d e w p X i W w g p p b U w b e W U g t g W W g U t w T i T d g d W w b p w h$a{)"))vYr0|ffc$daYfc"m)b h y $$d0fA)vYt$ca$}ocfPnhY&Yt g e w k U g W w be We i w b U g w y W w T te Ts te W T t w T n$i $y f$a)fhcj$y g e d X w X ` W we b u w X w w b i W w TW $c)cYq Yhh{PchfvfhYn& $$y 7ufYmcvvcfc"c$icinPfc&hid6ndff)BAnafPnafA$y g t teq g ie `e d X W Xe d T g w i t y t t g i y T ge u b W b p b k d T u W w Te d T we t W d gs t w T d W U g T y T g U te Ts X d u w We vcb
9 B $% $ 7 B FH(IHGFi% E
' 9 " B $ 9 ) 1 7 !A1 D' HI0@8#$ H2Di&$ !CR B 3 ' !C% 3 $ 1 ) '% " 1
ttovF td y gWU g ad$aYr0vq g y T t X t te Tq W w b t W g t i d T i w T W t W w b t t w be W T U pe d X t g S $nyfYhchycf}|$dnndcfychfnavYac|0I w be W Te b t t q d T u t g d `q T t g i y T g e w y T X w w b chfnhaU fYcT (ycY$a)vcn0ch$nm$ci g p te k u d b p g "P$$Yc"vq $)}YaY$mcb d y b u Td i w gd u w e $ W g e ft d T u e w pX i Wt nj0g g $y $t d gt b u u y U w b i W y g u w )ajr~ gq y i b h$"p y i g p te k u d b p b W y T w y We w p X i g Ue d W T p g W W g }"fh$f)"cf$n|v$a1ahYfnA1Yo t w T i t d y gq T gq e t d g we g Ue d W T p g w n$d$ahn&mvvfYc h}PYf&A$$y W w g t te w p X Yhi d y b cju t g i t X w w b i d b b U t gq h t d bq 0$$c$Y)awhY)vv W ag d T u te w p X i W w b cwP$aw))t W g g $y g W W ag g $y $|aj~ 67 a~$" d gt b u u yt W y g u w d } j| Y a n n}aB c &v a A r v a rff ) ) d | cr a $) d fajY ) cac Ar n$ $) n F zgw t ytt g e Y$i U g p p b ""caU YPqhe )$A&A"P$$Yc"$y0g g Wte g g q t h g w T g p te k u d b p w y Wt g $y )Yh~ nA$h"ffa}a"a$y $$yg $y a"ahvacs U w b i T de i w d T u X we p d g W X i W w g p g e w y W t g W w g p gqe U T gvfcr~I pe d X w W w Te d T e X g p te k u d b p w i W nPfc&v$y }"P$$Yc"$y Ig t gq y i b h$"p g i g p te k u d b p w i g X w w b "P$$Yc"y 7a$ci jE d TPq $A&y"fh$f)~)Byfc6){cy$ g w T g p te k u d b p U g T g d i w bs w b U t T u g w gqq g u u T w b W ag t g ie `e d t g w T t g U T u t 0h)vY"n{acfg c hvnc$h ga~$$c$fndq) y g i X w w b i W w TW g p te k u d b p b te w y We w p X "fh$f)"cfhmov$$i g i W we b u w i ge b k U g S g ie `e d g w T g U T u t g w y Xq g u u T W t )hcjm$y 7At)$d~))h)vYw$Anv)0m$~v$cfg chfW w be e i i q d y b u t g u y b d ` w g d y g W U w bs g p p b U X d X ie t w b $)T ocjqj$cYcaf$aYr)6{")vYa$hfcaU g i g e We t w T d W W w g p gq u pe t w be W U T g w i e w y ~chfvcYYua"v$"hlchfr)~$y 7$$1p g i b w T c$Anw ttovF
G 1 k n . ^ rig rig
ge u b W b p b k b U g w y We w p X ch$cYn"cc$wti
0 rig
(rig (ae),rig (e))
g y
td""cf0I7 Wg p p bt t g S g i tWg p p b |Y""cft g w w be W T Ueq u u T g w y h t y b t t g e $A&T chf&hh$$c$o)Yi U
g i g ` T p q ge b t t ycnAhe vaU fYcT d T nu g$ivgc`cT"heoWag"pp)btmycTjy wchefW&TUhhq$u$cT q p e b e u d y b u t d bq T T d i w g d u w $cjqdchndf$~ t g k U t gq d T u t X d W X p T d T u W w b 0$di0vonq0fY"ndnocft g i t0gi~0cc"hh~ca$h"ff$|a"$y $g t g ` T pe t gq d T u g X we p d g W X i W w g p g e w y W t g p p gq gq d T e t g w T t w be W T Ueq u u T t g ""ahvznqfa0$A&zchfnaPv$ni U g )BT q g w y T de y d W t w b w fW Yh$YfcaU ~ gW y p p b cf$}"caU
1 1
W w T e y t X d d T U gq )c&v$f|fYn|h$h
d y b cju
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
h(a) :
n rig rig (e : 0 n)
n rig
c:mn
X
h h(a) G 1 k n ^ rig rig
h
0n
n rig
m rig
c
h(ac)
h(a)
G 1 k n ^ rig rig
G 1 k m ^ rig rig
Bm k X
de Tq U t d bq T W t g q hnPdchnvI
k 0im
y0
F
x0 , . . . , x m
xi
l
p
F
t1 = = t m = 0 k tj = 0 i = j
(X, Bm ) k 1
X (k )m
p
B
B
(X, Bm ) k y0 , . . . , ym
Fp ^
(Y, Bn ) k
k p P(B) B AfndCor(U ) SmAfnd/U U Flt(p) (X, U ) U Flt(p) X U Ztr (X/U ) Ztr (Y /V ) (X/U )
cac} & 7c c d r c ~ Y| dyc r & ( Yj r)v zc 1n r&a & &7 F u{7
B
l : Bm k
X = Bm = Spm(k{t1 , . . . , tm }) k s1 = = sn = 0
Y
(Y, Bn ) k
F
ti = 1 sj
f
G Bn k
Y = Bn = Spm(k{s1 , . . . , sn }) k xi 1im B1 Bm k k yi = (s1 - ai1 , . . . , sn - an,i f m aji ti i=1
B1
^ k(p)
idc
h(a)
f
(- Bm (Spm(k)) = (k )m )1im x- i 0x k GY f: X f (xi ) = yi
n+1
k
X
f : X
AfndCor(B, p)
B } jv a h
X
Ztr (X/U )
SmAfnd/(B, p)
AfndCor(B, p)
X/U
homAfndCor(B,p) (Ztr (X/U ), Ztr (Y /V )) =
AfndPST(B, p)
CorB,p (X/U, Y /V )
X/U
X/B
SmAfnd/(B, p)
W Flt(p), W U V
SmAfnd/(B, p)
W Flt(p), W U V
Colim
^ ^ CorW (X U W, Y V W )
X/B
F : AfndCor(B, p)op
(Y /V )
Bm k
n rig
B
Fp ^
SmAfnd/(B, p) GY l 0 rig G Ab x0 k
)y
& mcd & c&d a j & & d& I1 r fa fj
AfndCor(B , p ) G AfndCor(p ) ^
" &A&
c h & " "h
Yf 1n
&
a " ) fljfr & jv& Ar& & ar n ry c d arjdfj &Y( y ra
AfndCor(B, p)
d a r fr | & a j a ~ ya z && F ucuii t y b t t g e U w be We t b u b d u Tq g d W w b p gq g p p b U t ge w acff0i zchfv)f1P|fYc"yv|""cawh$p t g U w T i w b u t g d d b U t g i t ge d b ` X W T U g y i w g W t d y g W U w bs g ancj0fYca10yhYccfnaEnT aYX a$aYr)6mou g i b w T g d `q $)$A&mY$a)vcT Tq g i q T pe g T hlocAv&Ap g d W U g u t y T w be We w p X i d T u q T ` Yfjnchfv$afcnaX g t teq g i b w YPv~)$A&T gq ge U b t t hvhYcT e I$y gfc}|)"acndafc6$y 7}f)jP~ t T g i W w g p g ` w T k U g i d y g W U w bs w i g t b u te i w W g g pd b w gWW gU d y b a"f)$fa|$cju g i X W Xq u p b U g 1Yh$"cvq gW b Yn$w w ~ q g y ie t X d t u d b U gq d y a$PfYAYcrhm$ft gqq g d y W T w w be W T yq T g w i g t b u te i w b vhaY$Y&chfnhn&r$$y 7rf)jPsc )v)x)vYtf)jcY$Pmnqe w be We t b u b d u Tq d T
1 '
t g i g We peq b fv"vhcaU
g e We i i T ge d b ` X W T U Tq g i t g k U t gq W w b chfv$)vYc)aY&ahz$d0lhlcft
U Flt(p)
gd pXd fY$u
We w p X i w t g t teq t g i b w v$a~ffPhv0)$A&T t g 0i W g ndYa 0aY ) 1v ) f t &
Y /B B F(B,p)
g$kdU|v$ci gq y b W w ge ahcb $
X
r
(
&a d &dd $ fj n
F
&
6$n d nYda i f r n Fxl{
B k
t W d gs t w T d W t w T t w be t d g g w y y i w g W w g w ge W g p i T q g U w g i X g w y W t g W w T e y t W T Wq y t X d g ad6ndffcfzchda)y$$zaYaah$w")caa$he ay${0w)c&v$fw&dt$fY|IS
^ ^ ^ ^ ^ (X B W )W (Y B W ) = (X B W )B Y.
g $y
X w w b i W w TW $cvcYX
W w g i X g p te k u d b p b te w i g t b u te i w b he ~"fh$f))P$y 7yfh$z)$h
g i tWg b g y g i W w b {d)lwcft
W g
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
X/B
Y /B
^ CorB,p (X/B, Y /B) = Colim CorB (X B W, Y ).
^ ^ ^ CorW (X B W, Y B W ) = CorB (X B W, Y )
p P(B) SmAfnd/(B, p)
SmAfnd/(B, p)
W Flt(p)
SmAfnd/B
d~
6$n
r }ad
)
g y
W w g p g W Te i X p pe g pe d X w b h W g g w fnh0"h|vfczc$0ix
et h Wd T u gdW y f$fnYf$nT }
B
F(B,p)
^ F(B,p) (X/B) = Colim F (X B U )
U Flt(p)
F
p
T Pq"gi|d$yagYW&TYthhqnTaX bUAhq|WwcTYWX"g"p"pcbaU e ` g
a
)
X/B
^ ^ ^ 0 CorB,p (X/B, Y /B) = Colim 0 CorU (X B U, Y B U ) = Colim 0 CorB (X B U, Y ).
0 CorB,p (-, -) W Flt(p)
W w T e y n&vt IufYd t g XW b w Wt Yn$Ag g u y bd ` g cYc"IS W w T e y n&h$t g i t W w g p Xq X t gq h t yq u g Yv0vh$v
t g W w g i X t w be W T U pe W w g ie t g i T w 0fa$he achf&vfaP$~
CorB,p (B1 /B, Y /B) X
0 1
0 AfndCor(W )
U Flt(p)
0 CorB,p (X/B, Y /B)
GG
CorB,p (X/B, Y /B) .
0 AfndCor(B, p)
g $y hnPm{q de Tq U W t g q e $y
t g i g We peq b U Tq W t 0fvhhchg X W b w W t g ge ` bq b p b i g u y b d ` g p fc$A0lhc)v)"c$k 7$)f)l" vAc$g gq W w b i W g g gq u p b U y i ge ` bq b p b i g u y b d ` d ge p g d u gq W t h$"ca$hc)v))$k 7j$)f)yahfhy0g
0 CorB,p (X/B, Y /B) CorB,p (X/B, Y /B) i CorB,p (X/B, Y /B) i CorB,p (X/B, Y /B) U Flt(p)
^ CorB,p (rig k X/B, Y /B) i CorB,p (X/B, Y /B) ^ ^ CorB (X B U, Y ) i CorB (X B U, Y )
ut g IfYd
Ztr (X/B)
^ k(p) SmAfnd/(B, p)
k k
B,p
Ztr (Xp ) ^
^ k(p)
k(p) G SmAfnd/^ p
^ p = Spm(k(p)) ^ ^ X/B k(p) ^ ^ (X, O)(B,O) k(p)
SmAfnd/(B, p) _
G AfndCor(^). p
B,p : AfndCor(B, p)
G SmAfnd/^ _ p
^ B = B k k .
p
^ ^ Xp = X B p ^
p M(B )
G AfndCor(^) p
k(p) = OB,p /mp
p
cm rYcEn
AfndCor(B, p)
G AfndCor(^) p
d &v & i
tse W T W y p p b U t X d d T U t g e d T u T w b h W d T u g d W y vf&d}")Yfca{0i he ccfnYf$nT }
^ k Bc k Bd (o, ) k G Bc k Bd . ^ k k
g p p gq gq d T e ""ahwhunq)
w ge t X W d T U X d d T U w y W vfYca|fYnwg g W t g g qe h fPe aIh$&r g i gqq w w be W U g t Tq g i g We y i X i W t g g d y ge d Xs we gq T W w b e d b k g k U Tq W g t gq T W X W w b t t gq T Ue W d g t g k U t gq vh$y "chfr0}hw$~fv$0owY$vYa6hvcYcYcw$d|hoi0vcYaucfqhnaPffa)i0$dihA b
|k |
"
1 '
"
"
w ge t X W d T U X d d T U w y g W te g g q vfYca|fYnYPIhe Iy g W g i X pd gs g i b w t y bt w y Wt "f6wc"nT )$wg Bdr& & { g $y af)j$$fvajzch |c$A&ccvfPccz$nu d gt b u u yt W y g u w b g i g i b w T g ` T we te b w y d T d g U Tq u p g d g W We cP$"aYyfv$y ( r ge W d T u g p e g y g i Tq t w T i X We T d W T d g t t T U g )vYcy"aha"Pn|YtcYYdaf{co b t T U y T d g w g p Td gt d y b e W )an{a$"ndf{$cju ve g e PhvYl)cYah"aY$hc~hffnla~)$aY$h|fhe h~ 67 a~$" g teqe W y w b h g d e p g d u Tq t w T t ge W d T u g y g i w g g y g d u Tq g t e i w Ba f 1)" d &A& jv& &A &v ( ) r)} Y r) a &A ntr r c v0 a a rad & d ) r Y } Ar h & 6 a Yc 1$ j } a ra $) ra ) c) ( ara rVf1a d ( c a & c n &7 F FF lfiI t g e w kU g y $$d0fW t w be We i w b U t g we T W d g U t y b t g p te k u d b p b t e t T )vYt$cam$hndmcfz"fh$f))Pe fcy 0g ( $y "Y"nV)vYr0t cfff~cvahf$mnYv$fYrIS w y Wt g p d T w be W U g t y b t g W W g U g i q T ue U we d u W T Wq y t X d g g
1 '
t g g gq u p b U g i g p te k u d b p w h$"ca"fh$f)"zy
t w T ni
W g
d y b u We y i X i w b h g W w g i X U X d u w be We t b u b d u Tq d T $)wv$0c)fa00f$lchfv)f1P{nqe
'
ttovF g de Tq U W t g X We Ue w S W w g p g W Te i X p pe t d bq T g pe d X g ee )fhnPwyfvP$$y ia"aY&P""hodchnyvYac|ft vhe X W Xe d u b d u T S e g i t d bq T gq y b U X i X p d gs W we b u w y W t YhfcY$} he Y)vch$)$|"Ya6v)${g b t T U g S ca|I W w T e y t g de Tq T U t t gq g d i w g W i W y t qe h W g n&v|fhnPnf0v|YaYX 7v"fIv$0g w x i $f7a$chf&d|"f6fhcjlg d y t q g w w be W T d X p d gs W we b u w y W t b t T U y T d g w g p Td gt W y g u w fclc$a"cY|f{$ajl)b X W Xe d u b d u Tq W w T teqe W y w e favY$)f$P{)cfPhvYlux 67 a~$" c & n) d v r n &a adY A & cm yF } dc h dI r ir Ar rnAd & Av& f r a j od ch c v I a dB z i ~Y&)j r "r rad & r) cac r& a a$ & & n c n j a c" d "$ & " l r &c )1 ifdj&)j }1 rna y wn & $ r & & && 0 r& r a j ad ) | & dB l & fa f &A& lv A &
X Y
"
"
n
r
u
&
Ar
d ) h &
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
CB
B
p
k
&
r}ad
)
mB,p = 0
mB,p OB,p Flt(p) CorB,p (X/B, Y /B) ^ ^ ^ X B U (X B U )B Y ^ ^ Zp = Z B p ^ p ^ Zp ^
m+c ^ Bk k Bd mB,p = 0 B X
X/B
mC,p OC,p
CorB,p (X/B, Y /B)
^ Y B C Y
B
AfndCor(B, p)
^ k k(p)
G Bc k Bd ^ k ^ k C k Bd (o, ) Y /B GB Bc k X/B C C B B Y /B B X
j
G Bc k Bd ^ k k
GB
p
G Cor (Xp , Yp ) . ^ ^ p ^
n+c ^ Bk k Bd G AfndCor(C, p) B AfndCor(^) p GB G Bm B Y X/B ^ i [(Zp )i ] p Y p
SmAfnd/(B, p)
Y /B
G Bc k Bd ^ k X k (Zp )i ^ GB G Bn B B B p p P(B) OC,p Bd k ^ X B C X G k(p) k U Z C
'
'
(
1 '
"
se W U g e g p te k vf $y"P$$u d b p b p b k w y We w d y b c"c"czwt$Y$)6s w be W Te U b t t S chf&Pff)T u t w T i g w g W w b U We b t g p te k u d b p d ge w d g i g U g i g ` T p q g n$y$y fcavt)y"P$$Yc"{v$Ya$yccAve (|$y dchnT t d bq g i w T p g i w cAaz~ t g ie `e d t X W Xe d T Pchff0fhYn& g i g p te k u d b p w y "P$$Yc"z$ chcchfv)f$1Pocu n&h$y 7$y w be We t b u b d u Tq d T W y T e X e t T p X kU )Aa$dYt g i g p te k u d b p w y W y T e u"fh$f)y{Fc&v$y X t T p X kU cAa$dYt g i g p te k u d b p g S ofP$$Yc"iI& g i b w c"nT g hq gW b w w Yn$}~n d y t ge w p g te y u g i b w T W t $ffh$o$y P$$u)$A&q0g g d `q f$chnT T S g W w T e y t w be We i w b U Tq W w g 0)fn&h$f1chfv)~hap ed X e Ya)$y ag W d y t tse W U g d y t W g te w p h t g d ` W w $ftYr($fafh$cYcfve t X p d gs t T p X k U t y b t t g i gq p g t w "Ya6q)Aa$dYt )$h$}ag (q d T u X d i w g ` w g W w g p g d eq w geq X T g u y b d ` g cyYa)$awfvhahvccYcyhq Td gW b w w b daYn$)$h fYhhyc$A&T g t teq g i b w w y X w w b i W w TW X h g p p g $}$$c$v)cYac"p }v
R CorY (R, P ) B Z Spec((R, O) A E) ^ k(p) (Z, O) GP A Z ^ k(p) CorY (Q, P ) B ^ k(p) CorB (Q, Y ). ^ ^ Y k k(p)
V Flt(q)
se W U g e g p te k u d b p b p b k w y We w d y b vfr0 ~"fh$f))"c$$wv$Y$c6s w be W Te U b t t q g chf&Pff)T $y hnPy0j n$"$y aY))y0g de Tq U W t g q t w T i g w gW w bU Wt t w T n$i g i g ` T p q e t W w g p gq y g t W g e $)nAve f|vaf|aFft t w T i Wt c$~g h t d bq adchwj t g ie `e d t X W Xe d T 0Pchf0fhYn& g i g p te k u d b p w i g X w w b i Tq W y T e "fh$f)$y 7$$c$Ah"~$n&vy X t T p X kU cAadft g i g p te k u d b p gq "fh$f)"hIh )v)w b e W e t b u b d u T q d T e g i b w )$A&T g hq gW b w w Yn$~ d y t ge w p g te y $fmv$y P$$u chfv)f$lPynqI g i b w T Wt c$Anv0g g d `q f$chnT T S g W w T e y t w be We i w b U Tq W w g pe d X e cfcBh$fchfvcaAh{atYa)I$y g W d y t tse W U g d y t W g te w h t g d ` W w $fwvfr0vawP$p YcaYve t X p d gs t T p X k U t y b t t g i gq p g t w q d T u X d i w g ` w g W w g p g d eq w geq X T g u y b d ` g "f6cA$dft cf~0"v$1afag vn}Ya)$aa"aY$hvahvc}cYcvq Td gW b w b dafc$w )uh g t teq g i b w ffPhrc$A&T w y X w w b i W w TW $r$$)zcYqf t w T i g ` T pe w b t n$cnAhscfE t w b pe W w g ie W ctY)Plg t w bW b cfcw e w p g u W g i W g t g t teq t g d `q $j|$vfYhhu0fachnT t y b t t g t t g i g W w T d Wq p g We peq b U Tq W t ){0{0fndfvfv"vhca"hwg g $y A q ~7 q ~ ` Yjcyi f Q h Q yUt g u b e g g p d b X k W gq d T u W w g q tq g p d bs g y T U bq g y T g w w T t gq W w T i d T ` g d w g W w g p X te T g pe d X g t e U g g d eq y ` X d W t "aYc$f{hicuahe nPa"f)6}cahc$nu0vn$dnaYmaiPnwtYa)wj0ocYahvcaYug g d `q Y$a)vcT Tq h$h W y bW d y b u g )ff$)|$y $y h$"hh g eq u pe gt kW b 0$fcu k u$a0vho0"f)$fY$v|0v$}7ffcfcYaft S t ge w p we t g p d b w t d y gq g i t ge w y p e t t y T W w b d g W ag t g d `q T t g 0f$chnoIS gt b u w b h g p p p g f)jAcc"aq~v gt b u w b A))h d y b $ce gWd g y )ffa)$cb w be t d g p pe g W W g U g i g U w g W te g q t d g T d W W w g p g e w e T d g teqe W y w b h g y g d u Tq t w T i g We y t Tq t w T chda""h{fav$wfhg da)BYYua"a$y $$y he dafhhvY"c&)c$f$|Pqc$wfv$fyhinc g W d g y b w be t d g p pe g w y W Yc$))vfda""v$$Aag e w p g W g i g d `q $zju lY$a)vcT g w y U w b i g Wte g g q $$zc$zYP{q d y b cju
1 j n D A{t1 , . . . , tn , tn+1 , . . . , tn+p } (f1 , . . . , fp ) A E
U g 0cBT
w be W T W w g t X d chfnYafY$u g w y g W te g g qe $1YPih7h r chc""ahh|nq7 g p p gq gq d T e t g i b w T t g d `q 0)$A&~Y$a)vcT g i gq T W X g p te k u d b p w i g t b u te i w AhndfP$$Yc"$y If)jPcb &i { h he cq)vhl"Yc$"$vzmt$}cb Iy e d T e ge w p we g p d b w d y gq g i We w y p w W ag gW b w w Yn$V~u r ) chfv)f"h|wnfhnA|v$)0w&Yv$fYyIj w be We t b u b d u Tq g i W w T w g W we T p gq y b U X i W T Wq y t X d g S g i gqq w w be W U g t Tq d T u g k U y T ` w be We t b u p b yhv$y zchffP{ndn1chfvf)j)U g i g p te k u d b p gq W t ""P$$Yc""h0g s r W h w be W T U pe W w g ie g W W g U bq y i b p g ( chf&vfaP"ffh$"A$y g g y h1cjycy nAg he f$Ynju k }avfc Tq g U d y b u t w b d T p W g g $y ay nAfd "P$$Yc")Pe fcy 1)t d g d T p g t g p te k u d b p b t e t T t g i W w b e g t k W b q W w g pe d X
^ 0 C
g $y
G ^ Colim CorC (X0 C V, Y0 ) G Cor ((X0 )q , (Y0 )q ). ^ ^ q ^
de b g W t g d t y b w qe v)c"|fY{)$7vh
t T U y T d g w g p Td gt d y b czc$a"cYyfw$ce
V Flt(q), |k |, <1
w y W w g pd b $Yc6s
W g
gYWwnT&veytygYah$nAhy$fhYcYr)~ft ( "P$$Yc"y d e w T p Tq g i g te d b W U Ts g g i q T w p b U gq p g t w t y b Inn|h$}g )t g p t e k u d b p g S U g )BT g p d bs Tq g i t W d g y b t gq g "Yc6PrlYfa)$ch$y "Y"nT r "vvfajag ix g pd g p p gq gq h W g w
CorB,p (X, Y )
V Flt(q)
1 '
W w T pe d X g e d X w X ` W we b u w )cvfc$y f$a)~hcj$y g W w g i X U X d u w be t t y U te i Tq d T u We b w b Y)aY$ychffafPwPcIvcc}cah
G B.
g p te k u d b p b te w y t T U g U t w T i T w "P$$Yc")P$c|a{n~ U g )BT W g h h b t T U gq d g We T d W g i W y t q ~cav~aYtcYY|$qhe Iy g i g i b w T g ` T we te b q g ic"niccvfPcc)a$y ffcjhe IInu gWd b u p w d T d g U Tq u p g d W y g u w cP$"aYajycb Iy $$)FcYq X w w b i W w TW
B B = B d X = Bd 0 Y = Bd 0 C X Y ^ k C k Bd (o, ) G B. p = oq
'
X Weq Te d b W U w bs g i X W Xe d u b d u Tq W w T teqe W y w x fvhnPf)frc611Yhf$)fmPnYPvhfi
W g
t w ge t X W d T U t X d d T U g y g i W w ge W b w b h ah0ffnf0fYnlvahf)l)$&r ""ahyvn$i g p p gq gq t w T g W b w w t gq T W X W w b t g y T Ue W d g t g p te k u d b p t gq Yn$~7hnd|))caPffac~"fh$f)"~0v b
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
CorBd (o,) (Bd X C
mC,q = 0
Colim
(o, ), Bd 0 ) = CorC (Bd Y V X
1
CorBd (o,) (Bd 0 C V (o, ), Bd 0 (o, )) Y X ^ C
mB,p = 0
mB,p = 0
Flt(p)
CorBd (o,) (Bd X C
^ k X0 k Bd (o, )
X0 /C , Y0 ) ^ 0 C V d B (o, ) A = (B, O)
^ 0 C V
Colim
|k |, <1
(o, ), Bd 0 ) Y
^ X0 = X B C
GX
Y0 /C
D = (Y, O) A{t1 , . . . , tn }
0
r D (
n CW n CW
Y
fj A[t1 , . . . , tn+p ] G Spec(E)an
^ Cn
W Flt(p)
^ Cn
)
)
CorY (Q, P ) B
P = Spec(E)
n ^ ^ CW = ((n k X)B W, O) rig
Spec((Q, O)) Z an B
^ Cn
k
Y
CorBd (o,) (Bd 0 C V (o, ), Bd 0 (o, )). Y X ^ C
P an
^ k Y0 k Bd (o, )
^ k C k Bd (o, )
Bd (o, ) V
G Cor (X0 C V, Y0 ) ^ C
^ Y0 B C
mB,p
n CW
^ ^ C n = ((n k X)p , O) ^ rig =0 W Flt(p)
GD
p q P(C)
V Flt(q)
GY
A
B
Q Z Spec((Q, O) A E) (Z, O) A G P an [Z] Z Z an
0
Spm((Z, O))
CorY (Q, P ) B
Z
(Q, O) GP
Z an
CorY (R, P ) B
P an
Spm((Z, O)) B
Corp (R, Yp ). ^ ^ Y (R, O) ^ k(p) Z an Z an Z G P an k k(p) ^ ^ Spec((R, O)) ^ Z an k(p) G P k k(p) ^ Z Z an |k |
n (
0
v s (
'
g i W w g p Xq X w y W t v$m0g h w be W U y d W t w b U d T e chfrfY)mcu U g cBT g i g Xt b u p b U T "cahq t d bq T g t b u w dchn~ t w T n$i g i g ` T p q W w ge W w b cnAhe ")vYcU g $y Y)aY$E)vYrYfcaPY$nT 7A&P""h" g W w g i X U X d u w b e W U y d W t w b U T q t d u i W T e i X p p e W t g q i w gd u w aY$z~j g $y aYW q g g i b w T g we T p b i g e w c"ny$vcAc$y $$y w y t d bq T g W te g g q $}Y)vclYPf d T u W w Te d T we g i b w T W d g y b w y W t n}chYn& v)$A&"c$)E$1g ~u t d bq T t w b t b dchn1cf)e w be t w g g q g i g w w ge te bq T ` g d y W q U g w y t w T chfW g }$ahPchnffn $$vn$i g i w be W T teq T p d b w T $z)vY&YhhnAYc$hq We b vc d y t se W U g d y t W g e w uvf($|F$p t w b W b g W w T e y t g d e w T )fnvj)fn&h$fYah$nAp h)f)afhg g d ` W we W t g $y tcf~ We T t w g i t gq e W U y i X d de t g W w T t b u p b U t g |h$vYr$Yfhw0fnY)j"c|0vq Tq g PAi We y d W t w b U w b h W w g p X te U X d u t yq e t$Yf)c70)"afhaY$hi7 c$y fca}t)t t w T i w g W w b U We b g $y fc$Anm$vcAcndcvAvndX q g W g i b w T g we T p b i i w T d ` t yq u gq W w T W g p p b ""cU t d bq T We w p X i w t g i b w T t W d g y i g We y dchnwv$$l~a0c$A&fYfa)$cb 7|fv$ft g w i g W w T t te b d U w be w y g w y W t $$y 7~YnYfPcYzch$$$${0g g $y g W W w ge W w b )vYcU g te y u gq e t t b u W w g p g de Tq U W t g e U g $y P$$h$vfY)j~a"aYhnPy0Fao g i g ` T p q W w ge W w b "ccAve (ahfcaU g $y af~Wg q gW g i g t d g we g ` T p q t w T i w g W w b U g i b w T g we T p b i w fYc hcc"he (vn$ry fca)$A&}vc")r$y t w b t te te b k )YhfPc$o g p p b ""co g i b T d g y q d T u g te d b W U Ts g c$wAnoWc$)b fnwfPf)fr)wft g p te k u d b p g "fh$f)~vq t w T i Wt n$0g }fcju$$ftmvWg g i W d b u y gq t W g ge w p g i b w T g d `q Ah$c$AnY$a)vcT g w y Wt $$g ~F g t b u w ge w p g d `q f)j~qch$1fachnT g w Wt $yA0g t d bq Y)vvu g i We y i X d W w ge W b mv$0fAahfc$y y w U g cBT g i Wd b u u yt g A)$rhq t w bW b cfcw d y b $)u g $y aft)j$$fvc))h d g b u u y t W w g p gq T ` X W y g u w b d g w T d g W We a"nYlYv$y e w p g $u W gaf)ju)ETPhqn`aXtg Xt b p bU q g T W g $y ghhagfW qq ge p g U w T i w b u t g d d b U g w y W h$wzan$cj0fYca$$lg g g t te g d `q iffPhqY$a)vcT t b g ycftE$w$yxyUYgjcbegig"paYdcbX$kfWhqdnTugYWPegghewe$wjAadft t u g t q h p g u W g i T p X kU w y Wt $0g g p p b ""co w be W T )vY&y X 7yacP$)vfhhe (q i g U Ts Tq g i w be t yq U w d y b u h e $cjv)g h q Te Ueq u pe chahh$"ht g gq p b g t T g U w W T t e p i e w g w T i w b u t g d d b U g w gh$u"caU"hqwnihqU$y|wnYftP$waXAgv$pyUc$)fYf)A$$y g g p w $~ Xd ` g i w f)aag gv)cbhc"pck 7yt0ucbYc`y0vqtvefU(d$ftyg"Pe$$uYdc"))$y0v$he c ( $y Y)cT I"Yf)~ e ` q b b i g y d t g d y ts W g y t p t k b p b p b k t g i We y i w id b i gdW w b p w r h g t ge W d T t y b t g y g i w g g y g d u Tq g t e i w g p te k u d b p b t e t T hffnu cfzz|c$f$PyPe v$z~$c"fh$f))Pe fcy $wg w y Wt
nN ^ ^ ColimW Flt(p) CorY ((rig k X)B W, P ) B
g $y d g y b d u X w g p T d W t g w b h t w be W T W b w t g U U g t d geq y ` X d t w ge d X k W g b w t T p X k U t t g i d y t tse W Tq g dt gq U U t g i U T qq y u gq i w g d a)$cY$vo$cY}))vY&Yc$Fau)0 Yvh$)aYahf$fa)"$dY0q$Fvf&PaYh$|) h$ $fv}aY$u w cb d y b u cj b W ag w g tq g d y W T w W w b vPaY$Y&|cft & ( g s ( 0"P$$Yc"cfP~0v$y vc)v)$y cAag v t g p te k u d b p b te t gq g W w g p gq T ` X t w b d T p W
'
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
U g cBT
T )b w
G Spec(H) G Spec(P ) ^ Spec(C n ) Spec(H)) = 0 0in Spec(G) Spec(H A E) ^ H G = (G H C n )red Y n ^ CorB ((rig k X)p , P ) ^
G Spec(H) G P ^ CorB (Spec(H), P ) Spec(C n+1 ) n H (Spec(H/t0 ) Spec(H)) = (Spec(H/t0 ) Spec(CB )) (Spec(H/ti ) Spec(H)) = 0 1in+1 ^ T Spec(H)an Tp (n+1 X)p ^ ^ rig an ( ) T Y CorB (T, Y )
T w b h w be W U y d W t w b cchfrfY)U d T e y w g W b e t w T g U w T i w b u t g d d b U T cqcgf)ufhenmn$)fYYchq gW b w w Yn$~i d T u g te d b W U Ts g cmfhYcYr)At g i w be W Ue d W t g d Tq W $EchfrPYfYhg WahfW)U w ge w b g q g $y fW g i b w T Wd g y b w y d g y bdW W y g u w b h gW w g i XU Xd u g u TW q t w T i g p p b $)$A&fa)$cIa)$cYfFaj)acYa0Y$ocYX Ic$o""co d y b $cju W ag g $y af)j$$fz$ajscuh d gt b u u yt W y g u w b d g w T d g W We a$AnYfft$y gq T ` We b hnaX1vcft g X t b u p b U Tq g ")Py hhafW t gqq g ge w p g U w T i w b u t g h$Aan$cj0fd dcr$ymg bU g w W e$pgju Yhhzf$chnT w W g i g t teq g d `q t y b t g w y d g y b d W W y g u w b h y U t g u b e g i g p d b X k W gq d T cfr$$"a)$cYfaj)iIaf0)F"aYc$fhAnqe d y b $)u Wg g$y hvfW gqq g ge w p g U w T i w b u t g d d b U g w y g W te g g q h$on$)fYYc~$$wYP$he y $t gt b u u y w ~ d y cbju h h gq U w y W w T t te w p X i ge w p g U w T i w b u t g d d b U g w v)g echU~nYfP$ah$|ancj0fYca1$y We vcb X d ` g i w h Y cag ge ` b b p b i t g y b d v)chqc"ck 70u$)f)`thq{$yfwttYWrU0&vef0gfPe$k$uYdc"c"cvv$he c ( $y )caYvc"|Yfc"z~ g d t se g w t p t b p b p b k t g i We y i w W w T w g W we T p g d W w b p w r h g geq T W X W t g X W e W U g d y t T S ge ` bq b p b k w g w be W T Ueq u u T g p p p Tq W w g te y i w )vh$ndw|fve vYr0$f"chc)v)"c$azchfnaPv$n"aqPw)P$ve
1in+1 nN ^ CorY (n k X, P ) B rig Corp (Bd p , Yp ) ^ ^ X^ ^ CorY ((n+1 k X, P )p , P ) ^ B rig
s0 s n+1 CB
wcb h g i gqq w w be W U g t Tq W t vh$y )vYrf}P0g w be W T U pe W w g i q Te chfnavfa$he (Pv d y b $cju
0in
g $y t)c We b tgkdUmg$yagi{t0vq$h w)veYtft)jcY$P{n$0i$ g b We b u b d u Tq d T u h W g g w x g u bW b p b k Wt )fc)${0g e $ qnPePev$u"heth$"cz|vaU al$""cU T U q p g g gq u p b U y i gq U w y g p p b de b W y g u w h g W w g i X U X d hc)$ajmcb cYa0Y$u g y $ccff0i anr ""ahv"nqe d gt b u u yt W y g u w b h t y btt g e U g p p gq gq d T gqq g hhafW ge w p g U w T i w b u t g d d b U g w v$wn$)fYYc$y y fhnT e t we W g g y
( s0 )p ^ CorB (T, Y ) ^ s : (n k X)p ^ rig G Tp ^
t g de Tq T U t t g i w be t w g g g t d u T h w be W U g t g w i e w y YvchcaY{chafW aY$n$I)vYr0$y 7$$}p d y b u gqq w W t $cj|vh$y w0g t w be t yq U we t gq U g chfvahh~0cBT
T /(ti = 0) T
w y We y d W t w b U U w b i T w ${t$Yf)y)"z~ g t teq g i b w Yhh)$A&T g i g X t b u p b U Tq t yq u g |af)j"cAPh$y~
^ n k X rig
CorY (T, P ) = CorB (T, Y ) B
R Spec(G)an Rp Spm(G ) ^ Spec(G)an P an Y
^ CorY ((rig k X)p , P ) ^ B ti = 0 P B ^ H Cn
Spec(G)an P an Y Y (Spec(G) P an Y )p Spm(G ) ^ N T Spec(H)an T H G R T (Spec(Gi ))iI Spec(G) Spec(Gi ) ~ Gi Spec(H) Gi k(Gi )/k(H) ~ Ri = gAut(Gi /H) g(R G Gi ) Aut(Gi /H) ~ Ti Ti H G i = R i T = i Ti ^ Tp (n k X)p (Spec(H)an )p ^ ^ ^ rig G Spec(H) Spec((T, O))
^ CorY ((n k X)p , P ) ^ B rig
0 i n
CorSpec(A) (Spec(H), P )
T
G ^ Cn Spm(G )
H
dn,i = 0
G CorY (( k X)p , P ) ^ ^ B rig
G (P an )p ^
an
n-1 ^ ^ dn,i : (rig k X)p (n k X)p ^ ^ rig
Q
H A E Spec(G )
R
n CB
(Spec(H/ti )
CorY (-, P ) B
0in ^ k(p)
h
^ (T /t0 T ) = (T /t0 n k X) rig ( )p s = (T /ti T ) = 0 ^ ^ ^ ^ X T = Bd k (n k X) = n k Bd rig rig k d CorB (BX , Y ) s0 GG Corp (Xp , Yp ) ^ ^ ^ dn,i = 0 0in dn+1,0 = p dn+1,i = 0 ^ ^ H C n+1
) (
&
1
G
Bd k
Yp ^
) (
g p p T d ` Te i gq se W T W y p p b U W w T i w g ""ndcnPyhFvfnY1)wn$aYd g p te k u d b p w y We W g u W w g p p T t y "fh$f)"zvtYj{)"cfmft d y b u g W te g g qe t y b t t g e $)1fhahh )Yi U r "v}v|g r chc"fhnPhcYc1v~nqhnd~){0vcYcYc$v$d0{0|1f1)U g p p gq gq W g de Tqq b d b U gq d T e t gq T W X W w b t t gq T W w b e d b k t g k U t g t W g g W y p p b
1
q T w b ` Te i g p te k u d b p w y T w nccch$"fh$f)x$x~v
gt b w f)ju ~~
G V.
g p p T d ` Te i g S ""ndcch$ g p te k u d b p g i d b i We y d W t w b w "fh$f)hq dcjnT 7t$Yf)U ~ 67 a~$"
Z GW V
Yr
d d &h
&
d&
k
" &A
n
n r 1n d a
e e
Y }$ & r~an ja 1n d fr rdc &A 1 Y d 1$) 1 r& r fr f r &A j
j
d1 fr ) a &A
nc 1n &A "a rfwar ) Bac c v& ctr ar
e0 : Z GU
ry f aEr n" 7 o ) aY1 & r) n Fzgw
nr ag r &r W h t0g"ppvq|t0giyte)`nT atuq)ghe)veYdveYdfWaXp)baX)`g$itg"p"pahq$y a$y fhhvY~c)Byc$w "aYc$f"c$flP|c~ g W i ` g g t g q g X teqe W y t w b T t y b g p d b X k W y i g y g d u Tq t w T he g e 00$fcu h){d$cncfW "fh$f))Pe fcy $wg t gt kW b k t gq t y b t t d y b y b g p te k u d b p b t e t T w y Wt e W
&1
t g g gq u p b U g i g p te k u d b p gq g h$"ca"fh$f)"yvy$y aYf){cj{af$n$)T {)"ahvacwz~jh$p d g d W w b p d y b u d g W u T i q W w g p gqe U Ts W y g u w t ge w t g U w T i w b u t g d d b U t gq d y t d geq y Ue W d T u W T Wq y t X d w y U y T g teqe W an$cj0fYca}h}}vh$ahYc}&dt0f$anzPhvYy w "aYc$f$mc$f$rS R IPIVF7&ED C g p d b X kW y i g y gd u T H H 9 8 % #
ttovF XdW w b p X i Wt )fY))"avg g"paYdc$kfW1gISjbYaX }gyc`chqcb"p)b$kWtg b X d b g $y Yfc"0o gdW w b p eU g g $y Yfc"p c chfnhflP0c1Yf$nT } gdW w b w be W Tq g d Tq h W d T u g d W y d$cbju y tddchqwj b t w bt b u W cf)jwg g i gqq w w be W U g t T vh$y chffAhq t w bW b )fcw U g 0cBT g $y af)j$$fc~Iff0i amnr ""ah1hwdchn"$y h$$c~ d gt b u u yt U w b i W y g u w t ytt g e U g p p gq gq t d bq T g eq u u T w d y b $)u W g
(T /ti T ) = 0 1in+1
g pe W w g ie w q g vfhcb ($y
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
^ X n+1 Bd rig
^ Z homAfnd/(B,p) (rig k X/B, Y /B)
&1
s0
U o
V
k
"
&
e: V
G U k Bn ^ k
^ Bn Z = V Bn o k k
"
"
j
^ k j : Z k Bn (o, )
^ k U k Bn (o, )
^ k Z k Bn (o, )
e0 id
Bd X
^ k U k Bn
dn+1,0 =
e
V
^ k j : Z k Bn (o, )
GV
f id
g
j
G U k Bn ^
G U k Bn . ^ k
GV
= s0
G Z homAfnd/p ( k Xp , Yp ) ^ ^ ^ ^ rig
GV
e
e
GV
^ rig ^ T = Bd (n+1 k X) k dn+1,i = 0
e
a ddc
$
a
"
"
|k |
Z V o
j
^ k Z k Bn (o, )
^ k Z k Bn (o, )
G Z k Bn (o, ) ^ k W Z
g0 id
j
g
G Z k Bn (o, ) ^ k
GW
( G Z k Bn ^ k
G Z k Bn . ^ k
j
IQ W ^ W = V U Z
(
1
0 1 (
1 i n+1
|k |
W g
h
W y bW d y b e cYoc W y bW d y b cYf$cju
g W b w w b h d g peq u pe t d y b e Yn$lc0avvhff$) g $y fW q g W ag
)Nn 1
g $y
q g aYW
t X Weq T ` X we t gq T w 0fvvc)av0vcb t i W w g w b )ve caU $y )fYc g t w bdW w b g g p w $~ U g 0cBT
t)chfaun0Pe Iy )cYg w b e d U q T X i W w T W w w y Wt $g g $y avc$AnT W g g i b w c
U
& wn r
d y b $)u W w Td i w g ` w ndc$g d T u e w p X nF$ai q g w w be W T d g we T p b a$chfnY1$vcAci g $y dgtbuu$)$ajm~ ag yt U w b i W y g u w W w g gq T bq W t g w be W t g vhnU hochf$y S 67 a~$" T a a d $ n 0 c ( 7 &c j wn ctrzar n F l{ &
0in ^ Z = U (X k o) X DBn (f0 |f1 , . . . , fn ) X X k |k | X U f0 , . . . , fn
ttovF g y g d u Tq g we p d g W e U g )$aY$P$v"YaYao W w g p X U d bs T w b h W w g w b a"dc6yzcahe cU g p te k u d b p w y t yq u y T g W te g g qe "fh$f)$|h$c1YPqhjh W ag
^ k Z k Bn (o, ) ^ k U k Bn (o, ) GV a = g0 id
g p p b ")o d y t t gq T W X W w b $f~0vcYacft
g p p gq y )"vi g p te k u d b p gq W w T te y i w fP$$Yc"~h)cfP$ve W g g p p b ""co
^ k Z k Bn (o, )
g""nTdcnPiyh{afW$}"ca{cfPns p p d ` Te gq d g y p p b U W w T te T g p te k u d b p w y We w d y bs X d "fh$f)${v$Y$c6|Y$p Wv$ybY$ul$i|YthqnPYdcfc6s1PhIahfffdnwt0g ""co)f1"cU ( Yfcayvyy hc)1yY0f7h$hYaY$ce e i d y X We Te b W U w b Tq w ge t X W T U W de b g W t g d qe h d g we p d g W d y b g p p b g W y p p b X d d T U gq g W w p gW T i X ag"f&Pe0p"hY)vqc{w~& pe t d b T T t w T i w gW w bU Wt c$yy af)g g q g $y fW t w T c$i d b W w b W g u gc$y)fdYAU)imyajEw~u t w nTi g i tW T p }drU)u"cbU e)Ty )nhPc)ho"Ynlvcnag t t g ` T we te b t gq e p d T u gq T w p b U W t t g i gqqe p Ts T 0zhhv"nrS g W d g b w be d g Yc$y)"chtda"p"pvvg$w$y 7vf)jPA~IvfYcaog ( y )caYvc"{Yf)"~ e i g t b u te i w w ge t X W d T U W t W w T w g W we T p g d W w b p w g W T Wq y t X d gq nYv$fYh b }$ n$y af)vg t w T i w gW w bU Wt d g y bdW W y g u w b c$)fYvch r ""ah}hwnqcffa)$c)vfY"h}$~g g p p gq gq d T e g W d g y b w be t d g p pe g w y W t
g $y
q gW W fvag
U g 0cBT
g X t b u p b U Tq g af)j"caPy$y fW q g g i WU T u p b et T c"caU fcy cnvfh)cy$qYPnhe Iy fY))"o r )v)oYvchhcYcaychfnaPv$ni"ahai$a)f1)U g ` T we te b w y g W te g g q g d W w b p g de Tqq b d b U y i w be W T Ueq u u T g p e g y g i g w y h g W y p p b
1
g y )fYc t w bdW w b ghc$chYffh{$y $cucYq) vfaivu $faY$wfYp {Yv$y ffaccchda""v{$i0g q ` w Te d W gq g X w w b i W w T W We W g u t yq w y g d i w g d u g d W W y g u g W We g W d g y b w be t d g p pe g w y W t g i t g g g w w b U t g W w T t b u p b U t gq t g W y b W g d W w b U w g {a$$cafcf"ca{h{YcYvYf)fd g te y y P$$u g i t y t t g i y T t g d p t g t d T u X we p d g W X i W w g p g e w y W t g g p te k u d b p q g W fYn{0f$u0wc|$hYaYv$y ${0~"fh$f)aflw hndf$h)vYffavYn& t gq T W X t g ie `e d t X W Xe d T g i g p te k u d b y"P$$Yc"p w y Wt U g $"0g czy $y cAaYEcqh g d T p gd w b g i X We Ue w q d g d W w b p d y b e fvP$$y }aYf)"$)F Yfcav~tY&Y$}"caU fd g $y ah{)f1)U W w ge q g W y p p b X d d T U gq se W T W y p p b i w g
0 1
g p p T d ` Te i gq h w be W U y d W t w b U d T ""ndcch$vI)vYrYf)wnqe
^ k j : Z k Bn (o, ) GW GV .
g X t b u p b U Tq t w b d X ie t w b af)j"c"PfcYaPfco
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
^ U k o
Z
^ k U k Bn ^ Z k o j
j
^ k U k Bn (o, )
^ k Z k Bn (o, )
e0 id
Q
Z
G Z k Bn (o, ) ^ k
^ k Z k Bn (o, ) U
y G U k Bn ^ k
G Z k Bn ^ k
GV
QW
0<
^ k Z k Bn (o, )
GV
e
<
'
V
|k |
'
^ k Z k Bn (o, )
Q
0 1
0< < ^ V Bn Bn (o, ) k k
0 1
1
V |k |
^ Bn Bn (o, k k
^ k Z k Bn (o, ) GZ g0 : Z
) ^ V Bn Bn (o, ) k k
^ k a : Z k Bn (o, )
^ k U k Bn (o, ) o
^ k U k Bn (o, ) o
^ k U k Bn (o, ) g0 id
G Z k Bn (o, ) ^ k
^ k Z k Bn (o, )
^ k j (Z k Bn (o, ))
a
^ k U X k Bn ^ k Bn (o, ) U Z k
w y Wt fg
fi
(X, O)
M |k |
<M < min(1, 1 /M )
|fi (x) - fi0 (x)|
1
,
fi =
|fi |
1
=(1 ,...,n
< max0in |fi (x)| |k |
fi t1 . . . tn , n 1
X k (X, O){t1 , . . . , tn }
1
< max0in |fi0 (x)|. GV
1
g $y
(
< max0in |fi (x)| (0,...,0) fi0 = fi Z j
^ k ^ Z k Bn (o, ) V Bn Bn (o, ) k k V ^ k Z k Bn (o, )
^ k Q Z k Bn (o, )
^ k Z k Bn (o, )
x Bn X x Bn (o, ) X
a
(
gXtbu")"h|$y Y"cyzw)vY cz~j p b U Tq g g W u p b U g i w p w g W w ge W b w t w T i t W w ge U b U W w b n${d)vamg "wcft t gqq ge W w g d X e hvvY)fjvi gi){$$f"f)6s gt T g w y g pd b g $y af)j$$f$Ycuaj1)&h d gt b u u yt ett y T W y g u w b W w g p g ` w T k U w y g de Ts g W We c$cdm${Yvcwfft$y d T cu W g d T nu g U Tq u p g d w b h g We y t Tq t w T )hf")ccYtyPqc~ g $y Y)i g gWd bt g W g d i w g d u g i t d bq T W y t q Yf$y$fdchnv"f
M = Lim M . T
, 0+
W w T e y n&h$t w be W Ue d W t g d T z)vYrPfY0fYt We b t h W g w x vcf0g i~ t d g T d W g te d b W U Ts g YcBdffhYcYr)t gt b u w b f)jscfh d y b e cu d y b $)u t d bq T g W t g g q dchnEfPe a}w g d eq gq y i b Y$hvh$"p y i
T /X |k | (T , )p ^ Xp ^
w w We te b k U w g t w g i W t p$ag~vfh)$d~Icfayg t w T n$i thgag uSu e W d b u u T d d T u ge w p we g p d b w Tq g w `e t X "fcj$ndAnv$v"Yc$P$ch0i b We te b k w b vfh)$dU cwh q g X w y d y b e aYd $$cw chffshn&h$ft w be W U g t Tq W w T e y
fij (T, O). > 0 Xp ^
g w y g pd b $~"Yc6s
d T u g ` T p q W w b ncc"he ({ci
tse W U g b d u t gq y i b tYrcfo0vp g i g W d y b U g W U T g g g We y yY$)yfc$yYtt g w y t d bq T W w ge W b w y T b w w b $wdchnahf)~n)&cft t w bW b w W cYn$|g w be W T W w g t X d u g w y t w b t te te b k )vY&Y0f$"$w)YhfPc$o nd$f$hhg ` w T d g i g d eq W t g $y $ajl)$h d gt b u u yt W y g u w b g i gqq w w be W U g t Tq g hv$y zchffPyi g i b w T g ` T we te b w c$An)nhfh)cA$y U g cBT d T nu gq u y b U gq d g U Tq u p g d g W We he t we se W U g b d u gq y i b h$$)vqcP$"aYyfv$y fvv)vfnY$wh$"p w $y U g 0cBT
N OT Spec((T, Symi N ))an i=1 (X, O){t1 , . . . , tn+m } (T, O) T /X N (T, s) (T1 , 0 s) T /X G G (T, O) I
a OT ,
U w b i g Wte g g q g i b w T W t g q )1YP c"nhe Iy
&
c rdr)& dyj ) a a ( ra )u &
s
a" rfad c aY c t & $ Aad) ) n a & &7 F l{
T X B k
f Q 0 q ~ ` ""ah$y 7yh)vYvYfa)a)1P"l)vY&dn$)$$w{n&h$""ah|IS g i g p p gq w i g ie `e d ge d W X p b X ` Tq w be W T W u T i T g w y W t g W w T e y t g p p gq g ttovF X d W w b p X i W t g g p p gq g )fYc"awy""ahyIS
0 0 0 0 0 0 ^ DBn (o, ) (f0 |f1 , . . . , fn ) = DBn (o, ) (f0 |f1 , . . . , fn ) = DX (f0 |f1 , . . . , fn )k Bn (o, ). k X X
)
g d e w d g i W w T q g i d geq y Ue W d T u t T U w y t d bq T W w g g i gqq x Yah$fcBBT 1|ahh$Pffn~c$~dchny)ve hhi7
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
^ x X k o
p P(B) T
mB,p = 0 p ^
X
a
Xp s ^ . ^ gij ColimU Flt(p) (X B U, O) (Xp , O) ^ ^ U Flt(p) gij X B U n+m j = B = (e1 , . . . , en ) OT i=1 gij dti ^ M Matn,n ((T B U , O)) M (ej ) = (j ) 1 j n T , = DT (det(M )|) s M Matn,n ((T, O)) M (ej ) = (j ) M Matn,n ((Xp , O)) ^ s M
g $y |h$hda)h{0g W g gq e t d g we W t g $y vnY~ chffAhq We T t w w be W U g t T t b wcYWnbv g $y hvfPYf&A"Pq gqq g W g Ue d W T p T wcbhfWgft h~hatdteYgjra"nYmt e U Tq W W u W w g p T t y W ag d y b e cv g $y hhafW gqq g g Ue d W T p g w ahYfn"$$y gt fcTg$rtfh)$dU ~| w y We te b k w t w bt b cf)e d y t te w p X i W w ge b $fwP$aw)v)t g $y aYW q g g$i}gcc"he ($y vn}aY$ff)g ` T p q g We Ts y i g X d y t t T W t t g i gU w g "fW g $y fW q g t g i t w be W Ue d W t g d t g 0)vYrPfY0f0vq W w ge ahcb
j = fij
t w T n$i
t w b e d U cchYu$ gqqe p Ts g w y g W te g g qe h g d eq X t b u u y t W t hhv"n$YP7hcY$hv$wg
0 G I/I 2
W g
g p te k u d b p b te w "fh$f)"cfh$y g S 67 a~$" d y t g t teq W t $f1ffPv0g g te y se W U g b d u W t y P$$u vfcf$|g gq y i b h$"p a ac r | "ad d 0 n" u A& n"
r
"
"
Y 1 }c 1n 1 n
Xp ^
s
g $y
g $y a"aY&P""hvta~)fvvc)ave W w g p g W Te i X p pe We y i X i w g w X Weq T ` X w d g t b u u y t W y g u w b h X Weq T ` X we g d e w d g i Tq d y b $yaj)cfvhn)$h}Yah$fP|$ce
Q j
B
fij (Xp , O) ^ ^ ColimU Flt(p) (X B U, O)
'
< minxXp |det(M )(x)| ^ ^ X X B U T (t1 , . . . , tn+m ) (t1 , . . . , tn+m ) j (X, O) T /X T /X U Flt(p) OT ^ Bn B U X T /X gij - fij
r
n+m
i=1
A X
GT
G n+m (T, O).dti i=1
fij dti ,
(j )j=1,...,n
G Tp ^
|det(M ) - det(M )| < (X, O) ((ti ))i=1,...,n T /X
t g p d bs t gq h W g g w x 0Yc6F0vc"i& t gq Te d T g i g de T X we 0v$chYn&w$Yhnvhq
(X, O){t1 , . . . , tn }
G (X, O){t1 , . . . , tn+m } Xp ^ T
s
G Xp . ^
s
n V BXp ^
n+m i=1 (T, O).dti
G Tp ^
G T /X
rp ^
X
$ G Xp ^
G (T, O) G 0. T1 n gij
g i gt T yfc g p p b ""co
nv
Y
1
'
a j f d la rw l c & ad c wn "c n r1a&v n ir c a ctrAr h & ra )q n$ n n F z{ r wr "vajaw$)flcf"I$y g p p gq y i i w g u X i d y b W w b t e r "v$y nr "vyh{$fyfd g p p gq y i i w g u X i e g p p gq gq d y t g t b u g g y g d u T t h W g g w x g p d b X k W We i y i X We ieq T Tq d y b u gqq ge W w g t t g W t g g t k W b caY$rYqiFc"aYc$ft$fvhhn&h$cjhhahfaff01l0$fcu lYaqIfag aho "aYc$fW k g W W g U h y i w g W w w ge g p d b X k y i g y g d u Tq t w T i g X teqe W y W w g p g We Ueq u g g W t g g e d X w X lc$f$h1c$lafPvhf"a"aYtahh$0m$y f$a)` gt kW b ffnju k E 1t)faz$y $y 1Fv$fy "vlIS q b We b d i w g g e w q W t g h We y t e g p p gq g ttovF g p p gq y i g y g d u Tq g we p d g W e U g c""ahyc$f1Py$h"ff0oj g $y Yf$nvg Iw gdW y T Wt t w T i g W e y t g W W g U g nfv$Yai g We peq Tq g Yt"hvP~y Yhn& cafyhhcfa"ahna{7 g d U we T w b U g t g i gqe U Ts W w g p gq T ` X W t g q t w T i g `d g w b n$~cYac )U g X eq u u T w b W g g |a$y v$ncfwyi g W w g d d y U X d g We y t Tq g faYY$0fYtAP$y dchnvvnYrch r )v)m""ahc$f"h$vYcyfv"f$"Pwcc)b y$y 0fvfaIfYnT t d bq T We T t w b g p p gq y i g y g d u Tq g i ge W d T u g d e p g d u Tq d T e W y g w q g t g We W g u e t t y
0 '
t g p d b w t gq g d i w g d W y g u w "Yc$wh~Yfv$ajl)b W g a$A&dfft$y Yahvhnwc d g w T d g W We h t d y gqqe T d T e g $y fW q g t w T i w gW w b U W g w p W w g p p Tt y c$y fcaAa1a""nYmft g W te g g qe h t y b t t g e U r fha{vcff0i awr ""ahzh"nqq g p p gq gq d T e d y t ge w p w $fh$$ve g p d b w Tq g w `e t X "f)$zh"$)vf$i b U g )BT W g We te b k U w b tfPc$dc7h d y b $ce We W g u W w g p p T t y vfj{a""nYA$t U g )BT g $y aYfc"|"YAf d gdW w b p g i Td yt q
^ (ti ) (X B U, O) (Xp , O) U Flt(p) ^ ^ U Flt(p) ai (X B U , O) |(ai )Xp - (ti )| ^ W Flt(p) W : (Xp , O){t1 , . . . , tn } ^ (f1 , . . . , fn ) G (Xp , O). ^ .
t g i b w T t g d `q c$Anf0fachnT
g i g p te k u d b p w y i w b u t g d d b yfP$$Yc"$)fYf)U
X{t1 , . . . , tn } (f1 , . . . , fn )
g p te k u d b p g S "fh$f)"yI
d y t gq e t d g w $f|v$hfYc ve
1
&
A f da1j ) a ra )j &
t
w)veYW&TYWwgt0Xfd$uygw$yaccYf{zc$h )v)"f)a$Y|vce b d g y bdW W y g u w b g p d b X k W gq d T ama rwaaY c cA r c ad ) t n & n
B sp = t ^ B k p P(B) ep ^
tvuh
G Bn X
g p p gq y i g y g d u Tq g we p d g W e U g c""ahz$|)$aY$Pv"YafIo
^ ^ B1 (b1 , )X . . . X B1 (bn , ) X X
67 a~$" Ya a n" FEz{ q gW W aY{g
s
d g W y p p b U W w T te T af$}"ca{cfPns g W We Yv$y y T t y btt g e U n~cff0i am r ""ahh~nafhnA$y g p p gq gq W w T w g W we T p g d y b cju g $y hfW tq g
^ ^ B1 (b1 , )X . . . X B1 (bn , ) Bn X X X s G Bn . Xp ^ b j - aj
g"fh$f)"z~cyfPe aIh$h p te k u d b p w y U w b i g W t g g qe U g cBT d T cu d g U Tq u p g d a)hf w)vfhhe (wn&h$ft fcyva"a)$ndwcy aYcyhndy"P$$Yc"p be t yq U w q W w T e y g i g t T g i W w g p g ` w T k U d T u w g W b gq T W X g p te k u d b eq u u T w v$n~I g $y hnPwdchn0j de Tq U t d bq T W t g q W y b )fW d g y bdW W y g u w a)$cYffm)b h U g )BT d T nu d g U Tq u p g d g W We aacP$"aY|fv$y
1jn X ^ X B U X ^ X B U U Flt(p) B1 p (ai , ) = B1 p (bi , ) X^ X^ e U Flt(p) ^ ^ ^ ^ X B1 p (a1 , )Xp . . . Xp B1 p (an , ) X^ ^
W w T e y n&h$t
tj
W w g p g q g d w y W a"a)ahaY${g
aj (Xp , O) ^
g W te g g q fhve Iy
We T t w b h g de Tqq b d b U gq d T e tcfc r )v)fhnPv)f)yh{nq$
1
g p te k u d b p gq W w T w g W we T p t w b d X ie t w b fP$$Yc"hvnfvcAcYaPfco gq T W X g p te k u d b p w )vcY|"fh$f)$y 0cBT U g
e
g i w be W Ue d W t g d T chfhYfY"hq
gW b w w fc$~
w be W T te d b W U Ts g w y X y b d W U w b i T w t g i b w T t g d `q )vY&YhYcYr)~$)$cYf~)"~$$)$A&fY$a)vcT
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
k
g i gq T W X g p te k u d b p w y W t $|hnd|"P$$Yc"$wg
ep s ^
ep s : Xp ^ ^
(a1 ,...,an )
Xp ^
T
e
G Bn Xp ^
G Bn X
9 X
s
t
b1 , . . . , bn (X, O)
1
sp = t ^
s
|k |
PT
GP Tp ^
e
ep ^
mB,p = 0
t
e: E U Flt(p)
j
GX
&v
Jac(f1 , . . . , fn )
^ eB U
E
E=
t
(Xp , O) ^
>0 Xp ^ |(ai )|X B W | 1 ^
(a1 , . . . , an ) (Xp , O) ^
B
k
|fi (a1 , . . . , an )|X B W | R>0 ^
^ (X B W, O) ((t1 ), . . . , (tn ))
p P(B)
" " $
&1
g p te k u d b p g "fh$f)"yvq
X W Flt(p)
B
|f|X B W | 1 ^
f (X, O) p
|f|Xp | 1 ^
U
<
mp
U g 0cBT
"
g $y
Y
ttovF de U X d W X d g W We hYfaf}Yv$y We vcb f g i
fi
&dda d n h c v fj Yf z r n 6$n d &A& jdn"n u r r f d & 6& d &dda f nv Y&A& r r fj n ryad & f d ) & iaa rEy0a u a d c A r a h c r v n$ n &7 FF zouI lv r fhenPhcYdcUygY&TP""h~aa$y )$w0chffaff)Yah$YaAP{c{h"$0)c&v$fy"aYcafIS g d Tqq b b i W e i X p pe g U w g X t w b U g w y W t g h w be W d g t t T g d e w d g i Tq W d T u te p h W w T e y t g p d b X k W g cYfWwcb"pXv0vYvchhcYcayI Xd i W t g g de Tqq b d b U g S W w g p g de Tq U T w g t teq W t a"aYvcha"~)YPv{g g $y af)j$$fvajl)$h d gt b u u yt W y g u w b t d bq T g t b u w te w p X i W w b dchn~f)jm~h$))t t gq q g gq d y t g i b w T w hja$y hf$~)$A&z$y W w g p g q g d w y g g w b a"a)ahaY$$p )ih d y b e $ci t w T n$i U g 0cBT
k(p)[t1 , . . . , tn+p ] gi X Y = Spm 1 i n ^ k(p){t1 , . . . , tn , tn+1 , tn+p } (f1 , . . . , fp ) gi OB,p [t1 , . . . , tn+p ] X = U {t1 , . . . , tn+p }/(f1 , . . . , fp )
g $y $t d gt b u u y Wgu cb r chc"pvqvq1nTq y w g p g g d e g T X vqcYWamg"tP$k$Ydc"pEw$yWnYWfg"pi)Tc$A&T p e u b w T W g i b w w $y We se W U g d y vcb qvfr0t W g g q ge w g t g W g q wa"pahqhahfWaftf0uthe Iy wcYf)b )vqipuWwgpg$heahq$uWgi$gfUcb6{gavg$y YfWcb"p "f)a$Y{IS 67 a~$" t bdW w g w g u t d y W w s U gd w g p d b X kW g a d"cdja a j r r&
0
(
&1
a j iI Y ila rf0a u c r o n$
ryad & f d ) & A ad c r h n FGooii n
SmAfnd /(B, p) AfndCor (B, p) B k p
&1
d$f~vYavg y t d ge W w g W t U w b i W g p p gq gq d T c~ag B )v)""ahhvnu g d y ge d Xs we W w g p g W Ue d W t W t Y$ahf6hw)"afhYfvg d y $t h t w T i Wt c$vg g y aYW q g q g pd bs T p X k U a"Yc6Aa$dYt
W = W W f
W w T e y n&vt g de T q b b U fhnPqv)fd)gvq f)a$YWsyrve$X$i ~vchv$fv"vhcaPcu g p d b X k i W y i w gqq g y t y g We peq b U Tq d T ut g 0fd t d bq dchnT W w p i ve$aXAw~ t gq T X g p t k d b t g g W te g g qe tq g t gq d y b 0vcYWt0"Pe$$uYc"pq0iwfhahh$y 0vicju t g i b w $)$A&T t g i g X pd b 0wa"Yc6s ut g IfYd g ig$vevqqgvYc)`afnaU cbf~Pq w g u ed b XW T t y t T ut g IfYd T d g W b w w b h g t teq g i b w daYn$1cBcffPhic$A&T w y X w w b i W w TW $f$$)cYq tvuh Xd w b X Wt cYfWc"piv0g|gp"vqyS nAYc$w{vfy$y g p g g q T p d b w W t g d ge w d g i g U g W d y t d ge W w g W t g q te y $fwahfawIhe Iy P$$u t w i t nTfWg h W g x agjalwu) g $i X d g we u pYa6sfWv)bfWycYmg $y fvfaoh$v"Yc$0g bW w g g We W g u t yq u g p d b w g i W t g $y P YY)Pao g w T d W w g Tq g d y t d ge W w g W t $f~vYavg g $y ahe W w g q g i ge w p we g p d b w Tq W h$$v|"f)$hwag t g i w g W w g q g d g a{a)ahaYyt w y d g y b d W W y g u w b h g y g d u Tq g i W y X i y i w be t t y U te i Tq d T ${c$)fYvcccaY$"h}~$rchffafPAPwnqe
B AfndCor(B) ^ bi (XB W, O) f n + bn-1 f n-1 + + b0 f k (x) |(f n + bn-1 f n-1 + + b0 )|Xp | = | N ||(cn-1 f n-1 + + c0 )Xp | < 1. ^ ^
g $y fW q g g e i T gqq g U g i g We y i X $y $c1hha1Yti
h g p d b w Tq g "Yc$mh1i
d g y bdW W y g u w c$)fYuA~ g i b w T g d `q c$Anf$chnT
d y b $cju g $y Tq We w y p w P~v$$1~7
g y )"afnh$""v"tE~i W w g p g W Te i X p pe We y i X i w W w g q ahe g p d b w Tq g i w be W T yq T g i y T g w w q W t "Yc$yPv$mchf&vcB~mn0$$cT ug
.
p
w be W Tq g chfnhfd g $y )v)g"p"pagh$}t$$~F q y i We y i X i w h gt kW b cffnju ynA$y )vnT k d T u g t w bq g u u g i b w c$AnT y i q g p d bs gq i b p w y W t Ia"Yc6ha"$w0g q g p d bs X W Xq u p b U gq W t "f)6Afav$")lh}g t u d b U y i w be W T yq T g i y T g w w q g YcEchfnvc&mEn0$$cT $y g $y "YA&|s chc1"v~vcc$y f$a)og $y $$)fcYq g p d T r g p p gq gq h g e d X w X ` W t g X w w b i W w TW
k(p) k (p)
g d `q Y$chnT Tq t w b d X ie t w b PfcYaPfco d T u X w w b c"$$c$i w be W T yq T g i y T g w w T w y W w ge W w b chf&vcB"n0$$n$|ahfcaU g y )vnT t w bq g u u t g w T t q g p d b s t T p X k U "n{h"f)6u)Aa$dYt t g i W w b 0wcft W g g $y $|)~u d gt b u u yt U w b i W y g w ~I ag W w g gq T bq W t g w be W t g ahnU hv)vY0$y j ~nhv0g T S d y t W Tq u W t g $y af)j$$fyajcInw BT rchfnavfnh$}}"aYc$fW d gt b u u yt W y g u w b h i y T g i w be W T U pe W Tq u g i g p d b X k gq d T e hynqI g i q ge W w g t t g q g p d bs gq i b p w uahfYi"f)6"v$y W ag g i q ge W w g t t g q g p d bs gq b p w iahfafYiYc6"ha$i "$y We q T d X w X ` t T U y T t w b w g t)b nda$c|cny)$a)ag g i W we b u W y b W w y)hcjycYrg g $y fvfj|h$|)"afhYf1"Yc$"yg g W e W g u t y q u W w g p g W U e d W t g p d b w g i W t g $y Yfc"0o g d W w b p eU g h t d bq T te T adchn |Pnr n hch YhnPv)f)|vfv)c g y fW q g d g y bdW W y g u w b h g ed X w X ` Wt accYfajm)))$y Ya$cf0g ")o$ g p p b g $y aYW q g g d e T q q b d b U g q d e b Wc"caU fcy c$)zw0g U T u p b et T Wd g y b w y Wt t w T n$i g i g ` T p q ycc"he ($h $ft "fvnd$cv&P$u ffPvwg d y W y t We T d y T W Tq g t teq W t g p p b ""co T w b t T U gU t w T i h Wg g w x xclcac${xif b t T U Tq d g We T d W d y b u g X teqe W T d g t gqq x w be W T d W t w b p X Tq d g peq u pe t g )ahzaYtcYY$)a fPvhfy YhvifI)vY&dfc"a$i Pzvv$"hi T d W W g p d g u gqq g h te bs g W y b g y d g u y t W w g p gq T b d u W t dff"fhv&aPc6af)t )$faj$fa"v$ccY$qg g ie `e d X W Xe d T PchfffavYn& Tq g i X We t teq g i g t k W b Pf$YtfYhhfffnju k uS 67 a~$"
B X
@ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$"
Xp = Spf(R/ /()) ^
DX (f |1)p = X ^ U p P(U ) p ^ X B W DX (f |1) = ^ X B W B
p
X
B
^ k(p) |f|X p | = 1 ^
^ R = ColimU/B, U Flt(p) ((XB U), O) /mp .
^ k(p)
k
f
Xp ^
g w y We $t)b
d y t d ge W w g W t $f}ahfayg
ai R/ /() = (Xp , O) ^ R/ / N N | N f i | < 1
^ k (p) 1 i n-1
f n + an-1 f n-1 + + a0 = 0
B
k (p)
k (p) ^ k (p)/ /() = k (p)
|f|Xp | < 1 ^ DX (f |1) B W Flt(p) W U =
X
(R/ /)[1/] |.| bi R ci R/ /()
q g fW
1 ^ (XB W, O)
B
SmAfnd /B X
1
B W f n + bn-1 f n-1 + + b0 ^ (XB W, O)
p P(B) k
f1 , . . . , fn U Flt(p) U
g y ve I W w g q We b vc g i U g 0cBT
C B
C B
Y
^ k(p)
0 B,p : 0 AfndCor (B, p)
B
k
p
Y = Xp ^
p P(B) k
AfndCor
G 0 AfndCor /k(p) ^
e : Y
B
mC,p = 0 SmAfnd /B X x B mC,p = 0 G Bn f
^ k(p)
F
d nYda
^ Fp SmAfnd /k(p) ^ G AfndCor (k(p)) ^
"
"
"
B,p : AfndCor (B, p)
k (p) = ColimU/B, U Flt(p) (U, O) /mp
^ X B W
F(B,p) X
f (x)
(Xp , O) ^
1
G Bm B
Fp B,p ^ X X
a i = b i + N ci
SmAfnd/B
Flt(p) bi
B
Fp ^
k (x)
(
g y
g $y aYfc"yv"fIh$v"YaY$c7n0Yhcs d g d W w b p g i W y t q e h d g w e p d g W d y b e g y T g U t e T g i W w Td wnd$p W w ge q t gq T U b ahja0vcavq t g e W U g b d u t g d y W U y d W t g y T W w g p g e W Tq g d g We b d i w gqqe )vYrcfiY$YrYfmnochfnhf~fvcYlvhh$y
: (Rat(Xp ), ra) ^ G (Rat(X), ra) , ra
W w g p g e W U g b d u g gq u p b U w y W t a"a)vYr0nY$|g av)$wg g i d y gW U w bs w y Wt $frc6A$g d y gW U w bs g $frc6~vq
g i y T g U te T n0Yhcs
W y bW d y b u g )f{$)yy
g e w b w T U g p te k u d b p b te w y T w b $y c$ca|fP$$Yc"cfh$ch d y $ft t w geq X T t g u y b d ahvc{j$cYc` d g d W w b p g i q T d X w X ` t yq u W t g q fYc"yIndaacwvw0$ ge U b t t hff)T q g w w be W T d g we T p b i w y e $$)vY&dvc")r$y afc6vq d y gW U w bs g
Y X : Rat(X) Yp ^
U w b i t w bW b c{cfcw$ W g g i t q g w w be W T d t g w e T p b i t g {h$$)vY&do0$hnAcvi t0fvYvfaw~$y $aja$vcY$I g We t t We W g u t g i W w g p g e w y i w g u X i g p q b d u g S g $y af)j$$f$ajz)b d gt b u u yt W y g u w d T cu d g U Tq u p g d g W We a)hf"1fv$y g $y aYcYPnY$u q g W y T g U te Ts X d w y W w g p gq u pe t W t $|a"ah$"vfg y T g U t e T n0Yhcs w y Iy 1he 1fhYn&P}$~a"aYd $caY$|0}Pe 1hhv"nmh$y vhaY$~hc"p W g g i X W X e d T T q g i W w g p g y b U g d w y W t g g i g q q e p T s T q g gqq g W g w p t we b Tq W t Pg ge ` bq b u b W Tq g v)chcjcfrhz$y chaj$cT t w bq g u u d y $t t w geq X T t g u y b d ` g i y T g U te T ahvc10cYczEn0Yhcs W y b W d y b u gq e t d g we W t cY}$)lvvfYc hg
K() = 0 ^ X X B U X Xp ^ t ^ Hi (X B U, (B,p ) K) ra K U =B ^ SmAfnd /k(p) G Hi (Xp , K) ^ ra t K t
W w g ie X g p te k u d b p gq g )P|fP$$Yc"yv|$y fY))"$cY$a){h$c$v0I d g d W w b p g i q T d X w X ` t yq u U w b i W t g q g p d b X k W gq d T g y T g U te Ts X d u t gq d y t g W U g de i g ` T pe d y g W U w bs g "aYc$fh{nqe $nafPnafhfyfYv$|)nAh{$aYr)6~hq gW b w w fc$z~ ge U b t t hYcT
B,p : SmAfnd /(B, p) (B,p ) G SmAfnd /k(p) ^
e I$y
w ag
q g d y W T w g p te k u d b p b te w y g de y d W t w b U g i We ` t q f$Y&"P$$Yc")P$~Yh$fYc${vccT I W w Tt b cfu g i t g i b w T t W d g y b t g i g We t We W g u gq d y t g y T g U te Ts X d u w y g p p b $v$)$A&wdc$)v0YtfytYj}hv$f$nafPnafr$}""caU We b w b h t y t t g e vc)rcjY$i o ch$vfdac h~g gq e t d g we W t
F(B,p) (Y ) = F(B,p) (Y /B)
U Flt(p)
g y
We y i X i w t g W w T d Wq p t g We peq b U g y T g W y p p b U t g W U T u p b e t T t~7Y)cYYt{Yt"hv)rc1f$}"cawYrc)U fcy YhfcB~$v)chc"c$)U t X W Xe d T t g i ge ` bq b p b k b
U Flt(p)
a &dA f n"n y 6$& d ndYa a ( Y iaya f rAq0n c ra 7 &
mC,p = 0 iN F
1
r $&)"d ) & rad & { a d c |aY c Ar v n t & n 7 F l{
SmAfnd /B X B B k p
"
"
" $"
1
$ )vYr0APn|$a"|h$cYn"c${cfYnPYn& vd6cYYw0cBT w be W U g t Tq t w T i g X w g p ge u b W b p b k d T u t W w Te d T we t W d gs t w T d W U g gcafPnaf$vy$y c)v))$ca1fX wc$vnYfcj"v}v)$"dac$y $y $dYw&dt$fY$|0cB}$h"ff~ y T g U te Ts X d u t g i g e ` bq b p b k b U g i y W q t w T i W w T W d b u pe gq d w y T d g y b e g e w k U g W W T Wq y t X d w y U g T g we p d g W w c ndYa B h| na y$ ~ d g Yd$p v ) f h &d h n$ ( n FF i{7 vjvuhF t w T c$i g i t ge d b ` X W T U g i g U w gq T e X g w y W t hf)cf&yyvcBh$y y$w0g w be W Te U b t t q g )vY&Pff)T }$y tcrvc0i We Ts y i gq y b U X X Weq T ` S de Tq U W t fvhn)X uj0hnPy0g w g q gd yW T aFaY$Y&w g p t e k u d b p b t e w i g U w g W t e g S "P$$Yc")Pm$y 7~aaYPg i gtd g w et T {da) ve fcy $y w U g cBT t d bq We dchnTuv$wpX$iw~ccheYccfnaw{ahn&vy $$ig g d b ` X W T U g i g U w gq T e X g w y W t h v r fhnPv)f){hqnq&aahvcF0cYcq0ohf)cf&yhn$i g de Tqq b d b U gq d T e t w geq X T t g u y b d ` t g i ge d b ` X W T U Tq t w T g i W w Te d T T d W w b U d y g W U w bs w y W t U h ge u b W b p b k d T u W w Te d T we W t chYn&Bdfcav$aYr)6z$vg n$ch$)fn")$wnchYn& vwg g p p b ""co 67 a~$" n 6$n d nYda 0 & d} a ) d r r ( fI y arjYYj y v r
G
r
jY&A&
"
2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H
Fp = F(B,p) (0 B,p )-1 ^ F
^ SmAfnd /k(p)
G B,p
F
p M(B)
hom(Fp , G) = hom(F(B,p) , G B,p ) ^
F(B,p)
0 B,p (0 B,p )-1 0 B,p hom(Fp , G) = hom(F(B,p) , G B,p ) ^
PreStrB (SmAfnd /(B, p))
p P(B) k SmAfnd /B
F(B,p) Fp B,p ^ (B,p ) : G 1 ^ PreStrB (SmAfnd /k(p))
Fp ^
0 AfndCor (B, p)
C B
r
ra
W w g i X g p te k u d b p g he "fh$f)"yvq T S 67 a~$"
F(B,P )
X/B Xp ^ (B,p ) Fp ^
^ Colim Hi (X B U, F ) ra
F(B,p)
U Flt(p)
^ Colim Hi (X B U, F(B,p) ) ra
^ Colim Hi (X B U, F ) ra
G
U Flt(p)
^ Colim Hi (X B U, F(B,p) )
G Rat(Xp ) ^
Hi (Xp , Fp ). ^ ^ ra
Hi (Xp , Fp ). ^ ^ ra
t
g $y $$)cYq X w w b i W w TW g $y af)j$$f}aj~nd$p d gt b u u yt W y g u w W w Td
t g We t g i g p te k u d b p w y e w p X Ytfy" P$$Yc"$I$i T w Xe U b t t T y T g U te T z~)vaYcmcYPns w b t d T u d g U Tq u p g d gq g W We cffcucP$"aYv1Yv$y n0Yhcs y T g U te T w y Wt fg W w g p g e W U g b d u g g gq u p b U w a"a)vYrcfav$")$y U g T gq T b cBAhnU hq g U w gq T e X g w vc&v$y A$$y We vcb
K t GI ra Hi (X, K) ra t I Hi (Xp , K). ^ ra
K
Rat(Xp ) ^
ra
t
(K)
G (I) ra Hi (Xp , K) = Hi (I(Xp )) ^ ^ ra ^ X B U (I) I
T w b z)$h r d y g W U w bs gq W t $aYr)6vwg t w bq g u u )vnT
g $y
yR" '"g$wgw C 0 t R s1 e(wg f ' " e' u'0 0b' 0 6"pRd$h $pu p X 0 rV"g$) r0h B B6q q $ R0X p p p I Q ` F c P { ` F P c x Q P D I I Q u s iRuh e gg7 f q 0 w G G G ' e 2u" d A0' ( b u gz Epeu 6g' x G i 0bT6g"f&y&V&0TTe6pT$6s1bp&y'$&pfTS I Q` E I D x` D F U Q F S c F I Q E E D S { E Dd F I Q D F cd D ` ` W F W` S c RfTp'a h D $I6U p cwb h Ddd D I Q fb6ec 0rt D ` ` S W F W` S c &pfhTgbe'a h 6$h$$p$"ewRrp"|ixi pfT6g"f&|$&0TTe6 D I U D W I I Q F I cF W t D x t x D r I Q` E I D x` D F U Q F S c EpB$wTEh&zFpTEB$6{1bzp$Bef6Bg6wTE h c x W P E D I Q E D S E Dd F I Q F cd c x W P I b $U w Rh Ddd D c I Q fb$pk0t I D` S W F Q I c x W P E I U W I I Q F I c F gfT$TgD $"$w6|6$p&hpp Hbx i Dd` P c ` E E U c F U Q F F E D I Q` F S D E E c D I Q P D E c v pfbgpTT6pX&p1hpfTTgTps&60RT"w6i S U Q E D E` U Y D S E W F W` S Q S E Dd D {` S W $pusTf601T1Tgb$p$1b6bTpa E S Qd c 0bppt FRD S Q E D E` Y D S W F W` S Q S E Dd F I c {` S W 6UpurTf$U1TxERfT6p$pfpebTpa ` aeYwGp$0p&pTE ` E` S c X E F S D a U Q E D F I Q ` PW%422RPGEC I HV S U T S Q I H F D GG p R@A pu' e 6 T7 ' p' eB ' 8 29bu'e T7R b 3 5b4 6'e T2 peu 0 1 R "g$wg G p R )6V6)hp ( ' peu 6g' epr |&%V# $ $ !"e'b`6GyD""gDfb6pc$sf$I$pfTeS h D60ES$1$bpBp&V1&ffRpH1TRf$p$60g60p"$TW F I c a UE x x d Dd E I Ed D I Q` F c U Y D E D I` c x Q E D E Dd` P c E W F W` S Q S E D U Yd D U Y E I Q x U E " R0B s d D I pQ ` F c h D U Y E D S I g 6g)' u "g g 0 g6$G fTeS $$ $ w e q G 1peu g e6p$e0 R2p' 6g' qpxG i D F I c a` U E W F W` S Q S cd S D c W D I` Q E D pe'b6GsTgb$0T$gpp0&s&gb0G1 I Q` E E D S jd t D F` U E cd E I c r i D Q I v pfG$&RD e0Tb$Ty6pip&p6"ec h $U I F` bpQ E I QS U c E U Q i c x W P 6p$pH60V&$B$wTE h D D x I Q E cS D &" 0$I GywgTE "g$wgTh $X&pV}wiwXpVisw$$k"$x$6rGVsX$m
0 CorB (X, Y ) (X0 , O)
T)
i` I { I jd I Q` F c E`d c a` S )66&f` e|0b'ffe` bTF D I j E` I U x E D F` Q S I D E W S { E D S` I D a S D F I` F I c E` c I Q` F ` S P E D D I U F D x 6$U y$$yyTbp&y$6y&2b$pg0fyefpnpfT&fR&h6$yR"&0c D U Y t I Q` F c U F` E D F F D P E I c $wpfT'TbGhGgRye$ c S S D a I Q S` Q g0up$t $Vwu sfppa $f$g0bpsbTg0g0|s"0H'0b'06"pRy6$pud6$VeTR"&0$e0$B'yuT$0RP D U Y` S W d c D` S F W x Q W I D D x x Q ~ i I Q` F c P {` F P c x Q P D I I Q D I U F I c F F D x c D Q I v c D S U Q h D I 6$U I U FE Q E c P U c I Q` F P D E D F F D P E I c D E E D S W F I j E I l p0gpu0b1GyGgRXe$yTT1TTf` gii D U Q S Dd D F I D W P W S u$ppybyg&1R$ F D D E E`d D Q I v gsGffs&p$Bec h $i1D I Q` F P D E cd E I c ` I { W E I Q a c E U Q I 0bTz6p$i66R&BppCd6p6t 6Gz1Gbd S U E E D E E` F RD U D F $g&dgD E D ` ` S E W F W` S c 1&f0bB1TRfe'a h D ` ` S W F W` S c &pfTgRfTp'a h $$z1g$6p& D I U E D W I I Q F I c F i Dd c` a` S I Q I E c D E Q U E D I I jd D U Y I Q` F c Ud c a D I U S U Q I c` D x` P S I Q I F Dd x Q ppHpfefTF h p$dpT0u$$T$0Q e600b'6fe'$$|$pue&"f$wwec h p$uRf$"pgP E S Q P I U D { I T0R6&$mil RRrrssR ddyrssR|yy|su sdygxsu
ara (E)(Y ) k B CorB (X, Y ) k B X Y k B k Y B
D D c x jd D U Y W I I Q F I cF $0eBf` es$s$$p$X0pp6i F I D x D a U Q P D TS $0R1TS Dd S D I c P E I c E E D F` F D F I D x x c E v U fHp$p6w6eV1TbTRug""eB$GE E D` I { I` E D x S Q I E Dd P D a f$6&fs"p$1bi0Cc E I Q` F P I Q E D S c pfTp8s$p6 E I Q` F P I Q E Dd S D P cd x D 60b608sfsR0f$"TS F U D I l id c W j U Y F I cF I $gu|i)p&` w0XpugD F I cS I D I ew&6p$D P D a 0Cc S c I cFE F I D x D a U Q P D p$6pGTS $0R1TS I U S c D I v c S D E Dd ` E E` x c F I D x D S a U Q P D S F U Q F t D ` a I Q I F E $2e6d$B'wzG|b6bTT"&p0"g6pR1T&pw0&}p$1D D x x Q ~ "0Hpi D Dd ` E E` x c F I D x D a U Q P D S I j &hf$fG$p2g"TS $0R}6U I gD D S { cd S c D I v c S D $6|ye6"$BeBTE D Dd ` E E` x c F I D x D S a U Q P D S F U Q F D U Y c S D S F I Q x I l &zf$fGf"&0y0"g6pR1Ty$pT"60uwgTp"nixi D U Y F I c I D F I` c x E I Q E Q $sp6g0feBspT0u$$U i E c P D P E I c S` cd P P I Q F E D F c Fd U E W S D v psR6p$Vbpfg6p&XGeb$G1Tw6i c I Q t D x t x D r up$0"sX$i D U Y F I D a do i U c D P E` c $hgf` VqppRe h FE GD D U Y E` U $T$$ E S Qd c c I l wpfenmisi D U Y S Q j E I QE Q $k0pc h6pT0u$$U si D & E U E E D U c E I Q` F P D E E Dd S U E D x E` S Q x Q E` I U F` U I 6T&hpHpfT1Gsfs$GiT$$p"pTfh$$$6b` e i e 60H6pT0geg6$pfTeTg0$pd$U F` i Dd ` E S D I` F E 0Q 0b6bTpa fsGD D U Y E I Q S F I Q f id D I I Q` F c S F S D a U Q I p i )
t E ra t p ^ R Y X Yp = (E)(Y ) = E() = 0 ^ ara (E)(Y ) = ara (E)() = 0 Yp = ^ Y Yp ^ R = (DYp (fi |f1 , . . . , fn ))1in f1 , . . . , fn ^ fi gi |fi - gi | ara (E) G ara (E) (Yp , O) ^ E Yp ^ t
F I D a W D x E` S Q x Dd t U c D P E` c g&` g"G$$p"f&e1Rfp h I U FE $1D ` E ti i t U c D P E` c T6fpD )` &pRe h U &ec E F I` D S F E D S I Q` F c E` F U c D P E` c D E S U D F P I Q U c D F U x x Q fgTGBpfTeTT&pgTeX&w$g608"&eX&y"0RP D P I Dd c a` U Y W D I U D S Q P I D F E Rgfe'b601X$6VT0R6HGD D U Y F` c S v U E d` t cd D P S U Q q i Dd c P Q $HbewB&Tuf06fRs$prpfeg&fd h
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
Y ara (E)(Y ) = 0
U Flt(p)
^ Colim (Y B U, O)
~ ^ fi (Y B U, O) (Yp , O) U Flt(p) ^ ~ ~ ^ f1 , . . . , fn (Y B U, O) V 1 , . . . , Vr B = U V 1 Vr R X
ETg0$pHfp6 F S D a U Q E Dd S c D F I D x DS a U Q P DS D $0"g6pR1Tibd E I QF Q 6peXi E D 1& F I D` b0Q ui F I DS I D I T&gp6gD D I` c x Q I $bpBp&|$U P D a 1pCc D &
F RD D U Y Ed D $HgF D Ed D I I Q` F c S E D I` c x Q $g$$0b'w1$bpBp& S D I v c S D F F` U id D I I Q` F c $B'wB2Gb$ieg6$pfTeS D U Y S DE Q UE D x t x F U D I Q $iG$6Gy"&g06t E I c E D c x` E Dd F I Q 6p$1peBf1bspTE E Dd D U Y S D E Q U E F U D I Q t D E I D F E fX60sgT0u$$T$guzppT6g$GD
B U fi
~ ~ ~ D Y ( f i |f 1 , . . . , f n )
p P(Vj )
G (Yp , O) ^
^ Y B Vj
S U Q $0
F gD
1in
^ (Y B Vj )p = ^
S c D E` S Q F P c D p6yTfp0yTE
bf i E S U D F P Dd U c E W E E` cd F I Q E Ed` c F W E D v $TRfh&e1GfpfpTVbpg&w6i D Y` Q $U$Ip6IecgPsD"xTf`$6TS0Qx0QG`I$UVFIDxWG`ecF0Ib`F&pg0t E E E D Q I Q W` Q E E c U c D P E` c ybgP Tpe1Rfp h U D F` P`d D I Q` F P U S F E I Q P cd F I c E`d` F U I &ygff$&Rg0b6TG6pgBfi0pTfb& i D ` a I Q I FE D U Y E` U D I U E I` Q x U c c I D d` F $f6$s$$6fp"ue"ugdbVgD D E D P E Dd E D F U Q &6w6s1b1T&0TF &|p6V1D c I Q pt S U Q t F D D I 6puRRi D I v c $B'wS D U Y S` cd P F E D d $sbpfgGrqo D P cd F $w16zRD S U Q $0
Rp ^ R R ra 1j r Yp ^ E 1 j r. G Yp ^
ara (E)(Y )
k
j
B
j : X U
k
G (U, O)
^ U B B
G X an 1
X0 , X0 X
B
X
B
U U
B
k U U
B
k
S
k
X
B
1
X
B
S k
U, U X an GB B (X Spec((B,O)) Spec((B , O)))an
B
X
B
S
B
Spec((B, O))
U X an an X0 X X0
B
U
B
X
U
X
B
X0 X 0 U
G"
` F c F U x x Q P D x x c S c` I U c I Q t E U E E D ` P D x x Q bTe&20Rs""ewpe&u$"up&6G$ h Rs""pRP S U Q D U Y t I { I D E I Q U YS c x $0s$6w6&gs60$pBgD i F I D a W D x E` S Q x I U E S Qd c F` U W I l g$f` gg"Tf6$T0"m$"0bpdb$&$ii D F D# Q I U `EE U c FE &hgDpm6VG6e|1D D $ F D# Q F U Q B R!py&0wCi D I Q` F c P {` F P c x Q P D I I Q D I U F E & pfT'bT06"pgh$$0z6$yGD F D D E E`d D Q I v c D S U Q gzTTbB$e0$B'huT6pRP h D I U `EE U c FE 6$TT6p2GD t E S Qd gwpfV% i0Tg&` gpfGbg6b` ed D F I D a W I Q` E Ud P I j DF Q I I l Tp$2ipi F I c I DF I Q P F e6TpgRD S U E D E E`d D Q I v c F S D a U Q I $TsTTfH&p6"erTg0$p$U F` Q E t F S c D S F U j 0G'TesT$ec yr i 6G1Gffs&p6"ec h $ifT0pg'Bfd S U E E D E E`d E D Q I v E D D` S Q W F c P c S$UTE U c D P E ` c W S I U t i i F I c ` S c a c S F I Q P S U D F P I Q I U F ` I { W I Q` F c` P Q E E jd D U Y F I D x D F c` W x x` D {` S W a I 0b'R&TT0c eV$sTef&1"f6bTpBil pReR$B$fpD )` ee'CpRs$g608"$Hb$6g& E I D`d W c E D U Q S D D x E` S Q x I U P I Q F` U I` d o i D U Y` S F W x E d c Q I Q x F E D S U D F P I Q 6gfbpu$0T0X&V"Tf$6T0h$60&sb$&6f6q0p$TR""x G6p$p$0s1$TR6p8 i
PICX (X) B G PIC (X ) X Y X an B B PICX (Y ) k B PicX (X ) X B X j: X Y Y j : PicX (X)
X
DH~&i F Dd ` S F D gb6bTfd F I c` Q E E c I ppfgP TphD S D F I Q I U F` I { W I l $UTRP6p8u$66R&ii S c D W I I Q I Q` F c P {` F P c x Q P D I I Q D I U F D x e6yg$$0&gpfTegbTRp6"pgy$$02$$Xg"&pc D E E`d D S U Q Tff"$pgP h c v i D E E` pTTfd P D a 0Cc E D Q I v 1&p$B'c h &B"Tf$6T0n$B6$p&spT0u$$U D D x E` S Q x I U W I I Q E I Q E Q i ` F cd D S S c P` q D E D U Q S E D W F`d c` S Q F P I Q D E W F W` S Q S E D U Yd D U Y S D c W D I` Q E D E I Q S U c E U Q gEbTefTdpgs&u$0T0&sTbfeT0Tp8&RfT6p$$g$g0eg&s&gfpTuHp$e60V 0 2 6 p 6 F cd S P` q D D U Q S 7( b`T'gDS"wecgi&Xu$pp f $0 s$ T 1 6 T 0 iT u R A wCeh n & 5 i ) 5 gB 1T2B@ 6 p'w eB " q6 $ 0 p T5 g 6' 0bb g b6p 1R0 2'e6 c0 ) e$ f epT 9 8 pr G6 0 P 6 ` F c d D S S c f` q D D Q U Q S 7( bTefT|we5&&pu$pp b p 0iu6e) T7zpeu e qB6 6 ggWb rR f
b6p
B 6 B RqH 7b 6sT g t f eue B u 6g bT s
OU U
t|U U
Hbx i 6& i "bu&|pfTTgTT0sb&&&"fp6g$6pfT'wS h $$1q$i i D x x Dd U I Q` F S D E E c D x ` U D cd S c d D I I Q` F c D U Y E D S F E D d o D D ` Q I v &s&f0$B'c D c I` E` Q a I U F E pp6bTppu$GD E S Qd % 0bX!i D d D I I Q` F c D U Y E D S D Q I v c D c I` E` Q a I &g$$0b'wS h $T1T6$e0$B'i0e6fTf0pu$U F` i E D S { E Dd S U E D E I D 0Q $6sf$TyGg& ` aeX 1b` T$$r0G6&H` feX GD ` E` S c F E D d j U Y E` U t D E I D ` E` S c F E t F D I i D S` cd P F I c I D F I` c x F E D D x x D U D` F S c D I Q P D E c 1RRD 0TfeR"e6TbpBG|"bd &ubGp6h&pR1Gwv i D x x Dd U D` F S c D S ` x D S p""gfu&sbGp6Tb"g$ cd D S F I Q x ` P D ~ XTp"R1H6i 0bT6g"f&&GD I Q` E I D x` D F E D U Y F` Q a I $0ph0Q D D $i&` a h 0$X$f0$B'sTg0$pd$1D I Q I D ` Q I v c F S D a U Q I U F E D x x Q ~ i Ddd I I Q "0H6pfb$U dp$I I Q` E I D x` D F D ` F P D # Q S F E pfT6g"f&y&XRx1$$1D I U P I Q FE j ~ 6ip&VGD iui c S U c I Q D S` c S F I Q w$eupbp00RP D U Y E` U Dd ` E E Q x` F E D ` P D ~ i D Q I v $f$6f$fG0u"bGgH6p$e0$B'c h E P Dd E I c t F D I 0cgzfs6p$wgugD i D U Y F I D a do 60yb` qi $"$6ewpsb$n0bT6g"f&z&yGD D U Y D I c S E Ud I Q` E I D x` D F E t E D S { E D I Q` E I D x` cd D $61& pfG6"b&gB& W F I` I Q ` x D E cd S c q i D S U Q P D I U F I D x D U Y` S W I W F E Tb`$U fTFpgP h "gTipp10$0R$$$fg6gprGD Dd E I c DE I D FE e$Gg&p1D D x x Q ~ "0H0i c x W P Bg$wE h fCt D Dd ` F P U W S S &sf$fTR6&b` D ` ` S W F W` S c a c &pfTTgbe'wv PW%422RPGEC DF I E Q x Q D I Tec0u"pgPy$$U F` 0Q i F RD E D Q I v c EFS D a U Q U D S c DFS D a U Q P DS FE 1&e0$B'wGp$0h&g$Ve6TTg0$pgV1D I HV S U T S Q I H F D w G w p BR6g b B e u'wb G "w w qp ' e w h e gfg7 3 b%' 5b 0'u p e5 6ep ' 6p " B 1 gp R ' R 'p q 3 BbP R26e$B h ' ! w 0 w G GG 6g' x|c%# $ $ e u6ewb u g2 w b u ' 0' p 'e D D` I U $yb$6yx
bB X
an
'n
!)pCf$Ty"bsf|ii F I c a` U E D x x Dd Dd c I l D D F S D a U Q D ` ` S W F W` S c E U Q E D D S U F P U S F E c &sTTg0$p&pfTTgbe'a h 60Gy$$6TG"TE D&sD&pQ6I"veciD0ec6f`GpQ0au$0pgs"0RTg&1R1T$u0b$R&"s& I E` I U F I c F W D x x Q P D F I D W P W S I Q` F` I { W cd D FS D a U jd cS DE I D I Tgp6pQ )"Ggudil w p 0 w ' p 'e ) Gu'w G b E 0 qp "b u u'e T7p' q 0r g ph 6ge6'
P 0
j U X X an X U X an
# % % # @ % 3 % # 4 # % # ! 4 A )&)5 CC$psxg'B0|rx$xp$'&$R
X = X Spec((B,O)) Spec((B , O))
Xb
-X
2 0 f h e R e5 G pu6e) T7H0' C 0 ggWb G 1G 6 f 0 q 0' b w u e) e g 0 gb7 t fp6e u'wb u R
B X X an an - X X X PICX (X) X an - X
e "g ) T6w 0 GG ' p'
P Ub P an Xb
an
U
X P
B
B k PICX (X) U X an
k
an Xb
red
Xb k
k(b)
X an
X
B
e: B
X
w
w
6 y 6gT g f 1 w 43uq'e tu ) 2 e q B 'bb b 1 p'6 6 p 6g' ' (prG6
X L OX U U
PICX (X)
(OX , U, id)
(L , U, t) (L , U , t ) = (L L , U U , t|U U t|U U )
(L , U, t) (L , U , t )
U
OU U
t|U U
B
GL |U U .
G L|U U
X
P an
1
GB
1
X an U b Xb B B
L L
P an Xb = P P
Xb
k
^ U = U B B
Ub
X an = j(X) U
X an - X
B
Y PICX (X)
PICX (X)
k
X
GB ^ e: B B B X = X B B = X Spec((B,O)) Spec((B , O)) X e : red ^ (L , U, t) (L = L (B,O) (B , O), U = U B B , t = t|U ) G Pic (X ) e : Pic (X)
PicX (X ) PicX (X) PicX (X)
e
X
j j
G Pic (Y ) X G Pic (Y ) X PICX (X) PicX (X) B ^ X = X B B X an - j(X) P t : OU 1 (L , U, t) G Pic (Y ) X X
e
(L , U, t) G L|U
X
X
X
& 5 i ) t wd 0
! E F I c a` U E E F I` Q U D E Dd S D {` S W )e'f$TVfpuh&g$Vfg6bTpa D F E D S d` t S D I` x S D F S U Q q y1Tb6$b"g$pr6i D U Y F` U W I Q $i6&&g0$t D U Y W I I Q F I cF $y$$p$ie$i S c D W F I D E W S F I D x Ddd D S U F c I t D Q I v c D S d p6gTG1T$g"bfg$Te6&p$e0$B'T$pfec h $$VGD D I U FE D U Y S` cd P F E D do i D x S Q I D F F D P F S Q c S S c $VfeRXGq&0p6sTGgR"VG06ewe6 D W F Wd x Q &yTgb60RP D fd DF Q I I l Tp$uii I D D x $ I d Q D x x Q gs"p%fpus""pRP D E Dd ` E E Q E I Q` F c F I D E W S D S E Dd E D F U Q F S U E E` S F E D x U x` I` x Dd &V1b$fTT60b'w0G1T6gfV1T&0T$TVT6V16yf$b"sfgpQ
|x| = min{|mi1 ,...,in |i1 ++in ; x = F Spec(F )an Z = (I , U, t) c : CorB (B, X) s1 PicX{T } (X{T }) k B Z alg I|X = I (Z) U Z = j X an - X X B1
i1 ,...,in
S U Q F I cE Q I $0e0uugD S U E D x S Q I D I U F` I { W I Q $T2"p$2$$66R&0t S U Q q $prui D S` c I I Q` F P c S d c W jd D E S U D F c S W I W E D bp6$0bpw8e$f` )2&Vw$g'wg$pi& F I D` gfpQ id cS W I W I D D Q I v c I U E c FE j eewg$pug&p6"eg$p6GD I t E` Q D F U Q B p8R&p!i D F I` Q #E` F E D F &V0fp&X1VRD F I D` F I Q b00RP t E Ud D r gb$i6i D D I v c FS D a U Q I U FE &s$B'Gp$0u$1D D U Y S` cd P F E D do 60feRXGq$i ` I { D F D D S d $6u &T$pfec h h 60Gyfr60T&G60H~ i E U Q E cd E I Q S W ` E I Q D D E E`d U D`d Dd E I c I D F I Q P F E D U Y E` U $f$6 Dd ` E S D I f$fGwg0a b` &VTffBgfbXb6e$BU gpRs1D d c W ` I U S c ` I { W W x S D D Q I v E U Q I U F E j ~ p&$de6$6g&"g8|&p$B'c h 6pTE $hGD iVi E I QF Q I F 60Tp$dgD D I Q` F c P {` F P c x Q P D I I Q D I U F E $0b'06"pR$$06$dGD D U Y E` U D F E` I D d $f6$BTGf&gD gf` D d D I I Q` F c D U Y E D S D Q I v c D c I` E` Q a d D U Y D F S Q x j &pg$6pfT'wS h $$B&p$B'B0e6fGp0g60dTTpuf` I F` bpQ PW%422RPGEC I HV S U T S Q I H F D Pq 0 6e) 7Xpeu w 2 R Bb q 0# w B 'r p 'e p e G p"T 6g 6w u'6 GG ) 6eru b g r0'u b w u ' qp' 6g' pr Gpr " Hbx i E U QEE D ` g60G$ h RP I Q` F` E Q Q S cd E I c pfTbT0up$yH6e$ F S D a U jd c S` U S F E I Q P I l Tgp6pQ )swf$TpR|i&i D W F` I U I Q` F P D E c &Tb$$z0bTfd pCc P D a E S Qd wpfec cS I DS I t F D D I w&6g$z0Q RRdi p6p00R6Ti$I E c F I cSF I Q P I DS D D & d D I I Q` F c D U Y E D S D Q I v c D c I` E` Q a I U D S` U S F E I Q P D F E D S d $$pfTeS h 60$V$e0$B'Vpp6fGp0z$Vb6T60R2XTGb` t1g6b"gTs$0r&i S D I` x S D F S U Q q t F I D x D S` cd P c I l i Dd ` E S D I` d c W ` I U F E TfeR"|i$0b6bTpa fup&|$1D D U Y F` U W I Q $V$$&|p&t D D E E` &sGbd U D`d Dd E I c I D F I Q P F gffXf6p$zU g00RRD I Q` E I D x` Q P D F E pfG6"b&&g&1D D U Y W I I Q F I cF $i$6p&0pp i D I Q` F` I { W D d c W j &dpfTb$6R$i&e1&f` ed E I QF Q pTpV i D W xS D c x W P E U QE I U &h"T8zB$wTE h 6pT}$u E S Qd c c S D {` F I D I l i D W x S D I Q` E S D x x` D I U F E D t E Ud D ` U i i I Q` F` E Q Q S 0bpg6g&` ir0g"T8pfGwg""bh$$yGg6f$z$H$Y e! 000bT0up$ cd S` Q yb0pa E c x W P pBg6wTE h D D x E` S Q x D U Y` I I j I Q` F c P {` F d c I jd F E $zT$$p"z$$$U $U npfT'bTbp6ec ey1D D WE Q x Q gT0u0RP cd U Y F` U W I fuD$z6&&kilHi U W { F d c I d F E j P &u6Gfe6pc je"1D pt S E D` I F E $UGuf$6{"1D D x x Q ~ i W P I Q I jd D ""pHpg6p6gW )u& ` I { $6gcB$wPTE h x W E I Q` F I Q P E d F I c { S W 6pfTb`&60R1Dbpeu`g0a F D ` S F I j D P I D F j S D S F I x D F v U d` Rfd$fTB$U VRgTGE` gD edT00Q"V&H"&TEfpt E S` c F I x Wd E D` I { E D P I c I Q E D S S Q DfeDbWpf$6x1R6p$60TT0RP E D Dd x D E I j S c W D I D I D x D `d F E 1&if$2gT6gD )dseVTS$6Ig0$gsF0"gS$bfHGD D$i6$Ip&e0i0T$$p"dTh$U XRP6TFGf&gD e60Tpgf U Y W I Q F I c F D x E` S Q x D F I j D I D E` jd E I Q S F I Q i S E E D c W F D x D S cd P F I Q $UG1bdp0gIg"Tf`eRVpTE F RD 6pfTeTEbpf` fTsfd$6b6sX6i E I Q` F c `d c a` S F E D t E Ud D r D Dd ` E &sf$fGEf`"x&0TS $0R1TS c F I D x D a U Q P D I$yxp8$Y U F I D S Q ` U F gD E Q I c E F S a U Q E D S U E F D ` P I D&p6"veyTgD0$pyfdy$TyIg&R)0QRP &e1&f` &&yE1Rh6U0$$ U c W U D D P D Y E` U E S Qd c t W P I Q I j 0bppg6p6gW )d D E I Q` F` I Q P E d F I c ` W a Dd ` S D S F U I U F &60b&pR"1Db0p{bTSp"FRf$bTFhT&ec$"E1D ` E t F D I i D S` cd F E TgugD}xpfeRPB1D D W F` P I j $ugf`6$U v PW%422RPGEC I HV S U T S Q I H F D Y ' e 0 ' h e g f 6y g'
0 w w qp ' u( 'e e6gp p" 0 B ' w g p 'e6 20r ( 6 bb w ' ' t6e gT" 0u6e) T7Vpeu 5 ' h p " GGw g ' Rh T ) 6u0' s1( T 6g 6y ue$ T B u '
I|U = OU G Pic (X) X I (Z) Z I OX Z
0 i ) y w GG 6 0x pn'e p w G R b w $ T p" e5 ) 6eru b g r0'u wb u ' qp' 6g' pr Gpr i D F I c a` U E I Q` F` E Q Q S cd c I l i E D` I pTpCf$Td0bT0up$"2uiuf$6{ E P Dg6Iec$pQuGE1TSpgP"$E1D6UpSpEfdsRDb`TF'dgDSSpcgP"&u$ppf"Tggfb$sfb$pghBTGG60R"Te'f$T"uewRW v I DS Q D Q D F E c ` q D E D U Q S E Dd D S F I D I D`d I U S`d c F W D F E` E I Q P D F I c a` U E D c F j i E F I c` S c I pfe'a b` h E U I D S E` F cd D S S c P` q D 6&6TbTefTdpgs& D x E` S Q x I U F` U I` E` F cd D S S c P` q D E D U Q S E Dd S U "Tf6$T0"h$6&6fHbTefTzpgi$u$0T0H1bs$TE D x E` S Q x D "G$$p"bd
Y X an
B1 PicX (X)
D6U0YHb`pfdgPsFGEDudqo0i0DTFIgD&` agWzI0b`TE6fdRP6If` j)d S c Q U E I QF Q I F 6pe$sRD F I c I DF I Q egT0RP t 6GGffV&p6"eHTgp6pz$U S U E D E E`d D Q I v c F S D a U Q I F I c` S c I h U I D S ` F cd D S S c P` q D D U Q S &6Tx'gdwef$u$0T0 b eTpCa f` w 'e bY u e5 b R ' p 6 gb
X
s 0
e)
e
u'wb u
w g 6 )42 bb p p2 q bY 6 T f p w G G w GG R 0' wb u ' p' 6g' prG6 F I c a` U e'f$TE D DE c D F I D x D I c P D &sG06s&g"g0$e6wyfd F RD P D a 1pCc
^ Y = Y B B
i
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
j
B
k
e
B
X
w
0
X
X{T } = X(B,O) (B, O){T } s0
1 PicB (X) X
s 1
GG PicX (X) ,
B j : PicX (X)
j:XY
1
G PicB1 (Y ) X
E D U Q S E Dd S U u$0T0H1bs$TE
X
^ Z X = B B X B c ([Z])
B
F k{-1 t1 , . . . , -1 tn } (I (Z alg ), U, 1)
I (V, O) = A F = nN I -n k(A) Z a1 , . . . , an xF
FbpQ ` 6 T w0 B
t
I OX
U
I (Z alg ) OX X
Z
X
j
I = I
GX
B k ^ Z X = B B X U X an - X
Z
V
j
B1
I (Z alg ) OX
1
G X an
X an
X ZV Spec(F ) Spec(F )an
c
Z X B
B
U X -X (U, I (Z alg )) = (U, O)
ZV = Z V
(I , U , t )
X
X
t
B
I -1 k(A)
t CorB (B, X)
an
x
B
B
Z alg
k
V Spec(A)
mi1 ...in ai1 . . . ain }, n 1
Spm(F ) Z = B |k | Z B
alg
(I , U, t) Z alg = Spec((Z, O))
I (Z alg )|X
X an - X
X
X
A Spec(F )an
(a1 , . . . , an ) 1 = I (Z) Z U U U U U X X Z alg
B1 = Spm(k{T }) k
I|U = OU
GX
X an - X
F
X
X
F
u
i $& i 0bT0up$fe$w66R& i I Q` F` E Q Q S cd E I c ` I { W D x E` S Q x U W F`d c S U F c I D D F I c a` U E W F W` S Q S cd c I T$$p"u&sTbbp$'6s$e'b6GTgb$p$"|il bf i D S` cd P F E D D x x Dd U I Q` E Ud P I Q P cd F pbpfgX1s"bu&|pfGbg6pgVRD t c P D P E I r EpyRXeci D U F` U I Q $Yb$&W$u06t 6p$eBp$XGD E I c d c xS Q I FE D U Y W I I Q F I cF 602$$p$ie6i D U Y ti i` $fpD )t D U $Y S D Q UE F D I GE$6GV&U|pQ t PpCc D a S c e6 S D P cd x D S D F F` U R0f$"TsTT$ii P D a pCc D U Y d D $gF D S` c F` I bpb$$U D x I d Q I F` U W I I l "p$%fpu$U"b$&$}gD ii D U Y d D $gTF D S` c F` I U D x $ I d Q I U P I Q D F E D d o bpb$$z"p%fpu}6B6p$dTG` gHqxi E I c e$ E F I D` v Q P d c` x Q d Q I Q` c U Y D I ` c E` c w0bgPzRD RsVDfe"p$I fpu0bF'61W$$UFbeGfF'E t S D`d U P` F S c I 1gfb6RTTe6B0i i i D x x Dd U F I D x Dd` P c Dd U Q P W C 00 "b"&s0"gfbgpf$pg& d` D x Q f"x0RP gfTIiGD S D` F D F E F D a W D x E` S Q x D v Ig&` g2T$$p"2wi S c e S D P cd x D S D F F` U Rp$"gdGb$Y S U E D` I { I $Tf$6$b` D x S Q I c F I D x a` F cd "T0$ydsHDpfTefDTS D U S D Q UE F U $YsgTE0u$$T&DhIi&i $Tif$6&fV"T0$yys0"gpfTefTS l S U E D` I { I` D x S Q I cd F I D x D a` F cd D E I c Dd ` E S D I e$Xf$fGwg0a b` FE GD D U ti i` $YfpD et zfegzDf$6&f`B"T0Q$I I$ygD$pc Dd c W ` I { I D x S U F x E I c 6p$ D & D E E cd P cd D U Y D S` F I D a D TT0fggfh$zTf&ygf` gS D U Y D S` $hfXr i P D a 1pCc D U Y S D E Q U E ` E I` c D I x c S D I $G6$GGbpd$p|TE pQ Dd ` E E` x c F I D x D a U Q P D S Dd S D S W ` E I Q P E S Qd c F U D I f$bTf"&0dg"gS $pgfhgg$fT6pgzwpfe&guil D U Y S DE Q UE F U D $G6$Gz& pxg1"TT2T&pc kri I Q t E D xS DF E DSF U j I D U Y W I I Q F I cF 60n$$0&|ppsi D &
X
P D a 0Cc
P D a 0Cc
E S Qd c c I 0bpgil i E Q E Q q E d ` E D I F Q 6IpT0rCig1Db6bTSpa f`rI0GE E D E U Q D Y S D E Q U E D F D x S D F I D x D I I Q E` c S D x t x D v i Dd ` E S D a I` F E 1bdrpTF6U0pG$6G&ggur0"g$$0Gew"g"'pf$fGwg0fGD D6U0YxSgDTE0u$$TEHIp&pF&guyIilpi Q U P Q U D D&xEpsfxSgTbewTFVH$I"ewHTypR6T$R1Ts&sG1$Tp exepg$0b'ww1$bpBp& c P Dd D F` c S D x cS DE I Q D P I DSS U P WS D DE F Q jd S c q i Ed D I Q` F c S E D I` c x Q D I ` I W D I FE &gpQf$$UTSy$$UXGD D U F I D` do $Yigfaq$i F I D x D S` cd P c I l g"TfeRBii F I D D a U P D d I Q S ` E I P F g"xgS $pQgShDfE6pgW&T6pQgRD D$2FIg"gfg6U D x Wd W I F` bpQ i D a FE p&` GD ti i b0D e` d c W jd D ES U DF cS W I W D D x FE E I e$f` )s$sw$T'wg6gpy&y"gTG&Gg$U P D a 1pCc
(i) (i) (f0 , . . . , fm )
E c P D v py6i
S D` F I jd S U E D P I D S S U P W S S c c S D I I Q E` c S I l gfTgD )6GsRgT$ge62wg$$0Gewuii
(i)
i
D Ed D I I Q` F c S E D I` c x Q D &g$6pfT'w6bp"0&y& D` I { I Q` I U W S D I U D x x Q f$6|0b$6gs$$y""pgP D S` S P W F U D I l i F I D I Q bR1i&uiu10b` a pgP E S Qd c 0bpt `E F D E D Q I v c EFS D a U Tig&p6"eTg0$pQ E1&ipc6 D S D Dd ` E E` x c F I D x D a U Q P D S I U F E &yf$fGf"&0Xg"gS $pg$XGD ` E D U Y E I D E Dd E I c G$V6gT2f6p$ S U E Dd c Qd F E D I Q` F E D U Y c 6GyfeP fXGg0b1$zwv PW%422RPGEC I HV S U T S Q I H F D G P 5b q 0 B 6 Rq6e 0 g ' peu Re qpx dc%# $ $ bf i E Q E E D ` P D x x Dd U Dd U Q P W F c Fd U E W S Dd gUpTT1& h Rs""gfu&yb6pR1&V'w$Tb&t D U Y W I I Q F I cF 60y6$p&Xpp i D U Y DS Q P I D U $ipg6gz0Q D U Y c S v U E d` $2wB&Gf0t D I D` F I $6IgfTpQgP D Y S U t` E I` % $U$0QeGbXi D I` c x Q Dd F I D` F I Q $feB0&fib00RP D U Y S` cd P F E D d $bpfgGrqo i D U F c S W I Dd $S6gDTe$gW0b$t E S Qd c D E Q I l wpfesG|ii R&6fdfewGRg$$0 D P` I` I` c F S D P I U S U Q d D U Y Dd S U E D Q I v c F S D a U Q I g601b6G&p$B'VTg0$pu$U F b`pQ 'i $GErg"xgfeP f$I$pTEewF&y0t S U F I D Dd c Qd S D I Q ` c S U D I Q F I D` F I Q b00RP D U Y ti i` t W F W` S Q S D x ` U D cd S U Q $1fpD )pRfT$0T6"gf&&g&if$prq i E I c D E I D F E D ` Ud ` U e$sT6g&XGw$b60Y E I c DE I D FE 6p$sG6&XGD D U Y S` cd P F E D d $feRXGwb` t E Ud D 6f$Xir i FgfTIpg$UY I D` F Q P ` D I v F D U Q I U FE j ~ &XD$B'csGSpa$0B$HGD i&i DE Q I l T0uBi$i F D Dd ` E S D I RVf$fGwg0a b` FE GD d c jd i pW&` )t& 6& i pfTFbG0TS6Bfdy&yp$T6Bfy&66pfTee6fs0w% i I Q` ` E Q Q c D D a U D S cd D t E I Q` F c F Q I E Dd P D a E I c DE I D 6e6yG6& F RD D Y d D D I c ` E` c F D a Q I $U&gTF$Bv'0` fSeX GSp$U0y$U F E P I Q td D I I Q` F c D U Y E D S D I` c x Q I U F E b`G`pQ6wyp)$$pfTeS h 60$6bp"0&26GD D U Y S` Q a S U Q $fpp$prq D D c I` E` Q a I U F E &pp6bTppu$VGD t I cS F I D x cE v U 6p0pTz&TE S U Q q $0rsi D d D I I Q` F c D U Y E D S D I` c x Q I U F E &u$$0b'wS h $T1T$$feB0&g$V1D D v i D Q I v &p$Bec h w0
X an X an - X = V - (X V ) Y X
i
# % % # @ % 3 % # 4 # % # ! 4 A )&)5 CC$psxg'B0|rx$xp$'&$R
Spm(A/(a))
D(f0 |)
f0
X0 = Spec( J -n ) n=0 I -n F n=0 Spm(F ) V -(V X) I -1 = (ai ) i Spm(F ) W {x W ; |f (x)| -1 } W - (W X) W - (W X) {x W ; |f (x)| -1 } W/(f ) = {x W ; f (x) = 0} W X
(1)
(Y, O)
(1)
U =Z= (1) 0 < < minxX (max0jm |fj (x)|)
k
Spm(F ) (V, O) J(V, O) = I J -n n=0
U=
Spm(F )
i=1
D( i |a) Di U Di X
n
(i) D(1|g1 , . . . , gm ) =
D(|f0 ) D(f0 |f1 , . . . , fm ) =
X = Spm(A) k |k | D( |a) U
(i)
Spm(F )
(1)
(1)
U=
X
i=1
n
(1)
i=1
n
j=1
m
(i) D(f0 |f1 , . . . , fm )
U = n D(1|g(i) ) i=1
= min( i )
D(1|gj ) =
(1)
(i)
A
(i)
|k |
a A
Di
:[[1,n]][[1,m]]
(i)
D U Y S DE Q UE F U D I Q t E D xS DF E DSF U j I $sgT0u$$T$guzpg1"TTT&pc |&i
i=1
I Q` cd D D I S D U 0bF'gSs$$Ug0a$pQSTFiP60Q&XF&guuili I U D I d c jd F DS I D I pW&` eIg&6g0$gD E D & E D E E cd P E Dd D U Y F` Q a I Q 1GpR1bs$Vbpp|p&t D U W I Q F I c $Y6$Ip&XpFp&i I Q` U W S c E S Qd F pf$I$Tds0bpcXE1D FS D a U j v Tgp6pQ 6i E S Qd c D E Q I l wpfesT0uuii c0Q6t I F&0TFV6pQuXFRD U Q S U D$ W x D I` Q F U Q I j U E Q UE P I Q F U D I l S c e S D P cd x D S F U D ggp$"gi& Sg8 Fb0&pFugD YXSgDT0u$$Ty60&$gugii I j U F` a Q pQ YbpQ0zIp&t D$HFe0YweBDTdgFRD U Y I c U S c x S I D F I D x D S a U Q P D S Dd F I c S W ` E I Q P I 0"g6pR1Tfewg&GpRu i P D a 1pCc E D I` c x Q E D I U P c P D E c P Dd S D F` c S F D E S Qd c F v U E d $feB0&H1&h$gp6w&spifHTbewTi&wpfe"&Gqo
X gi A A/a U D(1, gi ) = D(1, gi ) x X i [[1, n]] n {x X; |hi (x)| = 1} i=1 D(1|g(i) ) .
|gi (x)| 1 ZU
U = n D(1|gi ) i=1
X = D(1|gi ) D(gi |1) D(gi |1) -1 h i = g i A U hi hi A/(a)
< |c|-1
S c p6
S D P cd x D S D F F` U i Dd ` E S D I` F E ggp$"gysTT6i$0b6bTpa fsGD
D U Y S DE Q UE F U D I Q $gT0u$$T$guhp&t
U = {x X; |hi (x)| = 1} U = {x X; |h(x)| = 1} h A Z U (A/a) X X
D U Y S DE Q UE F U D I Q $VgT0u$$TX&gu|p$t P D a 1pCc S c e6 S D P cd x D S D F F` U R0f$"T2sTGb$i
-1
A /(a) P A [T ] |al| 1 X =U bi (1 + ac)
X
P (h ) = al P (h-1 ) A D( |a)
-1
i=1
n
D(1|a)
bi hi = 1 + ac
i=1
n
lA
b i hi = 1
h
h-1 (A/(a)) P A /(a)[T ]
h
D(hi |h1 , . . . , hn ) n = 1
A
Z
c
A/a
bi A.
c, bi A.
1 + ac
X
A /(a)
U Spm(A)
f0
(i)
P (h-1 ) = 0 D( |a) A Y
(1) n
D(|f0 ) U (1) D(f0 |)
f = a-1
Spm(F ) W {x W ; |f (x)| -1 } W X
aA G (A/(a))
U
< |l|-1 h-1 A
(i)
V Y an J = (Y, I (Z alg )) Spm(F )
D(hi |h1 , . . . , hn ) X = D(|f0 ) n gj = fj /f0 V
(i)
1
X
W V (W, O)I
(i)
bi
n=0
(1)
n-1
h
Z =
(i)
bf i W S F I Q x W F E D D x x Dd D v pTp"&V1s"bsw$i d c xS Q I U c D I I j pBT0$dp$6ec )d E I c 6p$ I Q` E I D x` Q P D E S D` x D S U c W j F` U Q S I D d D` S Q E a` d c W ` I j pfT6g"f&&Rg&"b"T$$e$f` z6&&T$ ggfpTf` f&e$fk$U 0bT0u"pR1&gb|$uT&ezGD dfegW I Q` F` E Q x Q P W Dd D U Y D S F U c F E j I W F`d c DFF D ~ GRHi E I c F S Q U E Q P F I Q E E U E E D ` P U c W ` E Dd E U Q F D U Y E` U 6p$Tpu$$T0g2pT6G$ h Rh&p&fs6p$T$$ I Q` F P` S F E D S E S c W F`d c W D F F D P S D {` S W a F U D I d0bT1TH$eTbfe0TGgRgg0$gudil
(X , O) 0 I =
I
i
P D a 1pCc E D S` c I I Q` F P c S fe6$0b08 U c W j W F`d c jd c I j U Y S D S F I Q x D P I Q F v U E d` t D S Ud P I Q P S U Q i W F`d c W F D P D D E S D I jd E S Qd c F E $e$f` sTbbp0W eyupQ gTp"&6p$"&Tf0T6bg6pg6prq pTbfe0RRs&Tpa b` ewpfeVGD I Q` F c E`d c` a` S F c pfTeTfefT2wv D {` S W a ` U Y F 6bTpu$gD D I Q` F` I { W D d c W jd F E &dpfTb$6R$&e1&` eHGD I Q` F P` S F E D S cd F I Q t Dd ` E S D a I` d c W ` I z0bT1Tyfp&pf$fGwg0fue$fz6U P D a 1pCc
I O X t
I
D W E Q x Q P cd S c gT0u"pRBe6
D D c x jd D U Y F I D` a do &speBf` es$Vgfwq&i
,
D U Y F` U W I Q $Xb$&1&|0&t
D U I Qd D F P c D D F` U E D I U F` I S U Q $0$pfT0$RsTb$Ty$$b$T6p8
,
c I Q up6t
(Z )I
D D U Y` S W I W F I` Q D $$T$g0Vb0fd
F RD
D Dd ` F P U W S S` E D F I c E Q x Q P E D Ddd` x c cd D I ` E W &sf$b6$TfsTpT"pgV$ffb"e$pfT&
[Z]
D W E Q x Q P cd S c gT0u0R"e6 D &DpecBf` edgEwpfdVD& i$& i x j t S Q % i Q` ` E S c D W P Q j E c D x FbpQ &iFIgD$f`gP6I)0QRPsEfd$fGEE0Qu1gT0u"pRh$g& ` D ` E D WE Q x Q P U D IpfTFbGQ0QT$yfd&R6I0$IgW )ds6Ip$i"x"pQgP E1bdSe D c D&Dppc"f` )d$YHEpSTIp"xFRXDfecDbdiDf$I6VDR6pc$6pQuTEDSTS0QRPD$$U x j D U I Q F Q D S` F I x W W ` { P I I I F `0Q IPW%422RPGEC HV S U T S Q I H F D 2 Rh e
^ Z X = B B X CorB (B , X) (I , U, t) CorB (B , X )
c
q'ug
2e Gw0 i ' w peu p e5
red
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
GB e: B = X (B,O) (B , O) X
k
k
^ X = X B B
2
CorB (B, X)
-e
CorB (B, X)
^ I = I OX OX U = U B B I
t F D D I 1RR|i D Dd ` E S D I` d c W ` I U F I D x Ddd D S U F c I F E &Xf$bTwgpa fe$fd$Hg"bfg$Te6HGD D W I I Q D F I D a W I Q` F c E`d c a` S F c g6$p&Tg$f` g"0b'ffe` bTyfd F gD t
G Pic (X ) X t OX
D U Y `EE U c E I Q U YS c x 60TT6p60$pBgD P D a pCc
c
(I , U , t ) I|U = OU
X G CorB (B , X) CorB (B, X) ^ Z = Z B B
` pTD &ib$2T08eVg$6p&HGD i D S S D Dd U x S Q cd S c D W I I Q F E i ` F P D # I` P I Q F E D d o $b60&X1q$i
G O X
D x E` S Q x Dd S c "G$$p"Vfp6 D D c x j &VppBb` v D E Dd ` F P U W S S` E D F I c E Q x Q P E Dd E D F U Q F S U E d I I Q I F E &f$fT&Tfs0pT"0RV1b1T&0T$Tw$U |p$VGD
i
F I D a W D x E` S Q x D &f` "Tf6$T0"sbd W F`d c jd D D E S D I j d S c fegW )&XTwgpa f` ese6 D D x E` S Q x Q E j D E E cd P cd F E &iT$$p"pTf` VTT0fgyHGD
...
[Z] e =
G I|X
G Tor1 (OB , OZ ) OB
I
i=0
G OX
(-1)i lgOZ
B Tori B (OB , I|X ) = Tori B (OB , OX ) = 0 O O i=1 I|X OX
D {`d x` E D 6bf$"bTyTE i ) f$yp8Bwi Dd U x S Q c v F` U W I l $$&ii S U Q $0
i>0
c I Q 0t
D$UTwpfd YES Q S U Q $pu S U E E F cd F I Q $T'$X0GE F RD
i>0
SgD$$0&X$0swpfec I I Q S U Q E S Qd D U $Y D U Y W I I Q F I cF 60$$0&pp
Tori B (OB O
I|X OX , OZ ) = 0
I Q` E Ud P I jd D E`d` F U I Q pfT6bg6f` eyTfffT&g0t
Z U =
CorB (B, X)
[Z] e =
(I )
lgOZ
I OX O X
X
Z U =
I
,
G CorB (B , X )
G OZ
,
OZ
G Pic (X) X
X
lgOZ
Tori B (OB , OZ ) OZ OZ O
GI |X
,
c
G0
[(I , U , t )]
= lgOZ
OZ
,
G OX
[Zi ].
G Pic (X ) X
G Pic (X ) X
,
G Pic (X) X
OZ
G OZ
, .
[Z]
Z
[Zi ].
G0.
Z
(I )|U = OU
[Z]
X
1
D x x Q E cd S c D W I I Q F E ""pT"fp6g$6p&XGD
D U Y W I I Q F I cF $6$p&Xpp
i )
D F S U Q P D F P c D D F` U E c TT$0RsTRp$Rs$T"wv
i )
[Z]
'e e
w
e
p g u'wb p $ $ qpxdc%#
F gD
D x F E E U D F` U W "g&T$VTb$&$ i E D G1TS i E D TS D I Q` F P U S F E I Q cd D E I Q` F c F Q I E Dd E I Q S c &mpfTT60RP g&z60b'we$B1b"p&wei
I Q` F cd D S cd E Ud D c I l 0b'g26f$s&"uii F D # jd D D x E` S Q x Q E j D E E cd P cd D D U Y I D W D I D E E cd P D F F D ~ gDpQ e$"Tf6$T0"pTf` GpR"s&y$u6gug&s$TT0fgyTTRHi cS DF Q I I jd D U wgTp$0Q eh$Y E I c D U Y` I Q I c P D E E cd P D I 6p$d$$p6egzTT0fg|6$U D & F D # I U W` P Q E E c P I Q c I gDpQ $ghfR&TT0h60&}il i S U E D F I Dd c a` U Y W D x S Q I D I U c S` U I` D x F E D S F U c I j U Y E` U $TsTgfe'f$"T0$s$$wb6&6f"TGE TsT&eg$U f$6 ES U DF cS W I W D w$g'wg$ps& D x F E E U ` Q P U E c I D W D "g&Tg&h0$wg&pdgug&s$I D U Y D x t x D F` U W I l $s"s&X6&&uiui D D` Q F Q x Q &b$0Tp0$ h D I U F I D x D S` cd P F` I S U Q ` U $$g"gfeRV6T$08$Y S D S W ` E I Q P P I Q F U D I l T&G60R60&i&|iui F U QF S U Q &p$0 DS Q P I D c I l pg6gdii
s
I Q` F P D E cd E S Qd c E I Q S W ` E I Q ~ 0b1G|fiwpfeXpg&GpHi
P D a 1pCc
F` Q E t F D D I bpT&RR| i pfTe"f&CT6$ec )&zbp$w|&H06d6gug&$I I Q` F c x` Q S jd D ` Q P U E c I D W D E I c 6e$ D D E E cd P c &GpR2wv i D D U Y` S W I W F I` Q U D c I` E` Q a U c D x E` S Q x Q E &$fg$pVfpu|$ipp6fGp0dp"Tf6$T0"pTf` I U FE $1D D U Y d DF FE $Ts1D ` F cd D S S D` F I j 'gsbgD )d t D Dd c P Qd I Q` F c U Y W D I U F E &bpg&bhpfT'01$$1D ` t F I D x W E` P W S E Ud q GE g"GR1T$b&i D D U Y` S W I W &$fg$p F I` Q U fpuec D Dd Q U D S S j &"b0&T&wpQ ed P D a 0Cc E S Qd c D E Q I l 0bpT0uiwi S U E E` F P D #S U E F D E` I 6GbT16GsR$6{ tpgbypT2gD E D S F I` F I Q E F D Q I v c FS D a U jd E I c E I DF I Q P F I Q &p6"eyGp$0Q es6p$s6U gpRy0GE E D v 1i D E Dd ` F P U W S S` E D F I c E Q x Q P E D &sf$fT&Tfe0u0R2& Df$yGgD )d d x DE I j F` Q i D F I c a` U E D S ` I c x cd D bp&0Te'f$T2gf$eBdf"& D` I { D P I c I Q E D S S Q P D I U F` I { W I l f$6sg6e66puG1Tpg6$s$R&ii F I D` F I Q gfT0RP D U $Y D S F I Q x D F I D W P W S I Q` E E U P E` c v i Dd ` E S D I` F E Tp"B0&g$npfG6gT&gwpf$bTwgpa fyGD d D U Y U E U E E D U c ` E` S c X F S D a U Q I c S E Ud D g60&s6TT1& h e` feqTg0$p6p0y6f$zfd E I QF Q pTpV
V0an V0 V U Zi s t-s t-s (x) 1 - (x) 1 - t t t Cycl(s) B
an X - V0 Zi X an an Cycl(s) = iI ni [Zi ] CorB (B, X) (Zi = 0) Zi Zi 0 CorB (B, X)
EwpfesGrqi S Qd c F E D do P D a 0Cc F RD
S U Q E S Qd c c I l $0swpfeBuii DS S W F U D I Q t F D I Tf`TgP"&gu0x1RRD i S U E Dd ` E S D I` F E $Thb6bTpa f2GD E S Qd wpfV% i D U Y d D $sTF D&iFgD"gfgWg6UsD"x0RP I x Wd I x Q D S ` E I Q P I jd Q g&T6pgu0Q egpt DE Q I l Guii I Q` F P D E D I pfTRTy$$U I0b`F'cffde` abTFyw&i Q E` c ` S c v D"xSp$2fypfTefTS T6g$0eBb` 1D Q I cd F I D x D a` F cd D D E I D D c x j F E D x E` S Q x D "G$$p"v i D U Y S` cd $VfeRP c I pQt FIgD&6p$D S I D I E Dd D U Y W I I Q F I c F 1b$26$p&yewpi DE Q I l Giwi I D S I Q D x S Q ` x D E D I U F I D x Dd U D E E c F 6T$}pwt "p$I h "Gu$$0"gf$gT2062gD p6h$$"GD D xS Q I D I U FE D U Y S` Q a S U Q 60b0p$prq S U Q $pu
. = maxi=1,...,n |gi (x)/h(x)| h = inf maxi=1,...,n |ai | ; h =
i=1
I Q` F c F I D E W S cd c` pfT'wgT$fa S U E D` I { I` D x S Q I cd D D F` U I 6Gyf$6&f"T0$fs&y6&6f` Ddd D U ` E W fb$&G1TS D"p$IuzD& x S Q cd E I E E` I x 6pQTT$$U2yFRD U ES U DF cS W I W D D x FE E I &sw$T'wg6gph&z"gTG&G6U F` bpQ Dd Q f$U&&"x h i D d D I I Q` F c D U Y E D S F S D a U Q I U F E &g$$0b'wS h $T1T6sTg0$pu$VGD D U YE U D E S a Q d D I i E D F $f`$6iF`&RDHFTgD0$UpTFdVpDT6Ig$GED F RD D U Y d D $TF D D I v c FS D a U Q I $$BesTgp6ph6U F RD D D# I $FR!pQ}6U F I D` gfpQ i D x E` S Q x I U S` I { W S Q j E I Qdd c E U Q "Tf6$T0"}$"f$6R$0pc 260bfe60V 6 i E D c F W E` Q S F I D D P Q S I g1pRpTdg&1R&$dil PW%422RPGEC I HV S U T S Q I H F D 1bb H eB 0 p w 0 w G w G e) e 6eu b R peu b ' peu e $ qxk|%H i I Q` F P D E U Q E D F F D P D d c ` P I` S F c Fd U E W S Dd F E D F I c a` U E D x S Q W F D pfTRTE h 60GyGRgy&ebg6b$'w6G1TfV1pCf$TsT0g$v
X (g1 , . . . , gn ) (V, L )n . V c : 0 CorB (B, X) B X X an k PicX{T } (X{T }) (U, O) U V
G PicB1 (X) X
D x E` S Q x Dd S c F` U I "G$$p"2be6b$&6f`
# % % # @ % 3 % # 4 # % # ! 4 A )&)5 CC$psxg'B0|rx$xp$'&$R
h -1 G (U, L ) (V, L ) t (U, L ) m = maxi=1,...,n |gi /t| s (V, L ) t - s < m-1 s m|ai | < 1 xU
P D a 0Cc F` 0Q i F` U I` b$&b&t
c I Q t I Q` F` I { W S c |0$pfTb$6g&Vepq
(g1 , . . . , gn ) : (U, O)n G G (U, L ) .
E D P Dd U Q D 1$w6 h f$$0&y& d D S U F c I D x E` S Q x I U F` U W I Q t i g$Te6y"Tf6$T0"$6&&pu'! $& i "bXr D x x Dd U
B1
Cycl(T s - (T - 1)s) CorB{T } (B{T }, B{T }) Cycl(s) Cycl(s ) Cycl(s) 0 (B, X) (g1 , . . . , gn ) (U, L ) (L , U, t) PICX (X) 0 CorB (B, X) Cycl(L , U, t) (L , U, t)
h(x) = 0
PICX (X)
|s - t| < m
Cycl((L , U, t) (L , U , t )) = Cycl((L L , U U , tU U .tU U ) = Cycl(L , U, t) + Cycl(L , U , t ).
xU
U = U (g1 , . . . , gn ) (V, L )n (U, L ) (U, L L ) CorB{T } (B{T }, X{T })
c
T s + (1 - T )s (V {T }, L )
G PicB (X) X
= 1-
an
X
c : 0 CorB (B, X)
PicX (X)
1
(V, O) (U, O)
Cycl(s)
n
i=1
n
ai gi
B
s 1
s 0
ai (gi /t)
s 0
s 1
OV
U
GG CorB (B, X)
s
GG
PicX (X).
gi
G 0 CorB (B, X)
Cycl(L , U, t) = Cycl(s) (g1 , . . . , gn ) (V, L L )n . (gi )1i n (gi )1i n
i
E I QF Q 6peXui
i E D TS
S U E Ddd D U ` E W S D x S Q I c 6Gfb$&G1T"T0$fd i E D TS E S U D F c S W I W D D x F E E I U E I Q E E` E` Q 6gTe$g0y&y"g&T$60GfTp$wP D U Y S DE Q UE F U D I Q t F D D I $zgT0u$$Td&m01RRm
.
e i )
1 - maxi=1,...,n (|ai | )m > 0.
c
gi /t
L|V
h (U, L ).
h (U, L ) - {0} x U (U, L ) >0
ni
X
.
ni
t-s =
s (V, L ) T s + (1 - T )s (x) > 0 t {Zi ; i I} X (L , U, t) (U, L ) t (U, O) n i=1 ai gi s n Zi s i U
i0G6&X0eBb` HGD DE I D D c x j FE F gD F I D` F I Q gfT0RP D U Y d D F D I v c ` E` S c F S D a U Q I $T6"e6` aeX Tg0$pB$U F` bpQ i S U Q F c Fd U E W S Dd E I Q E` U W F $pu'b$TsbpTf$$&VRD E I c e$ dgD$I6Ipf`T'cwS h D$D$iGED Q F U YE S F D60Yi6pT0u$$&i U E I QE Q U E` Qd c f0bp &2$0T0y&2$$gfTf0bp0yf$6u0bT6TF R$$U D D U Q S D D I I D` E` Qd c D` I { I Q` E I D D D I F` bpQ 6 i D cF D I EuegWd&U&ggDudgD$I6Ipf`T'cwS h D$$V1D Q F U YE DS FE D U Y cS DSF I Q x I l $Tp"uii D Q I v c FS D a U Q I $e0$B'Gp$0|6U F` bpQ PW%422RPGEC I HV S U T S Q I H F D Gy 6'e T2' p p' b uqx|c%# $ $
l/k U U V (P1 )an k
(P1 )an k
i10p'b$T"bz$hb0G1y$phi6i & 0i pfTbG0T$yfHb0pa F I c a` U E D x x Dd U I` Q E D c S U c I l i I Q` F` E Q Q S cd S` Q D D P cd cd &R0f$yfy F I D x Dd x` E c S` S P W I 0"gf$"bTywbR1zpQ F I D x D a` F cd D V0"gpfTefTS D S U D` S W F I jd E I c I D F I Q P F E &i$gfTTf` e6p$|U Tpgi1D D U YES Q v i E D Q I v c EFS D a U Q E D $Twpw$g1&e0$B'wGp$0& F I D` gfpQ i Ddd D U E U D S ` I c x cd D 0bfg6G6gf$eB"$ P D a 0Cc D {` F I D ` I l 6bTg$f|ii F RD S` Q a c E fp0CTB$t D E D U Y` I Q I c P E F I` Q E` Q S F D D E Q E` I l &60$0$eb0pT$T0uTf&giui D E D Q I v c E F S D a U Q D S F I D E S D` Q F Q x Q $s1&p$B'wGp$0T0$b6pTp"p$" E D` I { E D P I c I Q E D S S Q P E D D S` U S F E I Q P S U Q 1b66rg6e66puG1Tpg&Tf$TG6pgs$0 $& i T0g$z&"G$$p"pT` eywgTbfT&hppfTRTE h 6pTTGgR6pir i D x S Q W F U D x E` S Q x Q E jd c S D E`d` F U I Q t I Q` F P D E U Q E D F F D P E I c w$i bubfx ` E I` c F` U S F E I Q P c I l T6feXb$TpRBgii t
x U
Gi&pX
XpVisw$$XG$uX6&0X
(L , U, s)
i WSF I Q x W FE D D x S Q W F D v pT0&XG"gpg6T$i S U 6GE D I Q` F c E`d c a` S F D I U F E $|0b'ffe` bTs6$XGD F I c E E` E` Q P I eTGp6wi
T s + (1 - T )t L
E I 6ec6 D U Y DS I Q $hSTF00"x&60&hbpec qi D P I Q F` j E d o F I D x D S` cd P c I l i D a U D S cd D TfeR}ip0$g$n& D c W pFRgDSgf`"xgDS$ czE6p$"x"pgP d I c D x Q F` bpQ&si D $ F D # Q I j E I Q F S c t F I D x D U Y Q S ` P gDp6U h6pGp610"g$&T6bgW i D D E E cd P cd F E &sTT0fg"fVGD t Q` P U F E Q P S q wIpfTFTS6I0RXpcp$i D ` E` S $`TpcX 0e6IfTf0pg$U Xgf` 'T$ pyRXe6& $& i pfTbG0T$zfe$ D c ` E` Q a I j F I D a Q S t E c P D P E I c r i i I Q` F` E Q Q S cd E I c D"0QRP x x S W I Q F ecyDg6$Ip&E1D D Y F` E I i S` c I x Wd W D` I { D P I c I Q E D S S Q P D I $Uibecil0DTfewF0D"gfgb$ig6e66puG1Tpg6$U F` bpQ i W I I j D 0RP60Q$gW eds& D x ` Q "GE$$Sp"xu&sDTFbp&"XR$w6p0Gwg0fu$Vb$T6p8"G$$p"Ry$Y U ` Q S F D D P U c D E S D a I` I U F` I S U Q D x E` S Q x D P D U D x E` S Q x I U D F I D W P W S D c F jd D F` U W I "G$$p""$T&g$pgW e&b$&$"il 6
CorB (B, X) Cycl(s)
B PicX (X)
i F gD D I Q` F` I { W cd D F I D x D F c` W x x` E S Qd c Dd U Q P W &g0b$R&"s&Xg"g'&""bswpfesf$pg& F c Fd U E W S D v 'w6G1Tzwi E D S x Q I E Dd S` Q a c E E D U Y` S F W x Q W E W F` P`d ` Fd U x E Dd D U Y S` I D a S D F I` F` c D $y0$y1bfp0CTy$TR"ppy1TbRb6b$2y1bz602b$pg0fybeB6I D & E D` I { E D P I c I Q E D S S Q P E D P c dd U E Dd S U Q D S S D Dd U x S Q cd D U Y i 1b661R6p$6puTT0Rp$y` p6 h f$$r1bp6puHTD V&b6yp8Xf$! $& i "b"&p6g$f&rw$pg2eBU iil D x x Dd U D a U D S cd D E S U Q P U c a c I i S U E E Dd P P D S F I $TfR&RsTgD W F`d c jd S D S F I Q x D E c P D P E I c F` j E d o Tbfe0W eVT00"y&ipg2RXe$ibppc rqi E I c 6e$ D D c x jd FE $ypp"f` )1D pQ D U $Y S DE Q UE F U D I l G$6Gy&ii E S U D F c S W I W D D x F E E Dd S c D F` U I` Ddd D U ` E W S D x S Q I cd D w$T'wg6gp"&""g&T"fe6"$$6bfb$&G1T"T0$|f& F` I U x I jd ` b$$2pQ epGE U b00R1D I` F I Q P F E D x E` S Q x D v "G$$p"2i E I Q F Q D F I D W P W S D c F jd S` Q 60TpVi G Tg&1R1T$2pgW eXfp0a E S U D F c S W I W D D x F E E I U S c D F` U I $Te$g0s$s"g&T6xp6Tb$&b` S U E Ddd D U ` E W S D x S Q I cd F I D x D a` F cd D $Gfb$&G1Ts"T0$ifp0"gpfTefTS D F` F D F I D x c E v U &Rup0"eB$GE D c I` E` Q a I U E I c pebTf0p"$6e$ P D a 0Cc c I l i DE I D zi0Gg& F RD D U Y d D ` E` S c $TF ` aeX FS D a U Q I Gp$06U F gD D F D# Q I &sRD0$U F` Q E t F D D I i 0Ggugi ! $& i ""gf&60b'p$f2TbfT&d0Q e0Q D x x Dd U E I Q` F c F Q I E Dd W E`d` F U c I jd
U V an (V, O) (U, O) t (U, L ) PicX (X ) G 0 CorB (B , X ) (L , U, t) PICX (X) V X Cycl(L , U, t) = Cycl(s) s (V, L ) (U, L ) G (U , L ) L = L B B (U, L ) (g1 , . . . , gn ) s (V , L ) Cycl(s ) 0 CorB (B , X ) lgZ
, OZ ,
` F c F U x x Q P D x x c S c` I U c I Q '$y"pg""ewppf&g$gp&t $GsTTfd S U E D E E` D D x E` S Q x I U F E P D a c E D Q I v 0CH1&p$B'c h &s"Tf$6T0u$VGD ` E t F I D x W E` P W S E Ud q T&g"GR1T$s6fi $GsTff1&e0$B'c h fH6Gz&pReR$i&i"Tf$6T0d$1D i ) fT6$p"iwv E Dd S U E U c D P E` c W S D D x E` S Q x I U F E D x` E S Q x D S U E E D E E`d E D Q I v
k Cycl : PicX (X) B G 0 CorB (B, X). k B k
j r pQ 2ui
E I D`d W c E D U Q S D D x E` S Q x Q x Q I U F` U I 6gffgueu$0T0$"Tf$6T00"p$u$Xb$&b` I Q` F cd D S cd F I D x D S` cd P c I Q E` c f pfT'gzXg"TfeRdpifpi
max( t - s . t
D U Y E Ddd D $VfbTF
E S Qd wpfV% i E D F` E` Q P I l 1&bGp6w}ii
)
` E E U c F` G6e"0Q i
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
(U , L ) Cycl(L B B , U , t ) = Cycl(s ) s Cycl(s) e = Cycl(s ) X
S c p6
D U Y DSF I Q x ` P D ~ $Tp"R1Hui e i ) fegW )d W F`d c j S D x` Q S c P I Q F U D I g"f&CT$6eyp&s&guil
tt -ss
m = maxi =1,...,n |gi /t| m = max1i n |gi /tt | M max(1, t , t s s (V, L ) (V, L ) max( t - s , t - s } ) < M -1 min(m, m , m )
F gD
F RD
E I c 6p$
(g1 , . . . , gn )
6
i )
tt
F RD
I Q` F P D 0b1GE E I QE Q 6pT0rq
B
GB
PicX (X)
= (t - s) t + s (t - s )
ss Cycl
1
c S D S F I Q x I Q t D S Ud P I Q P S U Q q Tp"20'p$fR60R$0xpi
B1 PicX (X)
^ Z B B X = X
Cycl(t) = [Z]
(L , U, t) PICX (X) c (Cycl(s)) = (L , U, s)
t
G 0 CorB (B, X)
s (V, L )
G 0 CorB (B, X) c c ([Z]) X an - X
Cycl(s s ) = Cycl(s) + Cycl(s )
, s . t -s
t-s t
S U Q c I Q $punmpt
I Q` F P D cd E I Q S W ` E I Q ~ pfTRTE f"60T&fT60Hsi
T s(x) + (1 - T )t(x) >0 t(x) B1 (L , U, t)
T s + (1 - T )t (U {T }, L )
(P1 )an k = Spec(k[t])
(L , U, t) = (L , U, s)
U {T }
< 1
P1 k
t
D D` Q F Q x Q $f$pe"p6 h
D U Y F Ud P I Q P I l 60X$bg6pgiui
1 = [1, 1]
(P1 )an l V1 (P1 )an l
= [0 : 1] V X (P1 )an k V X
^ U k l V A1 k
an
F I c` S c I` F E eTpCa fXGD
D x x Q ~ i DE I D D c x j FE "0HupT6&ypp"f` X1D
(V1 , O) G (U, O) ^ U k l
F gD
t E S Qd gwpfV% i
V = gG gV1
P1 k
(V , O) V1 an X
(P1 )an k
G (U, O) ^ V = V k l P1 k V - {} ^ U k l
U G
X
an
S D S F I Q x E I Qdd c E U Q gT06pffepV i
V
F I D x D S` cd P c I TfeRyhil i
D U Y d D 60gF
i DE I D D c x j FE pT6&speBf` GD D I v c F S D a U Q I U D F E` D d` $B'Tg0$pz6VTGf&gf0t
G (U, O)
D U $Y D I Q` F P jd S c &zpfTRpc )He6
(V, O)
G
V
U V
DE Q I Guil
)m .
(I , U, t) Cycl(c ([Z]))
0 = [1 : 0]
(P1 )an k
e) D I $$U
D D x S Q I cd E S Qd c E` c f i Dd ` E S D I &"T0$hi0bpiepf$fGwg0a b` F RD E I c E S U Dd c 6e6iw$gfe'a D I Q` F P I Q D I &}pfTR6p8z$$U P D a 0Cc D x S Q cd D D` S W E D I U D x E` S Q x Q E` F D P Qd U Q x I Q E D S S Q "T08ufh&hfTG|$$uz"G$$p"0G2gRf$&&"6puG1TpgP D I Q` F P` S F E D S cd t E c P D P E I c r i Ddd U I Q` F P D E cd I Q E D S S Q & pfTTGpgR"6eppfb6I pfTRT}6puG1TpgP I Q` E S D x x jd D U Y S D E Q U pfGwg""f` ed60BG$6GE F U D I Q F &pRD D S Q x Q E` F E y$$p"pTGD t E c P D P E I c r i E Qd P F I D x D U Y` S W d c F E pg"R6pigbgig"g60$g0bpGD D U Y S DE Q UE F U D I $gT0u$$T&pQ
k q : DX (r |f ) (X/(f ))red DX (r |f ) G (X/(f ))red q DX (r |f ) r r n ^ B1 k (X/(f ))red k
gG
t6fdDRPb`0QpaS$UpQruipTecB &Efd$0QiED&DT0RP6DipQTES$6{ifdFI0Q& c S q DF D D U S Q I F I E D E D I Q` F P c S F W S D I U F` I S U Q 0b0R2$6b$$p8 D$zIpfTPSTGESyc&t Q` F ` F D d ` t Ud D i U ` I F GE&E6b6iir0D$Yf6$UsGED D"xG`$$SpQ"xbdiD$UYHf`ppa&bf`R0HGqo PW%422RPGEC E D S Q D Dd P c F E D d I HV S U T S Q I H F D B w e s $ 6p 1 0 b wb 0 'e u q 0 qx Ge x U c D I I E U Q E Dd D U Y E` U p$$pc h 60Gbs$f$6
I (t - o ))
eB n s 0V6' Tb b 6ep w $0 R bf E I c DE I D FE 6p$G6&p1D D F v U E d` t S D I` x S D F S U Q q &s"&Tb1g$f"TTH$pri
|k |
E I c DE I D FE e$G6&sGD
ipWSTFIp"g$X1s"bi Q x W F E D D x x Dd D v F U QF S U Q &p$0 E I c DE I D FE 6p$sT6&rGD D U Y S D U YS c x D 60rg60weBTS U c D I I E U Q E Dd D U Y S` cd P P I Q F E D d e1$$ec h 6pTifX$bpfgX60&sGqo
E S Qd c F E D I Q` F P I Q E D U c D I I c I wpfeV1upfTp8s$|p$$pgpQ
I Q` F c U Y W I jd S c 0b'61$b` ee6 ` I { W $6R$ D d D I I Q` F c S D I` c x Q Dd F E &g$6pfT'ws$feBp$ybi1D E S Qd wpfV% i DE Q I l T0uuii F U QF S U Q &0T6pu F gD D U Y d D $wTF F` i E c P D P S D F` c S F P I Q D F E D S E U Q I d o 0Q 6pgygVgpysp&sTG60$q$i F U QF S U Q &p$pu F I D x W PS Q c I Q g"gg08|0$t D U Y E S Q v i D S Ud P I Q P S U Q D P I D S S U P W S D D E F Q $Twpwpp$bg6pgB$puzRgT$gu&|T6Teu e"gTffb&B&gukprt jd S D E`d` F U F U D I Q E I c D E Ud P I` F I D x D F P` S F E F E 6e$uT6bg6f"TRfTG"GD D U YES Q $0v i E I c DE I D FE 6e$sT6g$X1D D U $Y S D S F I Q x D F` c S v U E d` T00"s&VpB&Twf6t E I c DE I D FE e$sT6g&XGD D x x Q ~ i DE I D FE ""pH0Gg&XGD D U Y S D S F I Q x D F` c S v U $VgTp"&Xbew"$GE d` ft D x x Q ~ "0Hwi D x t x D c I l "g"&|ii S U Q 6pu `EE U c E I QF Q G6ppTpV i t E S Qd wpfV% i F RD F I c I DF I Q e6TpgP D D D I I Q P DF I cE Q x Q P c &&R6$pgTe0u"pRXfd E I QF Q i DE I D FE pTpV&pT6&GD t D P I DSS U P WS D DE F Q 0R6T$gi&G1$Tp )epq i jd S c D U Y E D I I Q P E D F I c E Q x Q P D E I` Q x F I D x D F P` S F $H1D g$$0R1Te0u"pg&6fp"g"gTE F I D` F I Q gfTpgP t E` Q D F U Q B i D F c D E Dd U Q D D F I` Q #E` I Q` I U D I U D S Q P I D F E p8R&p0TeB &Xf$puy&y0fp&pf$$y6$T0R6X1D E S Qd wpfV% i DF Q I I e$uil i Dd U Q P Dd S c F I` D F F c F E D x U x` I` x D P D U Y D E Q U E I Q F f$$0Ryfsp6VbTT'XG6yf$b"Ry60sT0u$$T|0VRD D F Q I I l i E F I D x Wd W U D Tp$uigw0"gfgd$g& E I` Q x U c F I D` F I Q b0BpgfTpgP $VT0u$$Tzi0Tp"sBgfTy 6fe0Tgb0$bTz$GD 60wpi &$e6f&weXbH$Tg6g$R1TS D U Y D E Q U E I l i S D S F I Q x I D` S c j I d` t I Q F Dd I` E I U F E D U YES Q v D d c I` S c P Dd S U E D P I D S S U P W S c c S D I I Q E` c S I l e6 wg6$pTfpiii E I c DE I D FE 6e6T6g$uGD D U Y c S D {` S W I Q t d D I I Q` F c S D U Y E D S F E 60}6g0a 0seg6$pfTe$T1T$gGD D U Y S DSF I Q x S U Q $ugTp"u$prq i E I c DE I D FE 6p$yGg&VGD D S d T$pfec h fVGD cd F E ` aepGp$0Q es&s6bTfe6pc e6e6|U gpRXGwb` ` E` S c X F S D a U jd D W {` F d c I jd E I c I D F I Q P F E D d t F D D I id D I I Q` F c D U Y E D S F E 1RR"Ceg$$0b'wS h $T1T6rGD E D I U P c P E` c f &"$R06wpepi P D a 1pCc D S` S P W F U D I TfR$gu2il i Ed D I I Q` F c D U Y E D S F I Q gg6$pfTeS h $$pTE F RD D U Y S DSF I Q x D P I Q F v UE do $VgT0&sp&X"&Tq$i F I D x D S` cd P c I 0"gbpfggil
U =V W V = J V V = B1 (o, R) - k
J
V W V = (B1 (o, R)-B1 (x , r ) ) V k k A1 - {x } k k[t, (t - x )-1 ] (V , O) k W k[t] (W, O) I I I m = min= |o - o | (, ) I 2 1 W = W Bk (o , m) W W W k[t] (W , O) W0 = B1 (o , m) W o W1 = W -W0 (W , O) = (W0 , O)(W1 , O) k Wi = W i W i {0, 1} (W, O) = (W0 , O) (W1 , O) W1 = W1 (W0 , O) (W0 , O) k[t] (W0 , O) k[t] (W0 , O) W0 W W = W0 |o - o | = m = |k | 1 <m > R I T = I Bk (o , ) T B1 (o, R) k
I l i6i
E I c 6p$
` GE
F gD
` GE
t
` TE
D U Y S DE Q UE F U D I Q t i $VgT0u$$TX&guup&&& p i Tfeb0T0Ryfe6$0"g"sXi D S` cdd Q S Q P Dd S c t D x t x D r D U Y S D E Q U E F U D I Q t D a` F P D Q S D F` Q S cd D $VgT0u$$TX&gud0&p0b# T60T$s& D"G$$p"0T&p|$segT0u"pggfeegfi&6TgyTGb$i6i $T1Tp6gTHg6&fTH6T0R2 x E` S Q x Q F U c I U S c S D E Q x Q P F D E D S` cd c P E E Dd D S I D F W D F F` U S U E E W S c W E Ed D U ` E W S E S Q P E I c E W xS D e$"g8 E I` E D D F E` D d` t D Q I v wFb0Qs1&BTGf&gb0&e0$B'c h 6y1D I U FE D x x Q ~ ""pHi S c e6 S D P cd x D S F U D I Q t D F I D W P W S D c F jd S c gR0f6Ty$gu0pT&g$BpgW esepq 6 i E I c DE I D FE 6e$sT6g$X1D D D c x jd D U Y DSF I Q x ` P D ~ &spp"f` )$sTp"R1Hui D I I Q E U Q 6$p&p$I D U ` $YTSFRW"ewTb$iTfe$p6eTXfeg$f` $i x c S Fd U D S` cd U I c` S F W F`d c W I j v DE Q I l T0udi6i D U Y d D $gTF F RD F` 0Q i D U Y d D $xTF F` i ` F P D S U E F E D 0Q bTR# $TGut D` Q E E bgP Tpc ` U x$Y
^ a (U k l, O) ^ k l, O) a (V |f - f | Pan - U k a g -1 |a - a |
gG
D P c S F D x E` S Q x D t Dd c S c W E I Q` E I D D D I U F E R0T$$p"sbd 0b$pp6gTupfG6TF Rs6$1D
# % % # @ % 3 % # 4 # % # ! 4 A )&)5 CC$psxg'B0|rx$xp$'&$R
|o - o | > max(R , R ) J = J o A1 k
D x x Q ~ "0Hi
f (U, O)
l/k
= |o - x | R
U=
I
|k |
^ tr : (U k l, O)
B1 (o , R ) - k
^ a (U k l, O) f = -1 f = gG a g (V, O) (V, O) (U, O) ^ kl U U U k k U (A1 )an k
J
J
G (U, O) ,
B1 (x , r ) k
|x - x | max(r , r )
=
a g -1
B1 (o, R) k
DE Q I l T0uzi$i
J
E I c I DF I Q P FE 6p$zU gT0RsGD
D U Y DFS QE D I cS F I D x x cE v U $XTTpTi$ewpH"pTz&TE
F` Q E F bpTsgD
D&XDSTFIDRifd P D i
(T, O) I
' peu
' ' BfP 3 e &6 k
6g'
E I QF Q pTpV
DX (r|f )
0 RigCor(k)
N - {0}
tn .g(t) ^ g(t) (B1 k (X/(f ))red , O) = ((X/(f ))red , O){t} k X (P1 )an - 0 k G (X/(f ))red q : DX (r|f ) DX (r|f ) X r |k | r r R k[t, z]/(z - (W, O) k{R-1 t, (.m|I|-1 )-1 z} . z - I (t - o )
I
B1 (x , r ) k
(t - o ) .m|I|-1 .
k[t]
W =
(T, O) (B1 (o , ), O) k
f (X, O) r (X/(f ))red DX (r|f ) DX (f |r) DX (f |r )
I
B1 (o , R ). k
U
(B1 (o , R ), O) k
f (X/(f ))red g(t) q DX (r |f ) >0
DF Q e$I P D a 1pCc
F` bpQ
Dd cd D P S U Q D S ` E I Q P I l fzgg$02T&G60Rii
D$ F I c F I D E W S D S D d E S D ` Q F Q x Q S D x S Q W E I Q d d c E U Q i ) pgT$gb$2b$0Tp0$zg"p8g&ipfbppV i W F` I U I Q` F P D E c 0Tb$$|pfTRTd
E(r ), I ) = (V (r ) I|DX (f |r E(r ), O)
)k P1 , V k
r {r, r } DX (f |r ) c P1 k P1 I O P1 k P1 k k k k ^ ^ ^ ^ V X k U X k (P1 )an X k ((P1 )an - X) an (X k (P1 )an ) k k k ^ X V = (DX (f |r )k (P1 )an ) V ^ V (r ) = DX (f |r ) k a ((X/(f ))red , O) t ((X, O)/(f ))red t1 t2 ^ DX (f |r )k DX (r |f ) ^ a DX (f |r )k DX (r |f ) E(r ) ^ k DX (r |f ) DX (f |r ) |t1 - t2 | = |t1 - a| E(r ) (t1 , t2 ) E(r ) t1 = t 2 = a ^ |f (t1 )| r f (a) = 0 E(r ) DX (f |r )k (X - DX (f |r )) k u(x) = maxtDX (r |f )x |t - a(x)| G (X/(f ))red x (X/(f ))red (k) DX (r |f )x q : DX (r |f ) x
F Dd ` S F Dd S c W F I D E W S D S F E Rf$bTfe6TG1T$TVGD t` E I` )T6fV% i E c F I cSF I Q P I D psewT0R6TS D $I D d D I I Q` F c D U Y E D S D c I` E` Q a I U F E $$$0b'wS h $T1T$|pp6fGp0m$z1D D U Y d D F` U W I $nf&h6&&mil i P I Q F 6p&XgD D U Y F I D U Y`d x $"0$ff$"f` F RD E I Q` F` I Q P E Dd t` E I` i E F P I` F E` 6pfTb&60RfeT6bX% gw6fTGf$ F U QF S U Q &0T2$pu c I j U Y `EE U c E I Q U YS c x npQ G6e60$pBgD i F RD P D a 1pCc S D {` F I D g6bTg$f` E S Qd c F U D I l wpfey&ixi b0ec F I` Q U I Q` F P c S F W S cd D D S { c pfTRpwTgTdf"&B$6dd F RD S U Q $pu E I Q E Q q i E Qd P F I D x D U Y` S W d c F E pT0r60fRXg"$f$g0bpiGD D U Y S D E Q U E F U D I Q t cd D P S U Q q $XG$6G&gugp$gRi$prui F I D` F I Q gfT0RP D U Y ` E E U c E I Q {` S W $uG6eH6pgii D U Y a Dd ` E E Q x` F E D ` U Y D $zU Xf$fG0u"bH1$RP D U Y `EE U c c I $uG6eyhpQ E` c efui c I Q put `E t F D D I Ggugui Dd c I Q c` cd E c D S F I Q P I D S D bp6p0e&hV06sTpRgy$I D U Y E I Q U YS c x $V60$pBgD i I Q` F` I Q P cd S c 0b&pR"e6 D Q v c I` c x Q I U E S Qd c F` I { W I l &p$IBehD$feBp&60bp266R&ipi D Q I v jd &p6"ec eg I Q` F P D E cd D S I` D S F E D S F U D I 0b1GgBT$6bT1T2&gu pQ t W P I Q jd 0R60$IgW eD& 0b`0SRSnfzebfT&mi I Q F P c F D cd F I c E`d` F U I D S U D F P c I Q P D E F D S D` x D S Dd S U E E D U Y` I Q I c P E D W &d$gp 60R1Ghgzgf"g$gbh$GB1$f6p$pgz1g$I h I Q S Q P E 0&0Q RgDfd FRD F D x Dd c W D F Q I I l Ig"gfegnTp$iii E I c 6e$ D D c x j & pp"f` )d DF Q I I Te6il i DF Q e$I Iii l D F I` Q #E` F &fpf$gD F I c I DF I Q P egT0Rt E I c d D I I Q` F c D U Y E D S 6e6wg$6pfT'wS h $$ D Q I c FS D U Q I D E &p$BveHGpa$0d6UXTFG` gDdb` t& $& i I Q ` F ` E Q Q S c d S c q i I Q ` F ` I { W D d c W ` I Q i F gD D Dd c I Q c` c $Xbp6p0e&fd 0bT0up$yfsep&0b$R&&e$fz0GE DTp$BIii F Q I l S c p6 D W F` F I D jd D D c x jd D F I c F I D E W S D S I U D S` S P W E I Qdd c E U Q $XTbTg$f` )X&XppBb` )X&0pgT$TB$fTg&pfbp60V0i F` bpQ e i F gD D U Y d D $wTF D S` U S F E I Q P D E S Qd c F v U E d o Tf$TG6pgV&x0bp$G$qei F RD D U Y D S` $Tf& D E S D I` I U F E Tpa f2$rGD D U Y D S` r 60bi'i $& i "gpg6THbref$fGwg0a brpTE i D x S Q W F Dd S c E Dd ` E S D I` F I Q E D x E` S Q x Q x Q E D 1T$$p"p"p61v F I D a D S E S D` Q F Q x Q p0b` T$f$0Te"0$X
c j () = [idDX (f |r) ] j = [idDX (f |r ) ] j () = c [idDX (f |r) ] j () = c [idDX (f |r ) ] 0 CorDX (f |r ) ^ DX (f |r ), DX (f |r )k DX (f |r )
c
` F c F U x x Q P D x x c S c` I j D E Q E` I l i E S D` Q F Q x Q '$y"pg""ewppf&g$U sT0uG&uig1T$f$pe"p62
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
1/n B1 ) (X/(f ))red (o, (r /r)
cd c S D E a` I l ywgTf` f&dii
c I Q t F D D I up61RRu$i F RD t D I Q` F c P {` F P c x Q $|0b'gup0RP D D Q I v c FS D a U Q F U QF &V$e0$B'Tgp6pH$pTpt & $& i "bVfp1T$pc 2r e i D x x Dd Dd E S j D$UYsFIgf` a6dqo'iedgD$I6Ipf`TF'cwS h D$UYEDS$FGED D Q i E D` F S c E` Q S F I D I Q` F` E Q Q S cd D D a U D S fTTef0Tdh0bT0up$fs$0$g$ D S F I D D x E` S Q x Q E` I U E S Qd c F` U W I l TgX"Tf6$T0"pTf|$wpfeXb$&1&dii fegRiTewpRVGD Dd c W F D D F I c F E I Q P F E F gD
r (P1 )an k DX (r |f )
P1 k
D I I Q D I U FE $6puV$$GD
X
X DX (f |r)
U = DX (f |r)
(P1 )an - X k
I Q` E Ud P I j pfGbg6b` ed
E I QF Q 60TpVi
DX (r |f ) xX/(f ) B1 (x, u(x)) {t, |f (t)| < r } k |x - y| > u(x) x, y X/(f ) |t1 - x| u(x) > |t2 - x| |t1 - t2 | = |t1 - x| V (r ) E(r ) c [idDX (f |r ) ]
D E S D I` I U F` Q E ` U Tpa f|6bpTw$Y
D` I { D P I c I Q E D S S Q P D I U D S` U S F E I Q P D F` j E d o i D F I D a b$VRe$60TpRy6$if$TpRy$bppc q$0Tg&` gW D F I` Q #E` &Bb0& D D Q I v c D c I` E` Q I &d$e0$B'h0e6fGp0a 6U P D a 1pCc
U DX (r|f )
1 (V (r )
^ E (r ) B1 k E(r ) k s
^ 0 CorDX (f |r ) DX (f |r ), DX (f |r )k DX (f |r)
^ 0 CorDX (f |r) DX (f |r), DX (f |r)k DX (f |r) y
^ Cor(DX (f |r ), DX (f |r)) = CorDX (f |r ) (DX (f |r ), DX (f |r )k DX (f |r))
j
j-
j
I|B1 k DX (f |r ^
DX (f |r )
k
1^ )k P1 , (Bk k V k
U = DX (f |r )
DX (r|f )
(r )
c
c
(r ))
G PicB1 ^ DX (f |r )k P1 DX (f |r )k DX (f |r ) .
G PicB1
G PicB1 ^ DX (f |r)k P1 DX (f |r)k DX (f |r) k y
E(r ), 1
xX/(f ) B1 (x, u(x)) k |f (t1 )| r |f (t2 )| < r
DX (f |r )k P1 k
E (r ), 1
DX (r |f )
j 1 ^ PicB X (f |r )k P1 (DX (f |r )k DX (f |r)) D
j : DX (f |r)
c [idDX (f |r ) ]
k
^ DX (f |r )k ((P1 )an - DX (f |r )) k
s
^ DX (f |r )k DX (f |r)
U = (P1 )an k
j
j
k
G DX (f |r )
B1 k
I l ii
D D W I I Q S Q P c &sg$$0&wpQ RBd
P D a 0Cc
` I { W D Q I v c D I` c x Q D $6g&y&p$Bey$bpBp&yfd
s=
t1 - t 2 . t1 - a - t0 (t2 - a) |t1 - a - t0 (t2 - a)| = |t1 - a| t0
I Q` F P D E c 0b1G"d
S U Q cS I DS $0w&6g$ P D a 1pCc 0 i ) Fgb6bTF Dd ` S e P D a 1pCc
i )
! e i 0 ""$t c x x D v
D D ` ` S D U Qd c I jd F E D F I c a` U E D x S Q W F D B&s&pfT$0pfe6pc e1V0p'b$Ts"gpg6Tv
I ()|X k X ^ 1 g= X s X O X k X ^ X
j-
bf i D F I` Q #E` F S Q U E F &Xfpf&VTpu$$T"XRD WS D T0g& D Dd &ibgP gP h $R"&0c I U F D x D U Y I Q` F` I Q P cd S c W S U E E c F E D ` P D ~ $hpfTb&60RfseVT6Gp1gH6i $p& $bfXf$&&"x h I cS D D S `d Dd U Q I U FE 6HGD D U $Y Sfpp$sfR$0"&T6q'6pfTeTfefTrg&VR6TGf&gD erg6bTpV1T$fT6bg6pg$pueeGbX$g"gfeR$6gf` 0RP ` Q a D cd D P S U Q F v U E d o i E I Q` F c E`d c` a` S F Ed D F D D P I D F E` jd S D {` S W a D F E D S d` D S Ud P I Q P S U Q t` E I` % i F I D x D S` cd P c S I D a I Q F D# Q F D P D R!pRg$ D E E cd P c v GpRw$i D & F D # Q I U F I D` F Q I t I Q` F c E`d c` a` S F D I RD0d6b&pdpQ pfTeTfefT$$U FGD E F Dd ` S F Dd S c D W F I D E W S D S F E gb$fTbp62TgT$TsGD D Q I v c D c I` E` Q a I &p$BeXpp6bTpp|$U D cd D P S U Q F v U E do &"ggV$puX"&Twq$i
0
E6IgDfbdWpcu$pps&r'w&y"pg""ewppf$u6U sT0uG&gii ` E D U Q S D ` F c F U x x Q P D x x c S c` I j D E Q E` I l E I c 6p$ D I Q` E Ud P I j &gpfG6fRb` )d 6peV PW%422RPGEC E I QF Q I HV S U T S Q I H F D ww ' g ue) b yu A u q p 0 e) yu ' 5b3 w 0 Rg 'w 'e6 y0x 'q'R Rq e ' R 7
(P1 )an k V j X X = X0 V X0 I () X I () X. ^ X k V ^ X k (P1 )an k
' 5b3 ' h 0'u }
X0 X
ps6egf '5$0G 6 g pdp gh0' }
1 V X X0
D S c ped Hbx
! D F I c a` U E I Q` F` E Q Q S cd P D a c D I` F I Q P I E0p'b$Td0bT0up$"X1pCs$U fTpg|il G w q 0x ' 'sgH' G Pq i peu 0' i Y'g peu 0 Re qx Gw ' E U I D S E` F cd D S S c P` q D E D U Q S E Dd E I c 6$6g'ghwefs&u$pp1be$ i D W S F I Q x W F E D I Q` F` E Q Q S c v i E F I c` S c a I 0gTp"&X1h0bT0up$w&eTp'f` h
j () = j I|DX (f |r )k P1 , V (r ) k X0 X0 X (P1 )an k B1 F, 1 g = I|DX (f |r )k P1 , V (r ) k X = X0 V E(r ), 1
F RD i ) Tgb6pTp"p$$$Vb$$p8 0 i e 60Vbp0|0&t D S F I D D` Q F Q x Q D I U F` I S U Q F RD I D F I c E`d c` P W E I gebpfguGg D U Y F` Q a I Q i F c Fd U E W S Dd j r 1'w$Tsfd0Q 2i D U Y F I D a do 60Bgf` qpi D U Y F I D U Y`d x 60Bg$b6f` F RD E I Q` F` I Q P E D 60b$6pg1v i0DTFIgD$WRP1WT6bGp6f6e6V"0R$$0GewBpgp0fRg"$f$g0bp1D S D` F S c cd E I c D x x Q P D I I Q E` c S I Q F D E Qd P F I D x D U Y` S W d c F E D U Y S D E Q U E cd D P S U Q F U D I l $gT0u$$TsfRs$pu&Bi&i D D Q I v c D c I` E` Q a I U F E &V&p6"eVpp6bTpph$HGD D U Y E I Q {` S W $606gi i D E UEE D U &H6G$hec d c W jd D S U DF cS W I W p&` eX$$g'wg$p D U Y S` cd P F I D x Dd c W F E D do $sfeRg"gfeg1q&i D U Y E` U $f6$ S U E D` I { W I D` F E $Tb$R&ugf$1D I Q` F P D E c 0b1Gwv I U FE $1D
I t0 = 1 |f (t2 )| < r E (r ) E (r ) k |t1 - a - t0 (t2 - 1)| = |t1 - a| r E (r ) |t1 - x| > |t2 - x|
# % % # @ % 3 % # 4 # % # ! 4 A )&)5 CC$psxg'B0|rx$xp$'&$R
DX (f |r ))
|f (t1 )| r |t1 - x - t0 (t2 - x)| t0 = 0
t1 - t 2 (E(r ), I ) t1 - a
F Dd ` S F U D E E cd P c Rf$fTd$TT0fg"fd
(r ) E(r ), 1 g
F I c I D F I` c x D E Q I l 0p6gbp"sT0u|iui
I|DX (f |r
^ F = DX (f |r )k DX (r|f ) I|F
I|DX (f |r )k P1 , V (r ) k
)k P1 , V k
^ PICDX (f |r )k P1 (DX (f |r )k DX (f |r)) k
I U FE 6GD
t F D D I RRi
E D F I c a` U E E W F`d c W E Dd D F x Q P D I { I D c I l e'b6G1Tbbp0g1bs&"pg$ud|iui D F D # Q I j I D` F` j E d o $gDp$U gf$0ec gpqi
j () = j I|DX (f |r )k P1 , V (r ) k
X (P1 )an k P1 k
F, 1
g = I|DX (f |r)k P1 , V (r) k
F, 1
g
X X
E(r), 1
g = c [idDX (f |r) ]
g = c [idDX (f |r ) ]
s ^ ^ B1 k DX (f |r )k (X -
|t1 - t2 | = |t1 - x| =
X0
g
X
R
G e) w u zb q 0 & $0 6g' 8 qx Gwpr
0 0 RigCor(k)
0 Cor(X, X0 )
0 Cor(X, X)
^ 0 CorX (X, X k X0 )
^ 0 CorX (X, X k X)
^ 0 CorX (X, X k X0 ) j () = c [idX ]
D U Y Ddd D $yfbTF
k
D U Y d D $wTF D S` U S F E I Q b6T60RP D` I { D P I c I Q E D S S Q P D I U D S` U S F E I Q P D F` j E d f$6XR6p$60TT0Ry$6f$TpRy$bppc wqo
1 ^ PicB k (P1 )an (X k X0 ) X^ k 1 an U (Pk ) ^ k ^ (X (P1 )an ) - (X k X)
(P1 )an = X U V U = k ^ X k X ^ U (X k V ) =
` i E S Qd wpfV% i F` bpQ $i
F gD
D U Y F` U W I Q $y$$&pt D U Y W I I Q F I cF $B$$p$sppi Dd c I Q c` cd E c F I c S F I Q P I D S D bp6p0e&uf06iewT0R6T"$I D U Y d DF D Q I v c FS D a U Q I 60Ti&p$B'Gp$0d6U
i
(I (), U , 1)
X
U V =
^ (I (), X k V
j-
X
1
U ,t
G X0 _
1)
c
c
^ PICX k (P1 )an (X k X0 ) ^
k k
G PicB1 ^ ^ X k (P1 )an (X k X0 )
G PicB1 ^ ^ X k (P1 )an (X k X).
^ U X k U c [idX ] t : O X k V ^
j = [idX ]
j
P1 k
G I () ^ |X k V
0
F` bpQ
D &
D S U DF cS W I W &$g'wg$p P D a 1pCc
P D a 1pCc
i )
F I c a` U E D x x c S c` U I Q` F c F U x x Q P cd S c F I D E` U c S F D E E I Q` F cd D S E D S ` I S D U D E D e'b6G"p0e&g&upfT'w&y"0R"fp6gTf$$pwTGpfT'gsgf$g&d$g&V1v
p ^ ^ (I ()|Y k (P1 )an , Y k V Y X U , t|Y k V ^ ^ k j1 a = 1 i 2 b = 1 i2 p = 0 , i1 p = a j1 - 1 1) a j 2 - i1 b = 0
F gD D U Y Ddd D 60sfbTF D` I f$6{ DRPIec$6I0QTEDSSpQRP"D$I$UyDTf`$USTFGE6IpQgPh2D1Ti60$qui S FE DS E U Q I do F gD t E I Q` F cd D S E Dd P I Q c I 60b'gX1by6p$hil i F Dd ` S gb6bTF I Q` F P D ` cd Qd U Q x F I D I Q E D S S Q P E Dd ` E E Q E D W E Q x Q P U D E Dd t F D D I i D F U x x Q 0b1# $zfzf$&&"Vg&puG1TpgVf$fG0uG"0R&g$XfuRR60T&20RP U D x E` S Q x Q E j D E E cd P cd &y"Tf6$T0"pTf` yTT0fgzfz
X
a
W S S c P Dd Tpgyf&t
b=c
-1
F RD
FS D a U I Gp$0Q 6U D c I` E` Q a I U S c pp6fGp0B$p6
E S Qd c D E Q I l 0bpsG|iui E c F I cSF I Q P I DS D p6ewTpg6gs$I D D Q I v c D c I` E` Q a I &s&p$B'pp6bTppu$U F` 0Q 6i F RD D U Y d DF D Q I v $wg&e0$B'c E I Q ` 6p&bxi D U Y S DE Q UE F U D I Q $BgT0u$$TB$gu}ppt D S U D` S W F I jd E I c U I D F I Q $"6gfTTb` e2e$kTpgP D D Q I v &u&e0$B'c D U Y E I Q U YS c x $60$pBgD S D P cd x D S D F F` U t` E I` % R0f$"TsTT$Y )T6fV5i c I Q t DE F Q z0p0G1$Tp pri S c q
Y - Y0 = X - X 0 V Y - Y0 Y (P1 )an = Y W V W = k a=c
-1
DE T0QuBIi&i l E I c e$ D I Q` E S D x x jd D D c S U d c W jd D Dd c P Qd I Q` F c E`d c a` S F D I $zpfGwg""b` eV&V$$p0z&ue$f` )X&Xfeg&f20b'ffe` bTX$$U I6&Dyb`pQ i { I F Dd c I Q c` cd D F I` Q #E` bp6p0e&z&2fpf$ D D Q I v c D c I` E` Q I &z&p$Be"0e6fTf0pa $U F RD D U Y d D $gTF D D Q I v c D c I` E` Q a I &&p$B'g0e6fGp0m$U t D U Y d DF D Q I v c FS D a U Q I $gTu$e0$B'zGp$0$U F I D` Q gfp&6i &|pfTT60Rfs60b6ec i 6& i 0bT0up$2f6p$Tb$TpR|pfT1G"d $pu2&g$hil D I Q` F P U S F E I Q P cd E I Qd D i I Q` F` E Q Q S cd E I c D F` U S F E I Q P I Q` F P D E c S U Q cS I DS I
a j 1 - i1 p a j 2 - i1 b i2 p i2 b 1 0 0 1
F RD
F I D x D a` F P D E D S F I D a` S P j E E Ddd g"g0buTpfTgW gV1bf
a 0 a 0 -i1 i2 1 0 0 1
F gD
! E D F I c a` U E E D E Ddd D` P` S F c x E I Q` F c F Q I I F I Q E S D {` S W a E I Q` F cd D S E D v i E U Q E E D ` )TpCf$Tfd 1bfgfRT'B60b'p$ugD 0Ggg0260b'gw$g60G$ h RP E D` I { W F I Q S D E ` U Y E S D` Q F Q x Q E D` I { E D P I c I Q E D S S Q P E D f$6R$spgT$$if$0Te"0$y1b66p1R6p$6puTT0R$ RD t gT&f` HpfG6fRbsfd F F D E D F I D a W E I Q` E Ud P I` E D F RD t t P D a 1pCc
i1 i2 j 1 j2 a b
i2
1
D S c c U D` Q F Q x Q D F F D ~ id "xT0Q82fd2S6ecsb6pTp"p$sGRH)$U I D"xG`$$SpQ"xHbdxFRDsWTb`TFIgD&` j)dHDTSF0IDD`gPffd$&RD fde`RPbd$"fGE f`$pQFeQ"xpQ$sD$I$UpDf`$UTSFEIpQRPiHDF`GIpRPHp$UTS$Vcwv PW%422RPGEC E D F F ` D c ` x` D S E E Q D a D I HV S U T S Q I H F D 0 be t6'
0 RigCor(k) Y0 G X0 Y GX
r $0 G B6p 6 g 020yb
1
6B pw) $
p i ) y 0 g BqR p b7 b Bb3 G 0'u Rew pX'gf ' p Rq2p' e qpr |%H $
X X Y0 Y X0 , Y X X (P1 )an k
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
t : OX k V ^
X0
^ U X k U
j1
j2
j1 a = 1 ,
G I () ^ |X k V
a
=1,
a
b
X = X0 V U (P1 )an k ^ ^ (X k (P1 )an ) - (X k X) k
^ (I ()|Y k (P1 )an , Y k V ^
j1
j2
G X0 Y (j1 j2 ) G X (p b) " a " 0 G X0 Y GX Y0 "-i "
Y0
i1 "
^ (I (), X k V
X0 Y0
+
-i1 i2
j2
p
Y
"
k
^ W Y k W
aj1 -1 bi2 -1 p b
G X0
G Y0
(j1 j2 )
p b
=
i1
P1 k X = X0 Y
Y0
G X0 . QG Y 0 U ,t W , t|Y k V ^ p i1 = b i2 - 1. Y = 1). V Y0 1). - p i1 + b i2 = 1. a V X (P1 )an - X U V = k ^ X k X c 0 Y0 = X0 Y p b V W (P1 )an k ^ ^ (Y k (P1 )an ) - (Y k Y ) k X (P1 )an k -i1 i2 = 1.
i1
& i )
D ` Q P E D P S U Q $d0$wgV$prq
E S Qd wpfec
u
47
w t b ' wB qs V V b 0 26 5 ' "uR 'f ye Rq02u C e R0 p e5 ' 9 T p R Bq p2B b 6 Rh t h 7y e x|&%V# $ $ i D S` cdd Q S Q P I Q E D U Y ` E I` pbpffppg|0Gy60T6fec E U EE 60QGD& h RPs"xDbfdy` agTf`fbF&ywSgTE Ipf`Tb`&6pgP2vwIpQfTF1GE h E6UpTgT$y2$gg&T0g$Xgs&0$g$2wv ` D x d D c D W E d` U c D Q F I Q c i ` P D Q E D F I D E W S cd c S D U P P Q D x S Q W F D P D D a U D S c G e 00' e 6' T p 6 e g R eH' B s 1 2ip hE p 0'u ph eR6 p}'e T ' Bb3 ) p e g 0 Bq p h 2" ( 0 w R 0 y e p p e5 0 3( y B e ) g'6 0 02 " r $0 h 2" e er d%s $
uwe f g ' q h ) ' 6 gsR ' 6g bb G' Rh h
SmAfnd /k
! e'f$Teb$Tsbfb6egW X1hpfTRTE h 60GE F I c a` U E F c Fd U E W S Dd S`d c F j F E D I Q` F P D E U Q DFF D P D F U D v GgRs&X&$w$i U pQ Dd c fe0W S D E Q U E I D` S c S S D D I S U D F P Dd D v gT0u$$T"|byw&wgus$$T1bw$i D U $Y Ddd D bfgF D D I` Dd D` S Q W F c E U Q E cd D I ` E W &B$fgf$bp0gTegP h pTdy$0bT$ D U Y E I Qd D $6pfgu$pc i $Xf$pp6gTyRf$6X6pfT6gg& D E Dd c S c W E F D E D` I { E I Q` E I D F D E D E D S F P D E E Dd D U Y ` E I` c D F c D W F` I U Dd U Q cd F I D` F I Q 1TuTx1b$&T6bps'xB $6$b$0ixgfT0RP D U Y ` E E U c D E Q U E I l i E D Q I v c E F S D a U Q U c D c E E c S c F D E` I $&G6eT0u$$T2ieg1&e0$B'xTg0$py&pH0eT06p6pRr$6{ E F P D S` E F` U Q S Q P F D E F` U Q S S c Dd c F Rb&w$$ $0RXRsw$$ $ef$pGE D I` Dd D` S Q W F c E U Q E D I U { I Q t I Q` F P D E U Q E D F F D P E I c 6bb$Xbp0gTegP h pT6$D 6h000b1GE h pTGgR6pir
&x2
1 ) 2VGXs0GX&}$s
X k
p
G&uT6&0w$XXX$|
i Dd ` a` S I Q I D W E Q U pfecfTF h p$G$6GE D I D x P $I$b`$W"b`6wSpc h I0$BIpfTec6fe'aX$6HS$0gb60RPH6T0RPh6VD$pfTW& tpf`TRTEGgRPX&pQ6Ipir i '0 i0i 0b$R&yfp0a Q I Q` F Ud c D I U U Q F Dd x Q E S Q I U I ` E I Q F P D DFF D DF U F E c I Q` F` I { W cd S` Q F Q F U Q F I S D a Q P S E F I x D ` c U D P E c W S I U E &UpS$0H0Dpg0I0Rw$UTgD"bd$fe"pcgT`egT$|$F1D D E D Q I v EF D a U E D &H1&p$B'csTSg0$pQs$ D F` E F` F D U TbTbTgdec IpfTFRfSTGEcfIp$2Ue1REfpRW$"$UgD"gf$"fGEGD Q` P` F D S d F Q c D P ` c S I F I x Dd x` F E S U E I D S I Q S U F I x Dd c U D P E c W S I t E $TFgpgD0a pRPw$TEgD"gf$f`e2ecgT`egT66U'g1DxSgDFT&ec B DSF U j I 1bb s
Ouvaf (X)
w G B g s d peu ' g pe t wG Rh t bb ugb2yB u
B V V
w B ' 6gp 1 h 7h w ' peu Rg d0
F X
e i ) w 2 b' ' G y 5ug R eB 2 7wT7 As e gwCg0Tpa pg$Tyg"gf$fec F I D S D I Q P S U E F I D x D d ` R 1T2 Cg g Re qr G6
Q B k
! Tg8G6ps6edpReR$u$$0gT|0Gs&we$Y g6g0Tpa pg6Gyf$fe&upfTe6"fd E F S D E I c S F E I c E U c D P E` c W S I U S U Q E I D E I Q E D S c ` U D P I D S D I Q P S U E Dd ` c D I Q` F Q I c E I Qd D pfgu$ec p"T0g$z&Tb&bpC2H$0XTfeT1R6sGiG1$Tp TGgRGH6pp6pfHpVggfpTR0gp6TD h fHe$ i D x S Q W F U W F` `d c a cd S U Q D S` c E E D P W I F E D D E F Q D F F D P ` E E I Q S Q I ` E U Q i Ed D` S Q F P D a E D P c E E Dd E I c E S U Dd c a F E D U c D P E` c W S D P D U Y S D E Q U E c S U c I Q t D a` F` E Q D U Y` F E` S W F P c S c P D F E w$bpCG2&e1RfpR$RX60G6$Gssw$p"0ep0bT0uH60TGTT0pgX$1D &6g6&fTr6pgXb$0b0g"x D d D U ` E W S E S Q P Dd D U Y E S Qd t F I D h T6T$$fe0b$0Tp0$sepfe'a bygig0Tpa pg6Gyg"b$fewG8ewTypCne1RfpR$n$U fp0b0"p$0Rz& D E U D S U D d c f i D` Q F Q x Q S c F I c` S c I` F D F I D S D I Q P S U E F I D x Dd ` c E F S D E I c S F P D a c U c D P E` c W S I j D` Qd Q x Q Q P D U c D P E` c W S E D D` Q F Q x Q S c D P I c` S c I j d ti i ` t D S F` c P D P D d c ` P I` S D x S Q W F Dd D S F I Q x I Q I Q` F P D E D F F D P E I c &pgTegT$&ib$0Tp0$p6ReTpCa f` eb0D ep0T$p6wXgi&ue6fR6f$V"gp$TXbTp"z0BpfTRTiTTRg6pir
yyxyysxdw
" $#!'w
yrwyX
yH y' d
dG rRBd i
c
-1
2yRrsiR % s RyyRyy|y
bubfx & i ) "p0e&2$2pfTe&2"pRw'i D x x c S c` U I Q` F c F U x x Q P c v
^ Y0 k W, t|Y0 X ^
0
I Q` F cd D S cd D U Y ` E I` c t D S` cd P E S Qd c F E pfTefT$G6fe$0TfeRwpfeXGD D` I { D P I c I Q E D S S Q P c f$6pg6e66puG1Tpgid 6puwpfe&g$yil S U Q E S Qd c c S I D S I
^ c (b i2 - 1) = (O, Y0 X0 V p
D U Y D S F I Q x D S` cd` x` E d U Pd c P I $T0bpff"bTw$gfpggV
c (a j1 - 1) = (O, V ^ X0 X U , t|V 1) = (O, V ^ X0 k W, t|V 1).
F I D` F Q I l b&puii S U E Dd ` E S D I` I Q` F P I Q D I U D x x Q $Tsb$fTpa b|pfTR6p86$"0RP S` Q a F U D I fp0V&guupQ FgD D E U E E D U c D U Y` I Q I c P I Q` F c E`d c a` S F D I U F D x &s6G$de$f6p$pgh0b'ffe` bT$$R"&0c E S Qd % wpfV!i F I cF a eb` gW D D Q I v &&e0$B'c D c I` E` Q a I pp6bTppd$U F` bpQ 6i c I l i E` F cd D S S c P` di6gbTefTdwegq D E D U Q S E Dd E I c $pu$ppfp6e6 F RD E D P I c I Q E D S S Q P E D D c x jd S Q j c S Dd U Pd c P I Q g6e66puG1Tpg1&Vpp"f` )wpuec b$gfpgz0et gTz$Tf$TG6pg$prq d D F I U D S` U S F E I Q P S U Q
p a j1 - 1 ^ ^ c (a j1 ) = (I ()|X0 k (P1 )an , X0 k V X0 X U , t|X0 k V 1) ^ ^ k ^ ^ (X0 k X) - (X0 k X0 ) I () I () t|V V b i2 - 1 ^ V X0 k V V
4 # % # ! 4 $'&$)
3 % ! 4 % @ % 4 # % ! # @ # ( % z4 'up p Cx$'gu 0$# 0 )$'x f x)g uuux y f 4 @ ! 4 3 5
Ob(Cor(V )) = Ob(V )
(X, O)
B1
Colim
|k |, >1
F
(
V
F
k
Q
5 6
l/k
F X
F
k
i2 p = 0
k
F (DX (f0 |f1 , . . . , fn ))
^ Y0 Y W , t|Y0 X ^
F (X) (A1 )an k (O, V
0
V SmAfnd/k V Cor(V ) V SmAfnd/k
V
G F (X k l)G ^ G ^ X0 k W, t|V (A1 )an k 0 Hra (X, F ) V ^ 1) = (O, Y0 X0 V V V SmAfnd/B G F (DX (f0 |f1 , . . . , fn )) = F (X) X Ob(V ) 1) Hi (X, F ) = 0 ra B1 Q
V
i )
RigCor(k)
F f 0 , f1 , . . . , fn
i>0 1).
43
TV
'&
X Ob(V )
"
i
^ (X D())k l = DY k l (f |) ^
S D S F I Q x D c S v U E d` t F I D S D I Q P S U E F I D x Dd ` c F E Tp"X&yB&TufCg0Tpa pg6G0"gf$bpGD
i Dd ` E S D I` F E 0b6bTpa fV1D D x x Q ~ ""pH$i
D U YES Q $Twpfd
S U Q q $0r6i
D U Y d DF D Q I v c FS D a U j 60gTi&p$BeTg0$pQ ed D U Y D F` U E cd E I c c S D E Q U E I l i S D S F I Q x I D` S c j I d` $y$Tz6p$"G$6Gi1gTp""b"| bt fb H
iI0
g
eu|'pR
P
iI
$
Rqpeu R
B 0 G 0'
i D F S D` x D S U E F I D x D a U Q P D S E D E c P Dd S D F` c S F S U Q E U Q E E D ` P D x x Dd Dd c S D E`d` F U I 0 gf"g$u&g"TS $0R$pgsfgp$pus6pTT1& h gy"bsfGbfT$dil D Dd c S c W E D` I { I Q` E I D D D I U F I c a` U E D E c D F I D x D I c P F U Q F E S c D D I I Q i &b$pp6gTyf$6dpfG6TF Rs6$0p'b$TsG06s&Vg"g0$ew$pTs1T6es&R$$0RP D F E D S ti i D D I I Q P F I D x D U Y` S F W x Q W F E TG&fpD )` &R$6pRg"g$TR"ppGD D U Y S DE Q UE F U D I $gT0u$$T$guhpQ t U c I Q` F P` S F E D S c E S c &e|pfTRfTGye6 E I c F I QE ` U Y E D Q I v 6p$spT$1&p$B'c h F RD S c e6 S D P cd x D S D F F` U gR0f6Tz2TT6ii D D Q I v c FS D a U Q I U D x x Q &y$e0$B'XTgp6p$y"0RP S` Q a F U D I Q fp0&g06t D E DF I cFE I Q P E D E S Q P D &V1Tew60R&X6T0Rbd D I ` E W $0bT$ ` $i D g$$0Ri$e0$B'sTg0$pd$U HTGz0Go h&sg"TS $0Rh6U D I I Q P D Q I v c FS D a U Q I j ooo U Q o t o D F U F I D x D a U Q P DS I S U Q D U Y`d P 6puX$bgP gpc FE GD D U Y F I D I` c S F I i $g$fewTgD 6 i T0g$h$T$eu sf$gTp"2$"ewG|p&' 0i "T0g$ifepq D x S Q W F U E D E F Q E Dd D U Y S D S F I Q x W I D x c S F E D I Q t i D x S Q W F Dd S c
(U, V ) k X 0 G F (U V ) X V G F (U ) F (V ) G F (U V ) X (A1 ) k k G0. X (A1 )an k
E` S Q F D` S D c f D D Dd x Q P D pTgbi h g0Cy&s&Rf$"pgfd DF Q I I Q e$|p$t S U $TE EFS DE I c S Tg8G6ewTF P D c c D P E` c W S I U S U Q q 0aCdUpReR$u$$0x&i D Dd ` E E` x c F I D x D S a U Q P D S I $f$fG$p0"g6pR1T|$U F D D Q I v c FS D a U Q I gi$e0$B'Gp$0|6U F I D` gfpQ 6g T 0 0 0 0 T f 6i 5bP w p Ab 6g 'b 3 R Tr f 6i | 6 p R 6g'X 5 e0 g 0 gu q 0Vw $0 ' w" q 5k' gpeu p pg5 b 6 Rh t h 7 ' xs ! F I D x D F c` W x x` F I D` F Q I Q t E Dd U Q "0"gTef$""bgfT&0d0$f$&&"x h D D` S Q W F c P cd S U E F P c D F I c F &bp0g'g"$TRp$RVpRW S U DF P I Q D $Tp8v Hbx
G F (X)
i i ) &$w6p0XR0T&"Gwg0a bu$VGD D D P U c F D D F` Q S D E S D I` I U F E
a ^ : F (X k l)G [l : k]
F RD
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
^ p : X k l p r = [l : k]id
E S Qd 0bX% i D I Q` F P` S F E D S c &d0bT1T"d E I QF Q 6peX$i D` I { D P I c I Q E D S S Q P D I U D x x Q P D f$6VR6p$6puTT0Rs$$"0R$U a D D cS U D WE Q E I cSF c &$$ewp|&gT0uG6pyd
gG
F RD
E S Qd c c I l i D S` c F I D x Wd wpfe"|i0Tfewg"gfgW E I QF Q 60TeX
X (A1 )an k F V
l/k (U, V )
(-)G
U
V F X
MVF (U, V )
F
GX r rp=
X =U V
Q[G]
X
g
V X = U V ^ ^ (U k l, V k l)
a
X (A1 )an k
rb 9 eb p2$ T $ 06g 6g'x ) 5 )
G
0 'b X g0y ' e$ b ugT7qT
ri |o - xi | R D (A1 )an k
F (X) = F (X D(ri0 ))
F RD
DF Q I I Q Te6d0$t P D a 1pCc D U YES Q $0v PW%422RPGEC I HV S U T S Q I H F D eB fY' Bb3
R2u' R (r|c%# ' $ $
ri0 < d
X D(ri0 )
Y (A1 )an k
D U Y d DF D Q I v c FS D a U j $wT&p$B'Tgp6pQ )d
D(ri0 )
D
l/k
F (X)
|xi - xj | max(ri , rj )
^ ^ Y k l = (X k l)
ri0 = d D()
D() r i0 < d G F (X D())
G F (X D)
^ D()k l = B1 (o, R) - l
^ Dk l = B1 (o, R) - l
^ X k l = B1 (o, R) - l
iI0
B1 (xi , d) l
i=j
1/|I0 |
t E S Qd 0bX% i
= B1 (o, R) - l
iI-I0
F RD
ri0 < d f (t) =
S U Q $0
iI0 (t
- xi )
=
i=i
|xi - xi |
E I QF Q pTpV
iI0
B1 (xi , ri ) l
B1 (xi , d) . l
F
i0 I
B1 (xi , ) . l
p
^ X k l
B1 (xi , ri ) . l d = mini=i0 |xi - xi0 | F (r) X ^ F (X k l)G r i0 < d F MVF (U, V ) k /k k k F I0 = G.i0 Q |k |
FbpQ ` D U $Y c I hil
B e
IPWV%422RPGEC H S U T S Q I H F D 1bb s
6
6
(X /(f ))red DX (|f )
Pqg
B
q6gp
|k |
e
6
peu
6'0g
wg T R G 6p 1p' m
i D F D x ` E` Q S F F D I Q P D E U E F I D x D a U Q P D S E Dd S U Q W F` P`d P P j d D D E c cd F E D D F I c a` U E I Q` F` E Q Q S c pu bTpTRd60RTg&sg"gS $pgf$0TbRfR&R0c es&G06"fVGsTpCf$Td0bT0up$wv Hbx
^ (Y D)k l = B1 (x0 , d) - l
i0
i 6 i ""gfu&f08$$T0R6b$0R1&6w$w'b$Tsv i D x x Dd U E` Q D I U D S Q P I D Dd U Q P W W P S D P D S F c Fd U E W S D
^ Y k l = B1 (x0 , d) - l B1 (xi , ri ) l
c I l gii F` Q bpTE D U Y P I Q F v U E d o i Dd ` E S D I` F E $"60&y"&GHqu0b6bTpa fy1D
G F (X D D ) G F (X D) F (X D ) G F (X D D )
DE Q I T0udil i Dd ` E S D I pf$fGwg0a b` t i "bBfsepq i D x x Dd Dd S c
E D Dd x Q P D D x E` S Q x U D I P Dd F E &gb$"0Rs&s"G$$p"u&s$p$ ybGD
^ D k l =
iI0
E` S Q F D` S D c f D D Dd x Q P D fpRfX h pCs&s&Rf$"pgyv
i i 6 i ""gfsf6e6""pgP D x x Dd Dd E I c D x x Q
F gD
E I QF Q I F 60Te6VRD
F` 0Q i
iI
D U Y Ed D $gF S U Q $pu
D E D Q I v c EFS D a U Q E D &$e0$B'Tg0$p1bd F RD
$
6g
T
0 T f 6p
' 5bG 3
) 5
Hbx D r i D F c D E Dd U Q E D E D S { i&pTeB &f$puH&$6y
ipDS$fdRP6I0QRPyD&`p8D$I6UDSpQgP6IgDFggug $& i pfTbT0up$w$i U E Q D xS D i I Q` F` E Q Q S c v t E Ud g6f$ I Q` F P c S F W S D I j F I D x D S` cd P D E Q E` I l i Dd ` E S D I` F E pfTRpwTR$$U sg"TfeRGTf$hi&0b6bTpa fsGD
DY k l (|f ) ^ Y F (DY k l (f |ri0 )) ^ DY k l (|f ) ^ G F (D ^ (f |)) Y k l
D x E` S Q x Dd D U T$$p"bs$Y D U c D P E` c W S I U FE SgT0y&XpB&Twf$&& 6 i "bsfVepq fbp0gp6TD h &|pgTegT$g6V1D D S F I Q x D F` c S v U E d` t i D x x Dd Dd S c i Ed D` S Q F P D a E D P c E D U Y F I c I D F I` c x E I Q E $XegTfeB60G$$U i Ee$z"Tf$6T00G$"b$&6f` I c D x E` S Q x Q E` I U F` U I D U Y D S Ud P I Q P S U Q i $z$fR60R"$puug $& i pfTbT0up$gyg60f$$eh6p&B& I Q` F` E Q Q S cd S D U Y`d c P I Q F U D I l ii D U Y F I D a do $sb` rqi c I Q pt S U Q t F S c D S F U j r i D F c D E Dd U Q $puTe6T&ec 20TeB &f$0 E E DS { S U Q F I c 1D&y$66pup0Cc I Q` F P c S F W S D I j P I Q D E Q E` I l 0bpwTgTB6$U B6p&BT0uG& iri D6U0Y26p60weBD i E I Q U YS c x c pBgD I Q F D U Y F I D a do $z0b` qri Dd x D E I jd F I c` S c I` D E E` c f$2gT6gD )eTpCa fdTfpfd D U Y S DE Q UE F U D I Q t i 60gG$6G&p$ p i 0bT0up$ fueppff$U $TGfgpwefegW uGD I Q` F` E Q Q S cd S c q i Ddd I D U Y` F E` S W F P c S c P Dd c j F E D U Y S Q j E I QE Q $0pc |pT0u$$U
k {xi ; i I} f (Y, O) X D() = DY (f |) G Y /(f ) q : DY (|f ) Bl (xi , ) iI0 r i0 < < d X D() X D( ) DY (f |) DY (f |) DY (f |ri0 ) F Q Y
4 # % # ! 4 $'&$)
3 % ! 4 % @ % 4 # % ! # @ # ( % z4 'up p Cx$'gu 0$# 0 )$'x f x)g uuux y f 4 @ ! 4 3 5
0 Cor(V )
^ DY (|f )k l =
(A1 )an k
D I 6$U
F` i D D I I Q P F I D x D U Y` S F W x Q W c S D E Q U E I j U 0Q p&g$$pgz0"g$TR"ppG$6GmpQ Y
D$UYyfbDTF Ddd E Qd f`0bpc D$Du$U0QTS0s&D6I$IgfGEpfded0bT6TF RD D D` ` Q c I Q` E I D D D &g&pQ6I"eczGDp$U0kI6U v FS a Q FbpQ IPWV%4U22SQRPGEC ` H S T I H F D T2 H ) '6 b 'RT2i
ri |o - xi | R D D
X
^ Dk l = B1 (o, R) - l
|xi - xj | max(ri , rj ) (A1 )an k
^ X k l = B1 (o, R) - l
G
B1 (xi , d) l
F gD
8 xGwpr
16)y
^ Y k l =
Y =X D
F (X)
B1 (x , d) - l
MVF (X D, X D )
[0
[0
B1 (xi , ri ) l
F gD
~ I0
c I l uii D E D F I c F E I Q P E D E S Q P U c F I D x D U Y` I Q I c P D {` F I D j &0pG6pg&6pg|p$f6p$pgy6bTg$f` gE t E S Qd wpfV% i D ` Q P I U S U Q &hbp6w|$6pu F RD D P I Dd c a` U Y j I Q` F cd D S cd S c g6gfe'f$W d0b'gye6
k ^ (Y D)k l = i j |xi - xj | d
~ I0
E S Qd wpfec E I QF Q I F 60Tp$RD E I Q ` p&x i D F I D` F Q U Y Dd D I ` E W &VgfTp$sb$0bT& t E UEE D ` 6TT1& h H~
~ 0 I0
~ I0
F RD
F I D` gfpQ P D a 1pCc
i
I0 k = lStabG (0 )
i
X (P1 )an - {} k f (X , O)
d
' h 'e 'RYu ' '
X
6
R Rq6e xGwpr
X
F gD
X (P1 )an - k
DX (|f |)
DX (|f )
X
q : DX (|f )
i0 iI0
G F (X D )
G F (X)
G (X /(f ))red i=j
wg T ' 0 b b p b 0 &u q
|k |
<
DX (f | ) = DX (f | ) G F (X D) i0 I B1 (xi , ri ) l F
'e
0xy 6 py
G (Y k l)/(f ) ^
G 0].
G 0]
d
B1 (x , d) - l
B1 (xi , d). l
I0
B1
x = xi
B1 (xi , d) . l
i
B1 (xi , d) l
.
Y
l/k
G
` F c F U x x Q P D x x c S c` Dd E I Q S W ` E I Q bT'w&y"0R"p0e&sbs60T&fT60H~ i E D F I D a W E I Q` E Ud P I` E Dd F I Q g&` g60bT6fR6ff00GE F RD
-a1 A = a2 0 -b1 0 b3
F gD
D Dd x Q P U &gb$"0Rg&
E I c W F`d` ` F P c S F I Q P cd F I c I D F I` c x F` I S U Q i 6e$sfb6bpwT0R"fegTfeBVb$T6p8 6& i T0g$v D x S Q W F D
DX ( |f |2 / ) = DX ( |f |2 / ) = DX ( f |f 2 , 2 )
F gD
|k | < < X0 = DX (f | ) DX (|f ) Y = DX (|f |2 /) X = X0 Y (X0 X, Y X) X0 X Y X Y 0 = X0 Y Y0 Y Y0 = DX (|f | ) DX (|f |2 /) Y = Y0 DX ( |f |) DX ( |f |) DX (|f |2 /) = Y DX (|f |2 /) X DX ( |f |) X DX (|f |2 /) 2 2 DX ( |f | / ) X DX (|f | /)
E W F`d c W E D F D i 1Tbbp0gV&VR2 p i 0bT0up$fs$Tef&1"fsf$pg&gHui I Q` F` E Q Q S cd D F I D x D F c` W x x` Dd U Q P W ` P D ~ F I D x W E` P W S E Ud c I l g"GR1T$|6f$ iXi D U Y S D {` S W a D F v U E d` $|g6bTpn&g"&Gsft D x x Q ~ ""pHXi D U Y S DSF I Q x D P I Q F v UE do $gT0H&p&r$G&qCi P I Q F 6p$rgD c I Q t F D I n01RRD i c pt I Q DE Q I jd `E t D x t x D r Gn0Q eT0"zXi D U Y E` U 60$$ F I D x D S` cd P c I g"gbpfggpQ t1RRi $& i "T0g$&|6pfTb&60R|f6bTpa F D D I i D x S Q W F U E I Q` F` I Q P E Dd D {` S W F I D x D S a U Q P D S Dd F 0"g6pR1Tb|RD E S Qd c 0bp}c I l ii F RD DE Q I l T0ukii D U Y d D $XgF F` E` Q P I bGp$wmil i Rp$W es&sTgb$p$ W P I Q I jd D W F W` S Q S D x ` E` Q S F cd S` Q "gfTf0T"fib0pa D U Y DFS QE D 60yG0G& D U Y ` E E U c c S D E Q U E I l i E D F` c E` F c E F I D` Q $TTeBwgT0u$$TipTeTib0GE D F I D W P W S D c F j d D E W F W` S Q S E Dd D U Y D F S Q E D F` F D F I D x x c E v U E F E T&g$"pRW e$1Tgb$p$1b"$"TTpT"$yRu"pTz&Ty1D D U Y D F` U E cd E I c c S D E Q U E I $"6Gu6e$|wgT0u$$Til 6 i S` cd P F I c I D F I` c x F E D W P S D P D S F c Fd U E W S D v feRVe6TfeB1i$wwg6web$Twi P D a 1pCc E S Qd c c I l wpfedii S U Dd c a D $bp'y& S U 6GE DF I cFE I Q P FE 0pG6pgxGD D xS Q I c v "p$iwei Dd ` E S D I b6bTpa f` P D a 1pCc D S` S P W F U D I TfTgp&y0Q S U Q $prq i D F c D E Dd U Q E D F I Q E F I D x D I Qd Q S D P D E D S { E Dd D U Y E I Q S F I Q f pTeB $f$pu$0Gg"g0$pfp$Ry&1T66sfs$6pTp6i 0$pfp$ D I Qd Q S ` U 60Y I Q` F P c S F W S D I j D E Q E` I Q 0b0R6$U T0uTf&up&t gT&V6pr$i Ed D F D S U Q q D U $Y d DF F gTgD P1pCc D a D F E` D d` &Rrbt 0i 0bT0up$dfepi i I Q` F` E Q Q S cd S c q F I D x D S` cd P c I g"TfeR|il i D U Y Ed D F F D E F P I` F E` $"fTRR6fTGf& F RD S U Q $pu D S Q P I D F I c {` S W a T0R62eug0xt P D a 1pCc Ed D W S E gTD$ S U Q S D a U QSF F U D I Q $0ig0$pT&pt D x x Q ~ i E F P I` F E` E F I` Q E D "0HR6fTGf&w0fpu& S$pu U Q D Y E D $UfdTF P D a 1pCc t E c P D P E I c r i E Qd P F I D x D U Y` S W d c F E pR6pi'0fRg"g60$g0bppGD D Y S D E U E U D I Q cd D S D { S W S U Q q i D F c B D E Dd U Q E D E D S { S U Q F I c $UsgT0Qu$$TF&dpt$ggPHg`g0aH$0r00T'x@&H1b6pu1&H1T666pue0Cc I Q` F P c S F W pfTRpwTgTS D I 6$U j I Q` F ` S F D S c pfTRPfTGEduFE1D D U Y D F S Q E D F` F D F I D x x c E v U E F E $G0Gn&gRudg""ez&Gu1D D U Y S D E Q U E F I D x Dd c W F U D I $|gT0u$$T|g"gfegu&guil i1'wd$Tsfdd0Q 26i F cF UE WS D j r D U Y d DF F $wgRD P D a 0Cc D D P I D FE` jd D S F I Q $g6g&RD )Tp"x p i 0bFT0up$|t i I Q` ` E Q Q S cd d D I I Q` F c S D I` c x Q Dd F E g$6pfT'w$bpBp&BfGD D x x Q ~ ""pHwi S D`d U P` F S c I D bf$RTTeg|c I il i D U d D $YTF F c x Q ` E c U Y F S D a U Q I j D P I D FE` jd D S F I Q RPp6"0RP h Tp6Tg0$pB$U Vg6g&RD )Tp"x i D S` cdd Q S Q P Dd t F D D I i Dd ` E E Q p i bpffppgXb0gugBp0b$fTT I D` F D ` P ~ b$yGEHR1DHi D $UY"G0Gd&2bTRuyg""ez&Ty1D D F S Q E D F` F D F I D x x c E v U E F E D U Y c S D E Q U E I Q t D F` U E cd E I c $gwgT0u$$Tn0p6Ggy6pir 6 i E D c F W E` Q S F I D ` P D P D D a U D S cd D E a` I l uegf0ThR1Rs&sp6g$fsTf` f&uiui E I c Dd` F P c S F I Q P F E 6e6fb000RXGD
DX (|f ) DX (|f |) X Q |k |
X 1 xX /(f ) Bk (x, u
D Dd x Q P Dd D U Y cd D P S U Q c S D S F I Q x I l i F` F D F I D x x c E v U &Rf$"pgfs$"fRV$02wgTp"|ibTgg""ez&TE
G F (DX (|f |2 /)) G0
F RD
P D a 0Cc
F gD
DTb`$TEd$UVSgTpQ"&X"v&Twf`&g1TF0$RDpTETIpS6{s1Tb`"bfdpgPfs$gHg0Tpa pg6GXGD F U c D Y D S F I x D F U E d t E D P c F I Q E D F c Fd E D F x` Q E Dd D U Y F D F I D S D I Q P S U E F E D U D F` E cd I c c S E Q U E $YyTb$UTBfEe$"wgDT0u$$TuIi0TS6bg6IpgPX$pQupF$$pciGD l i D Ud P Q S U ` ` FE D U Y F` c Dd D E`d` F U I $ibefyTffb&upQ E P D 0cgygPVE6e66FgugDg&iTFp"x"ggDfTcu wb`$t I c t D I S DS I Q I `S j I d D U YES Q 60wpv PW%422RPGEC I HV S U T S Q I H F D p w q yq u
0 F G F (X) DX (|f |) = G F (DX (|f )) F (DX (f |)) G F (DX (|f |))
'
uwe
' Be g 6 gsR ' 6g bb G' Rh h h 2"
A A @ 9 75 # 4 # 3 % ( ( % # ! CB )86$ ))'2 10)'&$"
F
X = DX (|f ) DX (|f |) =
D U Y W I I Q F I cF $y$6p&ipp$i
k x, y X /(f ) u (x) > u(x) DX (|f )
DX (|f ) q x X/(f ) B1 (x, u (x)) k 1/nx u(x)
a 1 , a 2 , b1 , b3 , c1 , c2
S U Q DF c P D FE $puTeg$RVGD e i )
i ) 6' Bb3
0 0
0
G DX (|f |2 /) DX (|f |2 /) G DX (|f | ) DX (|f |2 /) 0 0 RigCor(k) |k | 0 G F (X) c3 G (X /(f ))red G DX (|f |2 /) > (x))
E
G DX (|f | ) DX (|f |2 /)
f (t) = (t-x)nx gx (t) gx (t) x = |gx (x)| DX (|f ) =
DX (|f ) = xX /(f ) B1 (x, u(x)) k DX (|f ) X |x - y| > u (x) x y B1 (x, u (x)) xX /(f ) k
G F (DX (|f )) F (DX (f |2 /))
DX (|f |2 /) X Q DX (|f |) Q X DX (|f |) DX (f |f 2 , 2 ) 2 2 >1 DX (f |f , ) = DX (|f | -1 ) Q q DX (|f )
< 2 / <
=-
A A
A
G DX (|f ) DX (f |2 /)
G DX (|f |2 /) DX (f |2 /) DX (|f ) G DX (|f |2 /) DX (f | ) DX (|f ) G DX (|f |2 /) DX (f | ) DX (|f ) 0 RigCor(k) C= B1 (x, u (x)) k 1 xX/(f ) Bk (x, u (x)) c1 c2 u(x) |k | x X /(f ) c3 GX G0
R
DX (|f |2 /) = DX (|f |2 /) = DX (f |f 2 , 2 ).
DX (f |2 /) = DX (f |2 /)
|x-y| > u(x) u (x) |k | 1 xX /(f ) Bk (x, u (x)) X |k | >1 G (X/(f ))red DX (|f )
|gx (t)| u (x) = (-1 )1/nx = x
G0
C
C C
GX GX GX V
DX (f |)
G0 G0 G0 F
6 pQ
P D a 1pCc
i )
i )
i
c I Q u0$t
t D F c D E Dd U Q E D F I Q E E D S { E Dd F I Q 0T'xB &1b$0V&VpT$6s1bVp$
D&sD$wP&QS$F0ID"pTz&TE x x cE v U S U Q $pu b` a I Q` F P c S F D S D I U D F E` D d `` pfTRpwTgTs$6&Rf` ff` t t
DX (|f ) = D1 DX (|f |) = D1 D2 < X X
F RD
F I D x D S` cd P c I Q t E Ud D g"TfeRyz006f$Xir i
E I c E F I D` P v Q P D x 6p$w0bgzRD Ryi0$ I b0h$HGD d Q I U F E
^ D2 k l = B1 (o, R) - l
~ I0
t` E I` )T6fV% i
E Dd D F U x S D fX&ygu
D U Y i $" 0i pfTbT0up$"f&s0$g$fe$y""pg6bgpuil I Q` F` E Q Q S cd D D a U D S cd E I c D x x Q P D {` S W a I
{xi , i I} ~ I0 x = ~ 1 ||
i
Dd x D E I jd F f$yGgD )xRD
DE Q I Q G|p$t S U Q q $pr&i D U YES Q FIe`TSpCa f`DTET`ecd c c I E D U Y S D E Q U E F U D I Q t Ddd I D U Y` F E` S W F P c S c P Dd c j F E $G$6Gr&gusppfb$U r$TgTRpwebp0W x1D $Twpwv 0 3 ( ) '6 i 0Q % T6Teu esG6$GupQ upfgT0$g$Tf&upe6TfeBs&VfTTe6 % U DE F Q jd D E Q U E I j U Y I Qd D E D a U D S cd D E` a` I Q F I c I D F I` c x D S` F S c
7
DE Q I Q T0ud0$t
t E S Qd gwpfV% i D ` Q P I U S U Q &dbp6wu6$pu DF Q I l Tp$I ii
S U Q q i I Q` F cd D S D F F D P S U Q E D P I Dd c a` U Y j E D E E cd P E D Dd x D E I j $0x&wpfT'gsTTRgV$pug6gfe'f$W TT0fg&yf$yGgD )d S c D` I { W p6hf$6g& S U E D P I Dd c a` U Y j I Q` F cd D S cd D S ` E I Q P I $ThRgfe'b601W n0b'gBT&G60Ril
^ D1 k l = B1 (xi , ) l
F RD
i=j
E Qd f`0bpc D$Du$0T0s&6$gfGpfed0bT6TF RD U Q S D D I I D` E` Qd c I Q` E I D D U Y Ddd D $yfbTF D $I$U F` 0Q $pD&R6$IpQgPIgD"D$UfSTgWx0QgW0iTE0u$$UTdI0Q jUY&p$IBv'cTgDp6p|$U i D I F x Y` F D Q E D Q FS a U Q I F` Q bp& PW%422RPGEC I HV S U T S Q I H F D 16)y 0 0 ) 0 6 s 6g T T bu $H ' 5b3 ) B ) e$ bqRT2 xH H i D W S F I Q x W F E D I Q` F` E Q Q S c 0gTp"&V1d0bT0up$wv Hbx
DX (f | ) = DX (f | ), DX (f |2 /) = DX (f |2 /), DX (|f | ) = DX (|f | )
F gD
F gD
E D Q I v c U c D W U Y`d c i &p6"ed&p$ff$$p6& i pfTbG0T$s&sb6pR1&wgHui I Q` F` E Q Q S cd D Dd U Q P W ` P D ~ F gD
r : DX (f | ) DX (f |2 /)
E W F`d c W E Dd c I Q t F D D I Tbfe0f|0$gugg$i E I c E D x E` S Q x Q E` E D F I D E` U I 6p$"G$$p"0G$X0G$&b` 0 i )
E D F I D a W E I Q` E Ud P I 1Tg$f` gpfG6fRb` U D E Dd D U Y D S` F I D` a D S ` P D ~ &&fs$sTf&BVgfTgHi E I c E D x E` S Q x Q E` E D F I Q 6e$1"G$$p"pT&VpTE F RD D U Y S DSF I Q x D $VgT0s$ uegs"gf` $g& D cF W D x U D cd D I D F D F x Q z&}U gBT$0RP B&Tsfwt 6 i 0bT0up$gz&h0$g$uy$b"gy$0)T6fVi i ) &Rf$"pge2gfe'f$W c S v U E d` i I Q` F` E Q Q S cd D D a U D S cd S D I` x S D F S U Q t` E I` % D Dd x Q P U c F I Dd c a` U Y F I D x D U Y` Q F Q x Q F E g"g60$0Te"0$hGD D gb60RwHT&f` B6pfT6bg6fzfhpTE Dd x Q P D v i E D F I D a W E I Q` E Ud P I` E Dd F I Q F RD pQ
-id -b1 0 , A = a2 0 b3 e 0 0 id
F gD
4 # % # ! 4 $'&$)
3 % ! 4 % @ % 4 # % ! # @ # ( % z4 'up p Cx$'gu 0$# 0 )$'x f x)g uuux y f 4 @ ! 4 3 5
DX (|f )
a 2 , b1 , b3 , c1 , c2 , c3 , e
e : DX (|f | ) DX (|f |2 /)
X (P1 )an - {} k G
r
E
0 RigCor(k)
C =
R
^ X k l = B1 (o, R) - l
c1
c2
c3
,
E=
iI
B1 (xi , ri ) l
0 RigCor(k)
F
A
|k |
F gD
E D Q I v c E F S D a U Q E D F` I { W I l &p$BeTgp6p$X$R&gii D D I I D` E` Qd c D F` S j &$$bTpfe0$pQ ed D U Y Ed D 60HfTF F RD F I D` gfpQ i S U Q $pu
i0 i0 I
E I QF Q 6peX6i F RD
ri |o - xi | R j = i0
|xi - xj | max(ri , rj ) I0 = G.i0
DX (|f |2 /) = DX (|f |2 /).
G
id 0 0 R= 0 r 0 0 0 id
C
B1
G
< R = |xi0 - xj | D1 D2
S c p6
F RD
^ D1 k l =
iI0
B1 (x , ) l
F RD
I0
^ D1 k l =
k
~ I0
B1 (~ , ) l x
F RD
G
x ~
~ 1/|I0 |
! E D F I c a` U E E W F W` S Q S E Dd E I Q {` S W )1Te'f$G1TRf$p$f606gi i
D2 = DB1 (o,R) (f |) k
E I QF Q 6peVi
=
=0
|~ - x | x ~
DX (|f ) = D1
~ I0
f (t) =
D1 = DB1 (o,R) (|f ) k
~ ~ Io (t - x )
X = X D1
DX (f | ) = DX (f | ) G X /(f ) xi .
` f` ` P D a 1pCc
^ D2 k l = B1 (o, R) - l
^ D2 k l = B1 (o, R) - l
i i |xi - xi | |xi - xi | < ~ I0 x = x i i
~ I0
iI0
B1 (xi , ) . l
B1 (~ , ) . l x B1 (x , ) . l G o l/k X ~ I0
k
t E Ud D g6f$iXr
F U QF S U Q &p$0 P D a 1pCc
P D a 1pCc
% %% X % ~v v v wQ%A%#| ~ E eif4W( uQ x`' i6# dx v x y y y | y Q nb % x xWb AxHi y y W)ixy vy x xib HUH ti pcj)Uiy et y z z x ~ z y % uB 4WR !b|4~ vAUzfinxu!u#| y ~v v v ~ 4 y eH vz v v | v RXe%A%#)xRx xWb ' lUc y ~ z x v % u#ty Uz x v ~ v yz v WW ix ix y ~y Rx xWb ' !ki y ~ x y X% zz x x y Uei4Wxb i { y AUz Ef y v z n~ x0e B u %% y i 4 X iiW4eu#QRWRdiRnx dWB4R)ft%%n xu!u#| ~ 4 eif4W( %x y v x x y y uy | v ~ } | x y { z y v ~ ~ x y ) ~ v v v k67cfEt x y y y k' i6# fx x xib dx y ~ y yz idixbtlRR W nx v euwu z iAWUi g a g D I P D w I S P p ieu 4X(W#f4 eR(6Y F P p D P D D v I P T w V q g D S V s q D S yS I D y P s q s F P F F D S h D v I P T w V D D I V q P q P q PS V h D F P F D V V D h F q s X#Rrjdd!X(W#Q# 7t#rbXrUR4us #e(UQt#(Reu4R(4RXX) Wl!WQRW4XdUtQXr(RAReXiXF 4(hXXiic4Rd dk#wXdd7fkklkwldfjkldwd4r V q DS p rWe7g
1/|G.z0 |
g
g q g4r g q D!WQIRT#XVpR4WQ#e#e!XXwWng v P w D p D I G s D qS I D V S h D V q D D I p P h D I D I P G D S h h s V F rX#(i)XheH#Rnh#reeii(l)D D #r h D (pBsd)DW4UXIRphrqRPXAt4r g q (UQX(nRQ#e!h))#XR`EEg F T S T V F sg q PS VS F P G P I G s D F D F v w V P G w F D C V iD FI P xRe SFF s DF P G q o XXufX(n`d%g 4u4EQrWeiWrWHf!R#eRn#E)D dBg F q s F V q DS h D P h D q P G q V F g F q PV P RXRt V iD I P 4RnG
F f (t) = zG.z0 (t-z) X1 = DX ( |f ) X2 = DX (f | ) ^ X1 k l =
zG.z0
D F D #rHbWQV D FVI D p P F D fxR#(fe D VS V D G F G V q D D V hS I V #rQinlU#`rW!WQbXQF V WD
|k | X
V q DS P rU(g VS P (Eg
iI
D D D 4(ieeWXV FS P s D D G P I y D D q q DS FS P s y q PS F q D V D D q bReuhi4RQRh##UXe(UR(pReWXW## VS P UR g D D q q P h V q D D S I V T P T y D P q s V I D p P q (W##(iH(!W#rbXQi!RRhR#A%HXWR4R# VS P g DS G P V P P w I s G V q sS I s p qS V D V q D y I D p q P h I F V q D D S s UR(U4RXR!R#HR4crReQu%UtWkrW(XRRWQ#XlrW!We#eum F V I Dm F q s I V h D p s s D h FS sm T I G q V F QXW64uxX)RBhu)iQeRiQ#i#D j!WeXe(XQh7QrW!X#(iQE)U#Rnf#rbUWiR)D S F D G V D vS FS P I V F V q D D I p P h D I F D I P G D S h h s V F D I D S I T p D V F D I #rW4UQWRtkXQR wU g D V s D F D P F D V q P F F D I F D V q P (X%t)U#(f)trRXHQ#4f)UHrR
^ k l (g|
gRDQI#eih4q(PihlI#(PGitg4r g q (UQX(nRQ#!bfW#rbU#4utbR6h#4dQRW4Wtdo s q PS VS F P G P I G s I D S G G s FS Pm D q D I P h q D V D G q I P G a pS #Rn#UY s q P (# #xQ#H(!RXXF D D wh PI G V q D sF q PS V h s I V D I D q D V FS D S aSSS ReXiRxXWX#4`QbiEebee6Y
DX DX k ^
^ k l (|g)
F D V q s pS F F T V TS I G P I G F D F F F D S h D P h D S I T p q o Qru%U4fQieX4RQ#Hec4XX) ii!!RW`QW(pdg V iD
~ 1/|I0 |
F I P s s q o WxReu!dng
F
g V q D D I p P h D I D I P (!WQr4Ri)Xhet#(G D S h h s V F #rbUWiR`)D D I D I V q P D V F S F DS I P V h D p F D h s G F #rlXQ((!dh#UWbWeX(XiRf)i(4)D D s D h FS sm T I G q V F uWXbuWX4#dD D F I P y | v ~} {z y x v 4(QxREC Wi#6%4Q| iuwu g D I h q P h I P G sg RQ#ei4(it#(dt4r g q q PS VS F P G P I G s I D S G G s V q s q D V qS s V D G q o ReXUr(X#!bfW#rbU4#ulruWXreuAtpdng h#xQ#trW!!uQA#F D D wh PI G V q D sF I P #RnG D #r VS sm D P h T a pSY T V TS I G P I G s C Uu!te#(i#f#UXieQ#RQ#Rg AReXi4QX4RWku4lXrWbWkjbeUiXWUQ#RQ#hRW)URQfrRXfee6dWbd#rrWe4 D q PS V h I V F q P h I s G D V q D S p T V F D aSSSY T V TS I G P I G s C g F DS s I p V q P F aSSY V D aSY D V q DS p V iD
~ I0
s q P VS V D G V q D s F RUXWHrW!!uQF V iD
l
iI
D F s D V q D D y q s w h F v I G s V q D pS I h g F aSSY V D aSY F T V TS I G P I G F D XR4HrW!W(#uxc)X4utrWReQiT fbe6dWcb`)XWUQ#RQ#H)EC
@ @ 9 8 53 " 2 " 1 $ & & $ " BA7'764#''%0)('%#!
ri |o - xi | R |xi - xj | max(ri , rj ) i=j |z0 - xi | iI |z0 - g.z0 | gG g.z0 = z0
V WD
h D p )RBs
~ I0
DX (f | )
B1 (~ , ) - B1 (~ , -1 ) B1 (o, R) - l x l x l
B1 (~ , ) - B1 (~ , ) l x l x
X = X D1 ^ l (|g|) = (D1 D2 )k l
aS bY V q s p s D Ph DF P G q ru%B!(irdo
|k |
1 iI0 Bk (xi , ri ) =
F Q ^ ^ ^ (X k l) = ((X D1 )k l) ((X D2 )k l)
V WD
DF P G q X(ndo
I P G FS s #(bu
l/k
<
g=
^ = D 1 k l
~ I0 (t
F
- x )
^ (X D1 )k l = DX k l (|g) ^
>1
DX
^ X k l
B1 (o, R) - l
^ k l (|f )
=
-1
^ X k l
^ X k l = B1 (o, R) - l
^ = D 1 k l
=0
iI
B1 (xi , ri ) l
|x - x |
B1 (xi , ri ) l
DX k l (g| ) = DX ^
G
/ (X k l)/(g) ^
X (A1 )an k
^ (X D1 )k l = DX k l (g|). ^
B1 (xi , ri ) l
z0 X(l) X1 X2
V WD
h D p )RBs
I P 4RnG
B
k
B1 (z, ) l
(Xi X)i=1, 2
^ ^ X2 k l = X k l -
F
V iD
V WD
I P #RnG
Hi (-, F ) ra F
k
iN / Hi (-, F )
Nis
Q
Hi (-, F ) ra
k
~ I0
~ I0
ara (F ) = aNis (F ) B1 (~ , ) l x )
zG.z0
B1 (~ , ) B1 (o, R) - l x l
F
f
B1 (z, ) . l
=
B1 (o, R) - l
z=z G.z0
SmAfnd/B
iI-I0
iI-I0
F
t6f
|z - z | X =X
h D p )RBs
aSS be6Y
B1 (xi , ri ) . l
B1 (xi , ri ) . l
D p D I G s I s G (#WQ#0efu6Y D D #rWQV D F FS s T w h QbUd!#xQF q I D p P I V V D G q o g F D VS V D G V q D s F #fW(#RQXfpdXUXWHrW!!uQXF F D I P q F D h D p !QR#cedRBs F4(UQ4(6f)R4G q PS V h q Pm F D I s F D D h s G D I q P S F F s G D y q s w h D )UHWRb#!WQRnR4H(#uxd#q D q q PS V s I D qS s P D C g s T g V q s V q W4#ReXuQt4UR!(lru)eS ErcrRQ!WD D FI DV sI T q T y F D `x#WQ%xW#R` V iD F q s 4u4 V q DS P g V q D y I D p q P h I F V q D D S sm V F rWeRnrRQWR(ix#drU4UR`D D F q PIV q P #rd4(XQrR g DS I P y T V s F P F s I F DS G P V P P w I s G V q sS I s p qS F V I Dm F q s I eX(RWQ%h 4RX!eH4e#RQu!R#HR4f(ReQu%efQXW64uxXV h D p s s D h FS sm T I G q V F g g a s E (Bu)iQeRiQ##c)D diB(dg q g q !WQRW4XteeR(6Y D v I P T w V D IS P p D q PS V hS I V F D I s D (UQbXQ)Xde` q D D I s VF W`Q#4bf)D F FF D S 4XX) g I P G q VF #(74rtD D G P I y D C ag SS n#RQRhUY
Fx ^ Hi (B1 , Fx ) ^ ra ^ k(x) Fx ^ |fi - gi | ra /X X F H0 (^, Fx ) ^ ra x x / H0 (B1 , Fx ) ^ ra x ^ i>0 SmAfnd /X ^ SmAfnd /k(x) ^ Z SmAfnd /k(x) DZ (f0 |f1 , . . . , fn ) X F f 0 , f1 , . . . , fn C X B1 Y Yx ^ fi Z
g D S F I D p qS V F (U4UXQRefD D FS w G I P D C ag S !Xe#4X(h6Y F D V q D s pS T V q D D pS V h D G F D HQ(UR%U#r)frReXin)XI V q P F aSS V D aSY F q PS VS q P h F D D D I V q P c rRXAeUY i!e64(UQ#4RWe#rXQ((!XE g q g q !!WeUj(U#(XR(#Au!rReQu%UD D D D DS G P V P P w I s G V q sS I s p qS V F D P !!Rcg D #r WXiD DV V V q s q DV q P ru4XrRWh D F FS D P q F P q D V FS D g I D F P G G F V D G q P D V q D T h T I G q PS F F h FS s I s QbUR4!us 4(F 4 XbWhUS Err#4Wn(kRQrW)iQ#(UXQ4iQe#bujG g aSSY V D aSY F q PS VS q P h F D F q s eUdice64(UQ#4RWtef4u4 I s u4G V iD I s u4G I s R4G I D h s G D I h q P V D G q o i(e#!X`4(dpdng I P #(G F D FS w G I P P FS F D V !b##QR!(bj#kiD q PS V h D S D q s q P ReXiV #t#4dA(u I s R4G D V q D T h T V q s D S q g rW)i)Xruk4(b## D V P q q P S g DS G P V P P w I s G V q sS I s p qS V D V q D y I D p q P h I F V F Qu#h(R(U4RXR!R#7R47(ReQu%ein(RQW(r(ix#X)D D VS T q o g F V q D h s F P F F D h s G F #rU#)dWxrWi(V 4(7)iR)D F)Uj#X!Xe#4X((bj)f(b#U)X(WXreR#AufQXr)U4uxu44ejrW!WeuxuXjQb#Xje44!Xe#4X()EC D I F F D FS w G I P P FS F D V q D FS qS F D I y v V qS F D P q s D I V q D F D s I s G T F qS V q D D s V P V FmS V h D VI F FS q F D FS w G I P F D g D D #rWQV I #XF F V I Dm F q s I V h D p s s D h FS sm T I QXW64uxXl(BpRiQbuiQ#G q D V FS D g D I s a a Y D F v w V P G g V D #0XeieS (!XA%`X6p)#XRw hi6Y a e4r g q !U0E7g #QbUh)D g D D D C I F D F FS V F D D D 4(!eUXV D s I s G T F q e#uxu4US VrURQRXRe4Xi4#fWR4RQXtpR q D D s V P V q PS F q D V D D q I D p P I V V D G q P D S V FS I T V h s I s h D V F #reQbXXiRxuWi#HD D F I P C g D F FS V F #rXxR#RXXbUtD D q D S V FS I T V h s I s h D Ree#d#rbXeQWQ(QRWi VF )D #rXxRB)Wee#RXcR#A% A#fD D F I P C g I DS y T I D P q s q V F D I DF P G G F V D G q P #rjWX(n##XkWn!Ru #f#RQus D I PV IS h T I V T I D V VS T ei)XQiQ`tXU#d#g V WD D D 4(WXV T I Dm D P q F P F q D V FS D g I D F P G G F V D G q P QW6dR#Aus 4RX#XbWeS (tr4#`WnR4 hRuUXe(R#HR4G D D y s qS FS P p q I s I D h s G D I D V VS WWRb#!WQ!iXU#dT g I P G q VF #(7#tD D G P I y D C aSS n#(X(Epbe6Y
x mC,x = 0 l/k mC,x = 0 ColimU Flt(x) Hi (B1 , F ) ra U ColimU Flt(x) H0 (U, F ) ra C k ^ C = (C k l)red ^ F SmAfnd/X k l X F X Flt(x) Flt(x ) ^ X k l x x x / ColimU Flt(x) H0 (B1 , F ) ra U x i>0 k F = (l/k) (F ) F p>0 F C Hi (B1 , F ) ra U x Hi (B1 l , F ) ra U^ C X
k
g D S F I D p qS V F (U#eXQRUHD D FS w G I P D C aS !Xe4#X(!EpbY D S I T q T y V qS P G V P V I P G F D V q s pS F F T V TS I G P I #rbX#W(H(eRnt(Xf#(f)Xru%e#H)XWUQ#RQ#G F)UhWQIXr(pA#RQ4(i)duX Reg q XeubeRQRiAepeii#(bUeX#6Y D I D V q P T q D s I h q P V F D q o g a a g D IS s P I P h D s D h I P G I D FSS V D F D S I T q T y F V qS P s F D I F D I #`)#reQW4WRhxrU(G #uQ#4`)U4F F FS G I P F S F s F D VS qS D!Xe#w4X(P(bS XR4ri)0U#4e`a q S g q #rW4UQWRA#0!UnrW(XRRWQ4drW!U#euXQ4uc#UiD Y D I D S I T p D V F S V q D y I D p q P h I F V q D D S sm S F F s S V F D T q q P #r#4R V q s ` ruxVjYg I F F V q D y I D p q P h I F V q D D S sm V q P #0QrW(XRRWQ4rW!U#eurRXF F D D VS T q o )U#rU## dcg I F FV q D yI D p q P hI #0QrW(XRRWQ4F V q D D s rW!U#eSumVrqRXF P DS y P P P w P h D s D h FS sm T I G F D D D p D I s D V T p eR(U(!R#(i# #uWXbuWX4)U #r(#WQ#G b #W# us rs do q g DS y P P P w P h D s D h FS sm T I G s F TS h P F F (U(ReR!R4Rii# u)iQeRiQ#Rc)UWQRs F D I F F D FS w G I P P FS F D VS qS )U!#Xd)Xb##QR!RXe0#AU#Uda q S g q 4(p4U#ReF bWip#r0ReWn#utY(URWU Y D D S q yS s D h D F q P D G G s I g D s h P D h q D s pS T D q V F i4UR%U#r)#4D s D h FS s u)iQeRm
F|X / p (I )
I F s D h F S s m T I G D F D D G P h D D F S w G I P D D I D I V q P D h q P s I F S D S F I D p q S V F #Xu)iQeRiQ#fie#!RWi!b##QR!e`4(HXQ((!h4R!xAURe#exW(UtD a 4S g q Y#rfIXQ((!j#RW)(RQ44 D DIV q P I P G g V q sI D D G P h D F D s h P ie#!Rite%euWe V q D D pS V h D V P I G D I P h q D V F rReXiWQ#fQRiWfD F D pS V h D V P I G F D I V h I V F ReXiXX#)X4XXQ!us VrWD!W(USQ%bWQI0DQ(X#p WeeU#di4WXi4R6m #D q D p V s D VS P I q DS T D I D V h q P q V F D P g V q sI !(rux#4 V q D D pS V h D V P I G s D h FS sm T I G D D D G P h q rW!W(UQ)WQ#u)iQeRiQ#A!WU4(i# D h q D s pS T D q g gS V q s I W4Weu%e#rA##7eRD 7rux#4 V q D D pS V h D V P I G V q D D h s G D q rW!W(UQ)WQ#`ri(e#!XI # VS P (g V WD D F q PS V hS I V F D I F D d4(UQbXQ)X)U V iD F q PV P RXRtjg DS h P F F eiXX(s D #
U
g
F D VS F D D FS w G I P q VS q )XUi#h!Xe4#X(!p4t##4US D I DF P G G F V D G q P #rtr#4l(r q PS V hS I V F D I s F I s pReXbQXQ0QtuG
/X Hi (X, F ) ra SmAfnd/X / Hi (B1 , F ) ra X
D FS w G I P D D F q P I V q P V D D F FS D P q Xb##QR!Uh#rf4(XQ((!tihQbUhR#Aus q # VS P g DS G P V P P w I s G V q sS I s p qS V F (4Re#(Xu!(#Hu4frubXR%retD D F q PIV q P 4(f4(XQrR g a I F s D h FS s #XpRWXbum q D P # !(ih I D I T S F q P h V q s V FS F q P D h V F D R s V q s FSS V q DY F V q D y I D p q P h I F V q D D S sm V q P WQWbRi(RQeX4(ih W4XRs )e(RXbeUQ WpxrWRQW(Rix#XprW!U#euAr(F s D h FS sm T I G F D D V q D D s y T D qQ s I V q D E g RWXbuWX#f)Uh#rHrW!WeurW`#b$QQrWF (Ug q (UQX(nRQ#b`4(`RRXQR4uH4(#WQiHPg Qee`(#A%s q PS VS F P G P I G s D D y s F F s G s F q P q D V D I a D F FS D P q V PV I P (Xf#(G H P #6Y D D FSS V q P V #rXbUeX RhiD I F V q D y I D p q P h I s D h FS sm q V q D D D I 4trW(XRRWQ#XF RiQbu#hrW!WeeWQ#XV a s q VF D S 4h)4( D I v S F q P q P T V h S D V V D h I D q I P V q P h I P G g F D P q s F D I F S q T V q D D S q V F QW#eX4RWh R7(XU#iU!XWi`W#Q#RQrRih#(R#Au`)h#j#irW!W#rb##h)D D FS #rXb##G VrW!WQ)XeR4#ree##Rs c4ReXUr(X#XXiif#rHxW#rxuAWQ`#Wei4UWX4(i#WQ)UErF (Ug q ReXUr(X#leccU#(ih q D D V h D IS F s G D S G G g F D q q PS VS F P G P I G D V V D h D s I D I s D I DS h q DS h F q P h I D V h D D CY g E g q PS VS F P G P I G s D D P T g FS G D IS P T I P G FS V F D D FS w G I P P FS q V F eS (Qb##iQU(!W!t#(le!lh!b##QR!(b#lD D VS sm D 4(`REC 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( 2 " $ " 2 w#%#w' %
Hi (-, F ) ra / Hi (-, F ) Nis
&
2 1 $ 2 $ 9 " $ 2 " $ " 9 " " & $ 2 9 2 1 3 rw%rwnR$j#Br#%W!(#( '#r%eEr'nnn d
Hi (-, F ) ra
F
B
SmRigqc /B Hi (-, F ) ra
p : B1 X
D V q D S p T q PS V h D V P I G s XrWbWReX)WQ#C I D h s G D I D V VS T g D S F I D p qS V F WWRb#!WQiXU#dURe#exW(UtD a )4S g q Y
U
h D p s D s h P )RBAURWe
D P q s VI D p P q S #(#A%0XW(#R# H#r
Ouvaf (B1 ) X
F|B1 X
ra
/I
X
F
p : (Ouvaf (B1 ), ra) X
SmRigqc /B
p-1 (U ) = B1 U
ra
p (I )
X
/ (Ouvaf (X), ra)
p
F|X
ra
B
Hi (U, F )
F|B1 X
ra
Hi (U, ara (F ))
B=X
(Ouvaf (X), ra)
ara (F )
ara (F ) {ra, Nis}
F
ra
Ouvaf (X)
ra
(Z, O) gi
^ X k l U Flt(p)
a q Sg q Y
G
x M(C)
F|X
Ouvaf (B1 ) X
X
x M(X)
Hi (p (I ))
Hi (I ) = Hi (-, F )|B1 ra X Ouvaf (X)
x
k
I
g
D D FS w G I P D S q q D V FS D !b##QR!#rb###hXeWE#g D V q DS p 4(trWeE#g D F V q DS h D P h F D D HQrWeiWrWH!Wv
r
D D D V D D I Pm DS I T F D S q g #rieUX`UeW!QR6`eXi#rb## ' D V FS D S D s I s G T F qS V q D D s V P V DS q q PS F q D V D D q V F Qbi7e((U4uxu44eHrW!WeuxuX`e#4Re4Xi`##HD
D P !(4g
DV P q q o Qu#pdg V PV I P #RXf4RnG D T I Dm V qS P G V P V q hQW6freRnf#RXWD
D D #rWQV y q sI D VF #RQhtD
D q q D S P h s D h S I V s s C ##U#(i(V WeQXuA0g q PS V s V q D F T I G D q VS FS P w h q o g D D q q P h s I D F P G G ReX%xrWXQ###HUbR#xd4RW##RW!Qr#4F q g D F FS D P q RP Er0XXbeiR#A%s q # VS P (g F q P V P g mS V h D VI F V q D D DS V q D F F D V F D a Y D IS P p D V F D I 4(XRt7Q)#X`rUeWeXrQh)t(RS g q p#rdeR(dQ)X g V s V F T I D H g uQU#XQUXP 0#g I #XF DS P q D a E Y D V q DS p U(RpWt%#S g q #rdrWe4g D s y T V F iUR(Wd)D D T VS hS GS V s V #0XUibe#UQ4dAbliD F I P g D S I T q T y T I y D q P WxRe (#reQW4WRQRRXF F q P V P g I D V h sm I DS D I G D I 4(XRt%4WXiRWe!WQ#fUj4F F q P P g D s I s G T F qS V q D D s V P V V D mS V h D VI F S q V F 4(XVRt%Re#uxuWX4UrW!WeuxuQWUXib#('#D D FS w G I P D C !b##QR!fE%g mS V h D V F V S q D I y v V qS D P q F P F q Q)I#XcWDw#4u(X(WQ(eHR#Aus 4RX4 VS P g F D F s F D h F q s a E URXR4Wu#ku#S g q d4(!UQ#j4#X(!RXeS Y F q P S I G g F D w G I P P F VrW!W4(b#(P#uhXFR4h)i((P q D D S q q s F D s F D V q V iD D V q D S p g I D V h s m I D S D I G D I F m S V h D V I F V #rrUf7)#WQ(hWeX4AU#XHQb#XdiD S D I V T V S I G I s D p I D F T I G q PS V h D S D V V D h F G D 4q4iQXWT g7iQiTeX4RPQ#GbRQWXQ#(UQV #0XWiwWU#0lwg D F D I y v V qS F T I Dm F D P q F P F F D D D F q g I lQRWQrUd!QW6c#(#A%s 4(`#dU4dWX4D d4F D F D I y v V qS F T I Dm F D P q F P F F D D D F q g D q PS V h D S D q V F #iQRXrUi!X6`R4!us 4(0e#0WX4WD 'ReXiV ##4d)D q PS V sS h P F F g ReXueWQRs fC
p
s-s(T )
D s I s G T F qS V q D D s V P V V D mS V h D VI F S q D FS w G I P D F q P I T S F q P g mS V h D S V F e#uxu4UHrW!WeuxuQHiQb#X#4`!Xe#4X(hec4RQWbX4R7QV #tD
CorB (X, Y ) Z Z[1/p] / Cor ^ (X k l, Y k l) Z Z[1/p] ^ ^ B k l
F D )
V iD
a E RFS g q Y D FS w G I P P P g D F q P I V q P V D F D F FS F D P q Xb##QR!R!R4w #rf4RQXrR!tWfXXbUc)(#A%s VrqUS(PF4)ViDwDf(UHrW!#UU4fc(RS g q i4WXi4R6h#g D q g D v V q D D qS D G V F D a Y I D V h q Pm D C D D #rXV VS P UR 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( % X % y v x x y W ( RQwiez ixnki y y ' i64 0Q x xib !ki y ~ x x v y W)ixy
sN (l/k) : AfndPST(B, ) F [l : k] = ps ^ F (-k l) / AfndPST(B k l, ) , ^
nx
v u y yz v e%7dWxbt(w
x
v u e%7l
a ((S g q Y Q% x v X % z i iw%i UW ~ y v v y W#ec4WAiUz ui4k%4i(X f6 p7 zz v ~ x y R UeW'iw%4iX4ib i z v %n f
p
g D p D I G s F T S G g q F q D S h S V s T w V s D R(#WQ#0Xiee###!s Ef4UWeQ%AW#Q%A#W F D q h s FS Pm D V P g qS F V D D I P V q D s q D G T q V F D F P F F D S h V s V F T I D D S I T y s F s h D F q s ) #iubR6iQRUe0ikWQR(!R#44eS d4(Q# iduQU#)XU7(#reQ#W(URd(Wei4Rd wU g D v I P T w V D p D I G s D p v w h s S h D R!WQR#XpR4WQ#e`R#xRW g q 4r g q !WQR#XfrR4WQ(euAfrWe#RWc'e6i'`xrU((rRQWR(ix#HrW!We#euWe#cD g D v I P T w V V q s q D V qS s V q D P h T ag SSY V D ag S Y F V qS P G F D C g V q D y I D p q P h I F V q D D S s m q DS V F D D p PI G S h D 4(`R#(X4nW
U Flt(x)
D y s g VF RRAUS 'fD
F I P s s q o g s T g V q s V q Q(URdu)eS ErlruxD D I DF P G G F V D G q P #rcr4#tR# h D p (Bs
U Flt(x)
I s u4G
@ @ 9 8 53 " 2 " 1 $ & & $ " BA7'764#''%0)('%#!
Y
D
FI DV sI T q T y D D vVF F q g Q4WXuQ#W(hi!WX# 7 I D h s G D I D V VS T g a S i(e#!XdhQU#dUuXE g q g q XeubeRQRi D IS s P I P h
Fx ^
Colim
Colim
|k |, >1
~ ~ (f 0 , . . . , f n )
Fx (DZ (f0 |f1 , . . . , fn )) ^
~ ~ ~ F (DX X U (f0 |f1 , . . . , fn )) ^
^ Y X U (Y, O)
U Flt(x)
~ ~ ~ Colim F (DX X U (f0 |f1 , . . . , fn )) ^
Colim
|k |, >1
k
Fx (DZ (f0 |f1 , . . . , fn )). ^
B
k
^ - k l : AfndCor(B) Z Z[1/p]
/ AfndCor(B k l) Z Z[1/p] ^
x v z Wiw%iUf
y z # !UWy ec y
a r rRS g q Y u % y z x f tW' cWi r !RxnWUz WQ0nc ~ y v x x y
rN
ps(T ) [Z]
n
B
n - dim(Y ) r ap k
fi pr X = Spm(A{t1 , . . . , tn }/(f1 , . . . , fr ))
T
[T ]
e(T )
^ ^ ^ ^ e : (X k l)(B k l) (Y k l)
Z = e(T ) ^ T = (Z k l)red
al
fi (tp , . . . , tp ) = gi (t1 , . . . , tr )p . n 1
Y l/k fi A[[t1 , . . . , tn ]]
^ X B Y ^ ^ CorB (X, Y ) CorB k l (X k l, Y k l) ^ ^ ^ ^ ^ T (X k l)B k l (Y k l) /Z T
^ A = (B, O) Y B k l ^ k l{t1 , . . . , tn }/(g1 , . . . , gm )) Y = Spm(A
p
(g1 , . . . , gm ) (t1 , . . . , tn )
r
^ ^ (X B Y )k l
gi
T
r
/ X B Y . ^
fi A{t1 , . . . , tn } ^ B k l
F D P q )R#A%s
: Y
/ X k l ^
r
F q s D 4R#Ue#q DS I T F s V D q UQWXb7WnD fg
F q s u#
DS I T F s I UQWXed#F I s G D T P p q D VF R4dR%(ri)D D D F F s h s I F D T P p q D V F 0XX(eWbd4dR%(r`D F q s 4u#
^ (X k l, O)
r
F D h q s F FS G F D V q P iuQXb##H)UHrRXF
ti ^ (X k l, O)
tp (Y, O) i r fj (tp , . . . , tp ) = gj (t1 , . . . , tn )p n 1
ps-s(T )
(f0 , . . . , fn )
^ ^ ^ ^ (X k l)B k l (Y k l)
fj (t1 , . . . , tn ) (Y, O)
q #s dg q o
^ X k l
Y
/ Bn ^ B / Bn ^ B
kl kl r
mS V s V P h T I I s Q%Q#d!RW`QXRWh F q s D 4R#eU#q D D F F s h s D D dXX(eWb#rXV
DV P q q Xu4do D F V q DS h D P h F D Hx(UWiit)EC
X
l/k
Y
V D G q P D h IS P p I P G D F v V q D y I D p q P h I F V q D D S sm F s G V F g #rhdrRQWR(ix#rW!We#eudR4D Eq s D h FS sm T I G D u)iQeRiQ#Uw D F FS D S QbU!bRy Wn(w#bs WiheRRh#(g S I T V TS I s QXieQu%p D q D T q q P V q sV ` ##AW4#RdrRQjg I F F V q D y I D p q P h I F V q D D S sm s D h FS sm T I G D H D F F g D F P G FS q g IS s #Xlx(RQW(r(ix#Xd(!We#URu)iQeRiQ#A!IXX(s ErXe# (P Erieubih s G V F g q S V q D D F D I D w s g F V q D y I D p q P h I F V q D D S sm F V I Dm F q s I V h D p s D h FS sm T I G F D I D V h q Pm D I D S #WQ4(6e0#ree#G F(40D EcUrW!WX4X# #euEQrW(XRRWQ4!(!We#URx64RQQ(Bs RWXbuWX#d G g V F S F V q sS I s p q Rs Ef!XEUkQrubQu%US V D F V q D y I D p q P h I F V q D D S s m F V I D m F q s I V h D p s s D h F S s m T I G D F D G D D F D I S q D V P I P ijxrWRQW(Rix#XfrU4URfx64RQQH)RBAu)iQeRiQ#d#c)U4iHe#WQRc#RG g F T y q sS I V FmS V P F D F P p D P D DS I P T w V s F q s s V q D s q Pm F V D P F V q sS I s p q W)U#(#ubQXfUXRtX#RWjeX(W#Q0ef4R#RQrW!R#4R6cxiV (Wx(ReQu%eS FVI DmF q s I QXW64uxXV h(pBRsiQbuiQ#kbReX) #R(UR4uebWic#RnkrRXceue#eQb#Rfx(!WQr4Ri)XkteR(U((XiekXRn#RQu4G D s D h FS sm T I G F D F D S yS I F D y P s q s F D s D h I P G V q P F F g F D S F FS s F V q D D I p P h D I F D DS y P P G P V s V I P G G s I I s q S V D P G P h D F F V q sS I s p q WDe#trqWQVI(!(iiXHQrubXR%reS V D F V q D y I D p q P h I F V q D D S sm F V I Dm F q s I V h D p s s D h FS sm T I G F D D p F q P p s F P WxrWRQW(Rix#XdrW!We#eucx64RQQdRBu)iQeRiQ#H)U#rf4(RBt4(t #kj4 G A D 9 7#kf#EkkwtF`E#ExClj# k7 BEwjx@"dtdH75 6 A 9 8 6 3 4t6f D S G P V P P w D I V h I V #rb#RQu!R4kQ#Xi4XQF D q g I 1 h D V F ## 7#2Hej)D
OHStrNis (k, ) t
g F D S yS I F T V TS I s bReXcQieXR%p s F TS h P F F s FmS V P D D T y q sS I V DS I P y T V s h D q I ufeiXX(tUXu!hWe#(#ubXQleQRRXuWi##H4F ) ( 9 8 $ DS I P y T V s h s D I s V F G s I I D p q UQR(WQ%Whb4(0QRQ4e#lQQW(ddo 05 )7%" ! '
F G H1 (-, F ) ra k
wUke g DS G P V P P w I s G V q sS I s p qS V F Re#RQu!R#HR4f(ReQu%et)D s D h FS sm T I G D D D I s S g uWXbuWX4`e#r!XA%E4(!c 4r g q D v I P T w V V D qS h F D D D D P h T !WQR#XHi4eif`!Ude#(i# I P G DS G P V P P w I s G D h q sS I s p q g C g DS G P V P P w I s G F V q sS I s p q #Rnle#RQu!R4kuliubXRBeS k((U#(XR(#cu4kQrubXR%reS V iD h D p )RBs
0 /F /H /G H /0
F q s D V I P h V h s D VS F D D T q P F P F P G G g F q PS F q D V D I s G T VSS s V F s I D S I T p D V F D I 4u4hQ4RiDQR#WDXU##q4W##q(H4q(r#44WRe4XifR4XUUe#RQbfW4UQWRXQE g F V D y I D p q P I F F V I m F q s I h D p s s D h F S s T I G D I P G D S G P V P P w I s G D h q s S I s p q g D p I D F T I QrqW(XRRWhQ4HxD64RQQVt)RBRWXbumWX#fF)UH4Rn`e#(Xu!(#Hu4`iubXRBeS `RQWXQ#G D FS P #rtbR6m D q D P h q D FSS V q F V q DS V P I s G D s V F V ##iQIRW4WDbeUQ (Plx(UQu#rhu40U4uQ`FD D I P p P G g F V D F P F I s G D s V F V F D DS G P V P P w I s #r `US(R`I#RUQWV RP 4RX`R4de#RQh0U#(XR(#`u4G Di4ReQus%eS 7TXieQ#RQ#tsbBWndqf a h q sS I p q g V TS I G P I G V D D g F q s FS I GY V D F F I G D s V F V F 4u4bX46fFxiV (P 4RPXRs4ke#RQD D V q D S p T VF D #rrWbW)( g D w h s y V qS P V g D D h q D V FS g (#x4url(eR$Rs 'hiW4WQbiD ' D P I G h D g I s h V F Rp#(X#kSi)USReWd)D D q S V h I V F I s V q F T I G D I P h g VS sm D C (PUQbSXQ)DXpbdDQ(D)X4WQRiUS ErdREg F V Dm q s I V h D p s D h FS sm D g xI6F4RQQRBs u)iQeRUfc 4r g q !XI D v F q h q V F D g DS G P V P w I s V q sS s p qS V V q D y I p q P h I F V F us#4(P)tjRe#(XuP!(#Au4G!rubXIR%reiDrWRQWD(Rix#X)D P T w D I G g DS G P P P w s G V q sS s p qS D V q D y I p q P h I F V F W#QV!UhRs(U4RXVR!R#IR4`rReQIu%UiVWlrW(XDRRWQ#XdD F I Dm q s V D p QVXW6F4uxIXdh(BsRsiQbuiQ#!EC 5D!AA97531)' D h FS sm T I G D 4 2C 8 B @ 8 6 4 2 0 ( x xWb y z v y U0WWxy x ki v y W)ixy
F atr ra ra OHStr(B, ) atr hB : OStr(B, ) ra 0 hB (F ) 0 tr B ara (h0 (F )) hB (F ) 0
1 1 1 1
% y v y y v x y y x v W#%4i(XdWi%4WdxWW$(v %Wy ~ z hUW y
g F DS q F D VS S s V D F D VS S P h F D VS V D G s D V P W)U#fQ!eU#ulif)XU!UeRWtQQinuiQd!(ih q PS F h q g I D V h q Pm D C g F DS q F D VS S I s G D s V F V D D V v G P h P h D q q DS T s V F (UX4ei4eS 7f#WQ4(6cEue#4)XUeeR4fU4uQkicXWU4(ii(R##UeWnufD DS I P y T V s h s C g q PS V s V P q s F q s eX(RX%huE(UQ%QR#`eu# D hS q g D D h q D F g D S G D S h D g w hS p D q FS DS y P P G P V s I P G s D h FS sm F D S F F s V q P ib4eS fcW4WX4Rs 4(be#ii)uxeW4btfU(ReRnRQtb#Rnu)iQeRQ4RrRXF D FVD P F D xiV ()EC u y v y Wi r dix W z ~ y v v iAWUnle%w%c x v ~v v v %% 4WWy )nxu !u#| ~ y Q e Ru QWb HiRpx! i`neWUn r uiy z v z ~ v y x v yy v y v x x y y xx %ncdW'4W( %x y y y i 6# 0x y y z v y y v y z v Rx xWb kiU%W'iWR0iUf4uR Uz B~ v ( !neWunt r uWy z y x v v v %w %f f6 x v W 0uQ4W!W xixQwA%wpx w4 tn4uR cWxbtlRR i yy v x v y yz nx v euwu g D V q s pS F q PS VS q T s F q P FS s (Xru%e#(UQ#iAbc4RXeR4
OStr(B, ) OStr(B, ) AfndStrra (B, ) OHStr(B, ) OStr(B, ) =Z OStr(B) OHStr(B) B k
wU g TIV q P T VF (XQ((!WlD D D D C !!WetErg V iD D D I V q P DS h sm h s h q g D IS s V q D T T DS q D h q s q P G F D I I P h D P h #rtXQ((!tUei(#WeRWtRQURQrW!WeWlU#W4u44Rn)XQRWl!!RiAp D D w G sI y D TF P GF q sIV s `##RQ(p#dWX(nX4uxX0b URR#rbX#W(dQRRXF VS P g D S I T q T y T I y D q P F q P V P g D s I s G T F qS V q D D s V P V V D mS V h D VI F S q D FS w G I P 4RQul(U4uxu44eHrW!WeuxuXfWUX)4##4`!b##QR! q VF 4dD D p s q o #r dEg F q s D FS w G I P P F q VS q 4R#i!b##QR!RXbS #`##4US D D I V q P q P I D qS I D V I P #r0QXr(()W4U!QWXl#RG
(f1 , . . . , fm ) (x) (t1 , . . . , tn )
g D F FS D P q (Qee`(#A%s
q VF #tD
.
D DIV q P S h D #rhQXr(7Wng
q PS V s D I s V q D DS h sm D S I T p q o ReXueXbtrW!WeUWRd4UXRdng VF D D V q DS p #rcrUwg
^ ^ [(X k l)Bn , Bn ]red l l
2 " $ " 2 w#%#w' %
2 1 $ 2 $ 9 " $ 2 " $ " 9 " " & $ 2 9 2 1 3 rw%rwnR$j#Br#%W!(#( '#r%eEr'nnn d
r
s q o d4g
D#rDXV I D p P I V V D G q o g T I Dm V qS P G q R#(XQHWndR!QW6HrU(# VS P UR g D s I s G T F qS V q D D s V P V V D mS V h D VI F S q D FS w G I P q Re#uxuWX4UHrW!URQuQHiQ)#X4#4`!b##QR!# q PS V h q Pm s DS P p q D S D FS w G I P D ReXi4R6delU(RWn#rd!Xe#4X(`e h D p )RBs I #XF
: Bn l / Bn l ti
^ x X k l
(f1 , . . . , fm ) (x) = (t1 , . . . , tn )
n - dim(X)
yY
(g1 , . . . , gm ) (y) (t1 , . . . , tn )
tp i
x = (y)
X
Y
pr
B
OStr(B, ) OStr(B, ) AfndStrra (B, )
y y ii4
x y y y | v ~ } | x y { z x y v y v ~ ~ x u y yz Wi#6%4Q!Ri(iR0nx i4ii#uWB4R))Wxbt0RR
| #x4x dt4euW& xWA% 4WR %tWne XfQnH % v v y x z { z y x z ~ Rpnizix r xuWyndxi v y v x y v xy Uz ) r z x y nc Xu z x eze404xiyb W { y !Uz fi y e eb y ! 7f6c#" ff i nx
B k
%
uB
v e%wu
y q sI D VF #uxhlD
OHStr(B, )
D V q DS p 4(lrUE
k
OHStr(B, )
X
A1
Q
1 hB 0
Ztr (X)
OStr(B, ) OHStr(B, )
OStr(B, )
homOStr(B,) (F, -)
pu
/ OHStr(B, )
^ AfndCor(B k l) Z Z[1/p] = pu idY = pu idX k l ^
SmAfnd/k dim(X) > 0 B1
OHStr(B, )
B1
B1
B1
OStr(B, )
k
ra
g
I P #RnG
sS hS G S F F D D G P h D D FS w G I P q g F I DS s I s G D F P G FS q o ubibe#!UXHie#!RWi!b##QR!# 7fQ#Ueeutu`X(nXedng h D p )RBs V q D D D I V s q D S V q D g F D D G P h D g F V q D S p T F D FS w G I P F D F D h q D h F D I T q T y T V D F D h sm I P G V q s s V jrUeWQ#Q%4f4UXrW#eS WHie#!RWH(Wx(br!b##QR!)UW4WWX)X#W(Wfi)i(#RnrurBkiD
CorB0 (B, X)n = CorB0 Spec A{t0 , . . . , tn } (1 - n ti ) i=0 , X0
I s u4G T I y D q D T q q P sS hS G S F D D G P h D X(WWh#4REReWee#!UXie#!Riie z ~ y z ~ v Uz WtwteirQUwtuQ x Wi { x v y v ~ v z z ~ v y %v fWHi%eB4tuQ AAei tuxc
U V
x y 4WfiAWib
x x !e%we v y i)Wxy x hW
/U U
y | v Axb#~ ! u x ii 0W Uz %ixW&v ~ z ~ x y v ~ | 4Q4x u'eB4l%x v y v ~ v x v { y v v WWnuix E i%#~ u %i y # | x v #xxfwxn%R4 c
0
% v u x x y z z inlexiXenl%x x W%#b% % z ~ y z ~ v { y v ~ y v f %v )4 0Rkf R x v n4uRl c y v ~ v i%#uH ~} 4Q!R | i4QWy x uv k6 g T S h P F F s D I V h D G F D I V h D G F D s S ueiXX(hQXinWXQnXiUbrp D D s q P h V D D qS D G DS I P y T V s F P #heu4uWHW#eWe#leX(RX%h 4(F D q #4D S V q D g D S q D G V D F D s V T F D I v y #t#40)URQWiQ#Reus F PF F D a q 4RX0)RP S F h q g I s G T q q P I P D D F q g D T F P G G g C 4UXrWbS WF eX4UW4eS 0R46Y ##(Q(e#0WX4WD 'ppr#(P kfg D I v y X4WReus s F q s F I D s p V qS P G q g FS q F D q s F D s V T F s T w h bj4u4jQ#UR%krU(A# 7e#4dj)#A%f)URQi(AW#xQF F D g gS rU(D u F q s 4u# D F D q s F D s V T F D y s qS FS P f4!ufeuxi)RuUXe(Rp F)iUQR(WQ%WAb D DS I P y T V s h s V iD D V P q q o g F D F FS V D S q D G V D F D I v y Qu#d7W)QeelW##H Q#W(URs F P F F D F F D D V q s I V D VS S P 4(dXl#dXruxXU4Q!eU(ih j w iQnRG #!WQRW4XeiR7RQWeU#(WQAX4WReus h F D G P D D v I P T w V D I s G g D I vS y T I D I v y DF q D V X4W!iD D q VF #4d)D D q VF 4#dD D D #rcXV I DS y T I q DS I T w V D P q s D q q F P F q T q q P D F P G G F q o g D F FS D P q WeU#(WQWeX#XWr4R##Rs 4(4##(X(n##Xd(QbUAR4!us q # D q P D VS s F q s # ((XU#XF bi4Rd
A X0 A0 A
g D IS s F F D h T q q PS V s S P I G G g V s V F T I D I D h q P q T IS P p P G D V q s p s F D IS s qS S T I (XeuQQi)#(UQ%AeBQ##Rs EHuQU#XQ`eHWW4R4WfU(R#(#fru%BH)Xeu4e!UeWQ#G F D D D qS P F D F q P I s F P g c q q D v I P T w V D D h D F D S q w h D V F D R F D D FSS V D p D I G s V q P q PS V s S P I )#re#rdeRXnj4(X4uc4RlrF gr g !WQRW4XA`UeWWd#rc#rb##x)Xj)!W)k)UdbeUQtR4WQ#ibkrR(UQ%AeBQ#G G g V s V F T q I F D F P G D I r g Rs Ed%x#XQI 4l#XdX(nWQAWe4r g q #dR#X4AnEu)iQeRiQ#ii0e#eu`iWRQW(Rix#Xbdi!WQR#X WQ((ih D D p D I G s C g s D h F S s m T I G D h D D S s m D h q D y I D p q P h I F s V F D D v I P T w V q D V q P D VS sm D #rUupEC D C ga Eu(r g q g q ReXUr(X4G biwW!U(RU(!4Rn!e!A)0rubQu%US q PS VS F P G P I s V D D q D IS P pY IS P T I P G FS V F D V q sS I s p q VF D x y u uB X % ~v v v nxu!u#| ~ r enW74W( %x 7iAWUi 4 i6# cx v x y z y y x xib lni y ~ x y vz v v { uQU%A%4| E k6p7kEtA x xWb ' dUHxy Uz 4k #QA(Wxbtz y ~ z v ~ } y y | i4QWy x y yz v iixbtR n x v 4W%7u g F V q sS I s p q QrubQu%US V D F V q D y I D p q P h I F V q D D S s m F V I D m F q s I WfQrRQWR(ix#lrU4URcQXW64uxXV h D p s D h FS sm T I G D F D G D g G P h s D VS q I Pm S V q s pS F D v I P T w V D D I V q P T q P q PS V h D F P F D V V D h F q s (Bs RiQbuiQ##de#!WWD 7 4Riun0U#Q#R6k#r0ru%e#X(W#QAeXQrR!W ((UQ)F RXXXiWd4Rd g eH#X(UQ#4RWh#4RWXbH#QXrRWE#riih F D I F q PS VS q P h D q P h D F s VS D I V q P h S D D D S G 4(i4(be#!US D D h q D V FS g C g a DS q D h q s q P G F D I I P h D q V q D D D F F s G V DY D FS w G I P q V q D T h I Pm V F iWXbWD k4BU44Hiu#4(XQQRi#4frU4WXc(4ti!Xe4#X(!#HrW!ixR6f)D D IS P p D DS h sm V F D 4(HeR(`#heeiRf7 gia D I DS I T V q g F q s D q D V q P h V F h#UQWQ(eS 'l4u#0#WXr(i`D D D y s g D FS G dRRAUS '0#rXb##Y I s G D FS I P V h sm D R4iXeQRQRiF D I DF P G G F V D G q P #rlr4#(4 d4xQ#G D D wh PI F G D IS s s h q I s U#dXeubuXF #hu4G I D h s G D I D V VS T Wi(e4X!XU#d7g F q s 4R# h D p (Bs D # q PS F q D V D D q D V FS ReX4WQrW04#0QbiD g F q PF P G G F VD D q US Er4RX(n##X)iwf g #r 4uxRU#0(!WXieQX(RBxAe#RnA##W4uWXWT #q D q s I y F G V q D D V hS I V F q P s I D D P D q h s q D V g F D FI P WxRe
Hn Sing (Ztr (X )) SmAfnd/D Ztr (X)(Bn ) k : Bn (o, ) k 1 fi (Bn (o, ), O) k Hn SingB (Ztr (X an ))
V q DS P rWeR4g D w G I P P FS V I D p P D y s qS FS P p q D v F F P h##QR!RXbcXW(#Rt(u4eXe(R#`XXrG rU(heRXXbefD V qS P G D D F FS V F I D h s G D I D V VS WWRb#!WQ!dXX4('Y q PS F q D S D D D q q P h D V q s F P G P h D q F q s Re4!Ui#hW##RWXruQ(n!Rii4#H4R#
x n>0
@ @ 9 8 53 " 2 " 1 $ & & $ " BA7'764#''%0)('%#!
f1 , . . . , fn (Bn , O) = k{t1 , . . . , tn } k |fi | < 1 (i) (i) fi = =(1 ,...,n )Nn a t (1 ++n |a |)Nn /X : Bn
g
Bn k
D T F P G P h s s y r!RW!e!7urWT V PV I P G D T qI P q P #RXf4Rn`#QRn(#q
D FS w G I P D F I P s F q P I T S F q P !Xe#4X(ecQ(URf4RQWbX4R h D p )RBs D4(leXV F D D P g a D s I s G T F DS q q PS F q D V D D q I s !!R4i)e#uxu4e#4ReWXW`4#fu4G D q q PS V s I V qS P G q V D W4#ReXuQtreRnp4ti!Rs D I DF P G G #rtr#4F
X x X
U m B
CorB0 (V (B/B0 ), X0 )m =
X = X 0 A0 A
n 0 B
CorB0 (B, X)
B1
a
1
a : n m
B = Spm(A) A A0 A0 B B0
an
Colim m
CorB0 (V (B/B0 ), X0 )n =
(B/B0 )
nN
V (B/B0 )
D
CorB0 (V (B/B0 ), X0 )
k
B1
CorB0 (U, X0 )
/X
B1
k
k
n V n (B/B0 ) = V (n rig,B /B0 ) CorB0 (V (B/B0 ), X0 )
Bn k
B0
(f1 ,...,fn )
Spec(A{t0 , . . . , tn }/(1 - Bn k >1
a U V n (B/B0 )
/
X
Colim
/ Bn X . k
U V n (B/B0 )
A0
/ Cor (B, X0 ) B0 Colim D n |k | CorB0 (U, X0 ). B0 = Spec(A0 ) V m (B/B0 ) V n (B/B0 ) U m n 0 B B CorB0 (U , X0 ) = CorB0 (V (B/B0 ), X0 )n . fi >1 1
0
n i=0 ti ))
V (B/B0 )
/U
A
A0
U V n (B/B0 ) CorB (B, X an ) nN A0 A Bn k k
g V q s pS F D v I P T w V D s q )ru%U4!WQR#X`e0do F D FS w G I P F D h D p s D w h s y q PS VS F P G P h I s G q D V !Xe4#X(!7eRBj#xu(f(UQX(n!Riu4hWQRP a S (#S g q Y
SFQF4usHRXRt F q PV P f
A
/U Spec(A0 ) Spec(A(n) ) = (U Spec(E0 )) CorB0 (V (B/B0 ), X0 )n CorB0 (V (B/B0 ), X0 ) CorB0 (B, X0 ) = ( )an u Br (n) (f , ) = A ^ ^ B1 (n) (f1 , )A(n) . . . A(n) B1 (n) (fr , ) X an CorB (Br (n) (f , ), X an ) = A A A (f1 , . . . , fr ) CorB0 (B, X0 ) CorB (B, X an ) (f1 , . . . , fr )
I DF P G F V D G P r4G#q( W#AuQQU#dg I D q s I D V VS T sS hS G S F D D G P D h q V q s F FS q T ubibU#!eie#!Rih 0eiWh #drRXQ e44i % h D RpBs VS P g T VS pS V h D VI s I P G D D I vS q s D R URf(XUreXib#XF b4Rn#rXU#RA!W) ep#iQ#do )e 'WxWIel s D D v h P I G q ga u(r g q g q ReXUr(X4!e q PS VS F P G P I G s IeR(6Y S P p D D G P h DS y P P P g s D h FS sm T I G F D DS G P V P P w I s G D h q sS I s p q g D V q s q D V qS s D P h T ie#!RWdeR(U(!R#w 7p#uWXbuWX4l)0e#RQu!R4luii4ReQu%eS dhru4XrURAde#RW V V F T I C %sx4)X0DE7g D s y T V UR(W`F)D F q s 4R# D D y s g D F I P 0(uAUS d#rlxReus F G 4U4dDdg D G P D F q s D h h V F D S iDe#!RihUt4u#0UWW`#r F q s 4u4 D S q D q s q GF D I Ph D V g cDU#ih4R#4RPnXQIX(if#q#FD dug F q P I T S F q P h s D h I P G RQWbRiB#bWW4Rg k#R(U(!R#)D D y P P P w V F F q 4Rs# D y s g IS P D V D I S I D q I D #D(uAeS D#r!eR(p QF)Xte#eSQWQV!I#RG g P sS hS G S F D D G P h D F q s D h q ubibe#!UXpWU#!(iU!u#eiih # q PS V I V F q P h I G V ReXih4XQRiHRs4tF)D F s 4qR# DhDXFXF(seWg h s C V q s q D V qS s F q P V P (R4WQrUR!fRXRt g F s T wh (!#xQF D D FS w G I P !b##QR! q g D T q q P s V s pS q PS V h D F D V V D # 7##(b#u%U4()T ReXiXQifg D s V T D FS w G I P URQiXb##QR! uReX)p##ARAiD s q PS V h D F D q h q P V
(n)
mS V s V P h D y q sS I V q F I P s VS T q P Q%Q#d!RW`eR#ReQXp#cxReulU#)p(#
D!RQ(ub04UXWrW!!uQA#F s I y sS q VS V D G V q D s F g D s V T D y s qS FS P p q V q D D D I V s q V F euQWiRR4ebR(#hrW!WeeWQ#Xu4`D
mS V s V P UXuQ0(ih I P G D V FS D S r g g D IS s P I P h D I s G #RnkQbie)uE (U(tQUReeRQRWeRg F q PV P q V D D q 4(XR#%Wndf F s T wh RAW#xQF D D FS w G I P q D V FS D g g gS #h!Xe4#X(!p4`QbiEeS Er4U(D F q s u# D FI P xRe VS V D G V q D s F )QindrW!!uQF #(#r`4RQXrRwg I P G D F q PIV q P I F D s V T V F #Xeuxih)D
n rig,B U n B E = E0 A(n) A Spec(A0 )
(n)
0
D S F F P G V F D S h D g e#UXQ(n0kWka
> 0
D P q s D I v y (#A%AQ#W(URs
/E 0 / A(n)
q DS F T V I s h T I I s h q F q P I P We)XXu`QXRW#c4(QR4 g F q s DF q D VF 4u4i4t)D s F q s D FS I G V q s V T DS q qS D I P q s ed4u4bQ#0ruxiU#UQR4eY
si A(n) A0 [s1 , . . . , sr ]
(n)
s D q q g D FS u)##us #rXb##G D F D #reXV
(n) A0
F P F FS S P 4q(QbFbR#wxh I P #(G g D T F P G P s I s r!RWh b!u4G F D D y s g )RuAeS F q PV P RXRt g D D I v y `X#Reus I 4RPnG D DI P h q D s q o #rdQRW4Wddg DF P G q o X(nd4g F P D 4RXFt#q#V(qD!WQUSRXQWh#qFRsfV)D gEq DI sFF D T G F D F q PV P #r4(XulRg F D I v y )X4WReus D D s V T D FS w G I P q g V HURQic!b##QR!# 7kiD FQ#Reus D I v y D#D!XeS4w#GXI(P!q# gEkU4#d F DS q D F FS D I v y QeeX#Reus D q D V FS D S F P F F D S h s g ##jXbWReWRXX) ikBe4r g q !UjeEujG D D D I s g I4RnG P D#rr4#G ( I DF P G G F V D q P I D q s I D V VS #A%xQU#dT g DS q D h q s q P G F D I I P h D q V e#4!W4u#Rn)XQRW##!iD S q D G V D V D D F FS D I v y #4WXXbUX#Reus q PS V s g D V q s q P G F D I I P (UQ%4rT Q(R#4(XQX(ih D F D D 4(eeWXV F PF D q I D p PIV V D q P 4(4#hWR4RQXiG (E n
/ Spec(A(n) )
h F D G P D D v I P T w V D I s G g w iQnRG A!WQR#X!eiR7a
D h sm s i(b h D p s RB'Y I P #RnG g gS eRD 'f sS hS G S ReWee#!eF % VS P x y UR )eixW D D G P h D F q s D h h q V q s F FS q T DS q D h q s q P G F D I I P h D q ie#!RWdUl4u4iUWW#lruQQe#ide#4`W4u#Rn)XQRWd## g DS y P P P g (U(ReR!R#w 7 F D G P I y F D I F F q PS V h D V qS F D V D F q PS V h D VI F F D VS qS a S Y D D I V q P q o g F DS V I s G D q D D T FS pS V F D D p D I G s n#(X()Uc#Xf4ReXi$ek)Hi4ReXib#c#HU#UkuXS g q #r`XQ((!deXXuWW`WXeecdR#X#dC
A(n) A(n) = A{t0 , . . . , tn }/(1 -
n i=0 ti )
gS q D G V D F D F FS F D I v y #4hnrh#HXXbUc)X#Reus F4(PFHF)DFhDXVrqusxIXU4`XU!UeRW!e0`eurWHhUeW jU wjU iQnRG hX(W#Q`UfujFRQWeU4RWQH)D D V D VS S P h s D s y T V F D D D h F D G P D D v I P T w V D I s G g D I vS y T I V F DXI4WvReus y s Cg V WD DV P q q XR#do 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( f67cfElA y | v v y x x y a S Ab#~ !uWb xy i k RXS g q Y irQUwtuQRAAb4~ A y z ~ v y | v
A0
(n)
( E
2 " $ " 2 w#%#w' %
2 1 $ 2 $ 9 " $ 2 " $ " 9 " " & $ 2 9 2 1 3 rw%rwnR$j#Br#%W!(#( '#r%eEr'nnn d
CorB0 (Spec(A(n) ), X0 ) CorB0 (B, X0 ) dn,i = 0 0 i n
ti = 0 E0 A(n)
A0 CorB0 (Spec(E0 ), X0 ) = (Spec(E0 /ti ) Spec(E0 )) = 0
dn,i : Spec(A(n-1) )
= A0 [t0 , . . . , tn ]/(1 -
B1
n i=0 ti )
A0
(n)
A0
(n)
(n)
(Spec(A(n) ) Spec(E0 )) 0in
fi
0in
A
fi A(n)
(n)
|fi - fi | (n) A0
= (Spec(E0 ) Spec(E0 ))
A0
(n)
A0 [s1 , . . . , sr ]
(n)
E0
E0
E0
/E 0
k
V n (B/B0 )
^ ^ B1 (n) (f1 , )A(n) . . . A(n) B1 (n) (fr , ) A A
(f1 ,...,fr )
Spm(A(n) )
Spm(A(n) )
(n)
(n) Spec(A0 )
U
U A(n) A(n)
(f1 ,...,fr )
0
(f1 ,...,fr )
Spec(A0 )
CorB0 (V (B/B0 ), X0 )
u (n)
/ (Spec(A(n) [s1 , . . . , sr ]))an ,
/ Spec(A(n) [s1 , . . . , sr ]). 0
/ Spec(A(n) )
2 Spec(E0 )an
/ Spec(E ) 0
/ (Spec(A(n) [s1 , . . . , sr ]))an .
Spec(A(n) )
(Spec(E0 /ti ) Spec(E0 )) = 0
/1 (Spec(E ))an
|k |
U
E0
CorB0 (U, X0 ) U E0 /U A
(n)
SingB (Ztr (X an ))
1
E0
/ A(n)
U
DV q I P WQp#H#RG
DQ4l4#jxReurWeX#RAdu)b(F VS F D q F I P s V q DS V P q o g s T S q P DV P q q P V Qu#A(WD y q s I D D I S V F #ux`dQ#eUf)D F D pS V s D I F D DS V q D I T S F D D P (UQ%bWQk)UeWeXrXn f`U4
A0 I =p A0 /A G
-1
F D P )U##! D DVI Ph DVh s XX#(iXiR#iD qReXuQrWXQ#l4#t##(lXdg PS V s V q D F T I G D q D q q P D F q o D UHXPFRsWhuscVrqDDUeSih(mtD#v!uxlXdo 5D!AA97531)' s q sI DF q 4 2C 8 B @ 8 6 4 2 0 ( % y | v )Uz W!!b4~ Ax x
e E p : A0 [t1 , . . . , tn ] (n+1) Spec(A0 )
z
X 0uQ!A0 % W x v x ~ } y yz f #QdR!WQbi )( z v y X x wiy Uz b(% y | v (W#WRi #6Ww x n4 %Wy x y x v wx wx v WRw )( z ~ } y yz b4k 4Q'RpWxbt )( z x
E A0 E A E k A0
{ x v z Rx n4 Ei 0%xURr dn%RWB4R)f x v y v ~ ~ xy
k6pp7
g S hS D p D I G s F q P FS P I G D I F P q I D V h D T VS P P h s I P G g D T IS V V F D V q s pS F D D D ibhR#X#bc4RXe4X#XfR##)#X)UpiQ!!RWbH#(U j AXeXHHru%e#i!!WehEC
dn+1,0 =
wU D h q P V D T VS pS V h D V q g D p P I G S h D eR!ipXUreXi$eS '(#RQ#ci)fg D F I P #rdxReus I s G RAg V q D T q VS q T (!WeWT ##i F s T wh RA#xXF D D FS w G I P q D V FS D S VS V D G V q D s F #!b##QR!0#kQbiUjUXW(!RXXF
/ Spec(A(n+1) [s1 , . . . , sr ]). 0 CorB0 (V (B/B0 ), X0 )n+1 Spec(A0 )
(n)
g D vI P T w R!WQR#XV I P 4RnG q PS V h I V F q P ReXXQ4(ih WXV #h#RAg D q I P G I P G D VS s F q P D p D I G s D DS V I s G D I vS D I G s I s #Rn4(RX0R)RR4WQ#lbk#ceXXu4kXU!WQ#deujG
Spec(A(n) ) /U dn+1,i = 0
q PS V h I V F q P h I s G s q o (UQ4QX4(itu0dng D q V q D D IS s F F D h T q F s G V F g #4`rW!XeuQX)i)#hR4D Eq D F q PV P #r4(XRtEg V WD F D I v y Q#W(URs
E0
I P #RnG
D I DF P G G F V D G q P D y sV q s p g #rHWX(n##XdpRR(ux(R%Bs 7
E0 A(n)
I D q s I D V VS T #A%xiXU#dg
D D eUXV
CorB0 (B, X0 )n+1
I P #RnG
DS q D h q s q P G F D I I P h D q D V FS D g D F P G G F q o e#4di4R#4(XQX(id4#XeWeS EriX(n##Xdng
I P #(G
@ @ 9 8 53 " 2 " 1 $ & & $ " BA7'764#''%0)('%#!
(U/ti U ) = 0
dn+1,i = 0
1in+1
g
dn+1,0 = (Spec(A(n) ) U )
D FS w G I P D D FS I P V h sm S !b##QR!pUbQRXiRc#r V q DS V q P rWeXrRWh D D D #reeWXV
D FS w G I P q g F I P s D F P G FS q o !Xe#4X(# 70xReuX(nbdg F q s 4R# D I DF P G G F V D q P #rr#4#WnG (f I D q s D V VS W#A%xI XX4d T g V D S q D G V D D F FS D I v y Wd#4 Qee X4WReus F PF D q 4RX##
A0
(n+1)
E0 A(n+1) (Spec(A(n+1) ) Spec(E0 )) (U, O) A(n)
Spec(A(n) )
(Spec(E0 /ti ) Spec(E0 )) = 0
/U
0in
E0
1in+1
g
(Spec(E0 /t0 ) Spec(E0 )) = (Spec(E0 /t0 ) U )
D D FS w G I P q g DS q i!Xe4#X(!# 7de##d
D F FS D I v y XXbe`Q#Reus
a Y I4RnG P a s 'Y DF P G q o rpdng D D I v y F P dQ#W(URs 4(F F)X4WReus D I v y D D s V T D FS w G I P AURQi!!Xe#4X( # 7 q g D q D V FS D S s g ##iQbiU(Be4r g q !!WedetujG D D D I s
(Spec(E0 /ti ) Spec(E0 )) = 0 A
(n+1)
E0
E0
A0
(n+1)
= (Spec(E0 ) Spec(E0 ))
A0
(n+1)
1in+1
g
(Spec(E0 /t0 ) Spec(E0 )) = (Spec(E0 /t0 ) U )
V q s V T DS q qS D I P q s ruxihe#4el!QR#0eY
h D p (Bs
q DS F T V I s h T I I s h q D I Pm q o g We)XXu`QXRW#h!X(6pdna VS FS P w h q P UbR#xR I P G 4R!g
>0
D P q s D I v y (#A%`X#Reus s F q s D FS I bf4u#XeQ#G F q s 4R# D D y s g hRuAeS F q PV P RXRt
fi k si
aEY g ag s #'Y
D D y s g RR!eS ' V q DS P rWeR
a n6Y arsY D #r D #r
r
A(n+1) A(n+1)
0
(U Spec(E0 )) 1 i n+1
U
/ I/I 2
A0 [s1 , . . . , sr ] g
(f1 ,...,fr )
[s1 , . . . , sr ]
/ n Edti i=1 E O E //E
e
/A G
4 EO
f i A0
E O
/E 0
/ Spec(E ) 0
CorB0 (Spec(E0 ), X0 ) E0 /t0 A(n+1) /t0 /U Spec(E0 /t0 )
/ E/A 0
(n+1)
(0)
|fi - fi |
/0. E0 E/A0 E0 / A(n+1) A =
g # gr q (UQ)!bfQRnubWH4e#cQuQU#XQHehUeWeXrWXQ`XU#RA## 7ibeUQHUXu!f4euxih q PS V h D F s D F T I P s T F G F V s V F T I F D D DS V q D F F D D I vS q s D q g D FSS V FmS V P F qS s V I D D s h P RWe F D v P F D q PS V GS I h F T s D V q D s pS T D I vS q s D q g P qS F s F D h h D F D D G P h D F D I D eW!(UQeXWX)bRQ(UR%U#r)QWe#uAp## 7(jEeUX4 e!eiiie#!RW!!XXV q D F D S yS I FmS V P F qS s V I D h D I V q D F D FS w G I P P P g F D G P I y F D q PS V GS I h F D s FS Pm D V P Wd#e(UQ`Qu!lURQXWi!QXrWd!Xe#4X((!R#w 7`n#RQR`# ReXeQiQpewe(6iQR7g q g q (UQXped)WnReW(W q PS V h D F s F D T G P D p T Fx64RQQ)RBs RiQbuiQ#h)dih)U#HW4u44Rn)XQRWh)Xeux(!WeW`QieX#(X4hA#r 4WnW#!4ii!UeW!QR6m V I Dm F q s I V h D p s D h FS sm T I G F D V D F DS q F D h q s q P G F D I I P h F D F D IS s V q D T T F T V TS I G P I G F D D q D G T D q V D D D I P V F D F DS I P y T V s h F D h D q PS V h I V F q P h s C g V q D D pS V h D G F D ))UQR(WQ%WdWdReXi4XQRi4)(!WReXin)XI V WD F D T V P q V q P I D F F D g F P p D P D )WXR#trRQWXl)Uef (#(Wji V iD F D V S FmS V P F D F DS I P y T V s h F D F D S V s q s F D S yS I F D y P s q s F D VS q T q P q PS V h D F D V V D h F q s )X!UXR))UQR(WXuW#)#reQrUR4ubReQE4RReuuej#iRBEReXiXHXWi4Rd
8 8 8 9 4 (4 2 9 & $ &4 2 F 9 9C4 $ 0 @ 9 )6 PDIG)'#753HG)6 ED'%BA)6
(ti ) B
wU g T p P I G V F D D D D C (R#(X#H!!WeE#g D VS T q o 4(`##dng h D p s I D S h q P h VS P )RBHW#eW4'(ilUR F q s q PS V P F D V V D 4R#p(UQeRXiXWih D D y s g C `RR!eS fg D v V F F q PS V P F D q F I D !WQX(UQeRX`#4kxW(p F q s D yI D p q P 4u#t(XRRWh D T S G G s q P V D `W#rbU4#upRXf t D D V q D I I h T I D VS s a g g D D D p D I G s D DS V I s G D I vS D I G s I s g XrWQX4iQpXU#F bXE (U(!!We (#WQ#bppeXXuQWeX4bRjG ia I F D FS I G V q s V T DS q qS D I P s #XXeQ#!(RQWpe#4#UQR4q b6Y F)XUXWc(!WQURQQ#Qus D VS V D G V q D D IS s I VS I F D I P q F D D I q D I V D G q P F VS V D G V q D s F )QR4fedQ4XtpR4WxQincrW!!uQXF V iD #rH(R4WQ#f D V q s q DI G q g a V q D D pS V h D G F D (!WReXin)XI V iD I F F D FS I G V q s V #XfXeQ#truxiT F DS q qS F D I P q F D e#4ej!QR#e6Y V WD D F D 4(cbWXV V iD I D p PIV V D G q W(#RQXcWnARP I P G g #RFa F q s D FS I 4R#lXeQ#G V q s V T DS q qS D I P q s ruxiie#4#U!QR#!b6Y D I D F P G G F V D G q P FS FS P w h q DS F D GS V F D I s 4(dWX(n##XdWnReXbR#xWe#le#UQ4dH)tu4G F D I D h s G D I D V VS eHWWRb#!WQ!iXU#dT g D D D s I D p P I G S h D (!UheixWR4RQ#Ei)ng s q P R #`#xQ#trW!!uQF D D wh PI G V q D sF I P G D I D I V q P F q P s F P #Rnh#rtWQXrR!c4Reeut4(t
A1 B (a1 , . . . , an ) B1 >0 B1 -1 ti B : A0 [t1 , . . . , tn ]/(P1 , . . . , Pn ) B (ti )1in >1 a i A 1 (ti ) A |ai | 1 / A1 . |Pi (a1 , . . . , an )| (Pi (t1 , . . . , tn ) = 0)1in
F D I v y )X4WReus D D FS w G I P q g D F P G FS q P D F v w V P G w I s G !Xe#4X(# 7X(nbRRX4XunHuj1g #X`e#exW(UtD I F D S F I D p qS V F D D #riXV h D p RBs D I DF P G G F V D G q #rWX(n##XWndo 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( y (u uB %% (lAxb|#~ v!!7W y x y Uz b(X#i v y z yi y y (u W y | v x y z { x Rd!b#~ A!cb(Q4e fi u k6p7 y v y v ~ ~ x #%W%#()uwWy W#|xWyt(hUz iAb4~ Ax y | v v u y x y y y v v y z e%7nWxxiR Wd#u' untiUdR !c g D S F I D p qS VS sm q D V F D a s Re#UXxWRetRpWtcBrS g q #rXQ((!RXX) ii!!WehER!Xe#4X((bS XR4r#tD Y D D I V q P F P F F D S h D D D C g D FS w G I P P F S F s q V F
F D D G P h D D FS w G I P D D I D I V q P D h q P V F ie#!RWi!b##QR!hU4(HXQ((!h4Rl!X
q F q P S g D F q D V F #h4(@RX4i)D
F D FS w G I P P F S F s F D s q P g !Xe4#X(!RXeS XR(H)AR#S U gr q !X(W#QdeHujG D v I P T w V D I s F I P V P V s I q D I G q P D VS F s F q s Q#($RQix4X4(#RQ#X!et4u`4g V iD F q PF P G V 4RX(ntiD I P #RG D I v y F P F s C X4WReus 4RXg V iD F q PV P q 4(XR#7 g D S F I D p qS V F Re#exW(UtD
q #
D FS w G I P D D I D p P I G D VS y g F g s T g V q s V q !Xe#4X(eh#rHWR4RQ#ht(us Wu)bS ErlruxD D FI DV sI T q T y F D cQ#XuQ#W(H V D D F FS D P q WhQeehR#A%s V q DS P g g rWeRkS e4r g q !X(W#QWe4r g q X(W#QepXe#)(!(i!(RQQrR!D ReXiXF 4RXF ep#eQWQV do D v I P T w V r g D v I P T w V D D IS T V q D P h V q s I V q P q q PS V h D F P s D qS I D q
f0 , f1 , . . . , fn A0 [t1 , . . . , tr ] (U, O) / A0 [t1 , . . . , tn ] //E U
wU
g T I V q P T V F D D D D C g D s V T V F (XQ((!Wt!!WeE#ReuxitD
D T P P s WXF(nG(ih!b`D4(teSubihfxIReuslD7#g I s F P V F DDXRs4D#q4t)D F VF D I DF P G G F V D G q P #rtr#4lpR4 FDT#q#q(PQI(PPihdF)DDQURT#ee IS s qS q SV Pm q s V q RPeXusAXI(6F4RQIQD#4DXeSumDQUS#dTYg I s VV F q s F V q DS h D P V q P 4u4dQrUWAWih ((F F D D F q P G G s )U#rd4(U#4utI g D DF s D q D I P XR4h#4` !QR6m F D I DV F DIV g !QWXc)XQus 7 D F FS D I v y XQeeX4WReus s I s G D FS I P V h sm D bubX(XiRXF D hS I V s ibQX%A bp#(uAeS fUXWArW!!uQXF s D D y s g VS V D G V q D s F I P #RG q g D S F I D p qS V F f (U4UXQReD D I DF P G G F h q P V D q o 4(WX(n##X4R#pG dtg D FS w G I P D !Xe#4X(ej WQ44RYRe#exW(reA)D D V q I P G g D S F I D p qS V F
1 j r. M
F q s 4u#
s b6Y D I P !X(6m
D D D #reUXV
D D D V D hS I V s s 4(ieeWX`W eQXu!b VS P UR DFrGpqdngia (#A%hX4WReus P o D P q s D I v y s F q s D FS I G V q s V T DS q qS D I P bf4R#bX4t(RQWU#Uh!QR#q Dh IV ibSQX%sAiD4#tSXFbR#xR4 q V S P wh q P D TI q I P G WWQ#t4RUg D DF s D q iXRi## D hS I V s D q D V FS D S D I S V F ibQX%A`##`XeWneRQ#eUf)D D P !!R
E/A0 E/A0 k >0
w
1 $ & & $ " n%!r%#0)4
RigDM(k)
(fij )1in; 1jr E nr (j = ( nd (fij )1in; 1jr A0 n A j = i=1 fij dti
D D D #rdeUXV
g
M Matrr (E)
I P #RnG
$ 1 & " " %)R k e
D
(ti )1in
E
/A
1
Colim
|k |, >1
((j ))1jr j
B = DU (f0 |f1 , . . . , fn ) A 0 = A 0 B0 = Spec(A0 )
Hn SingB (Ztr (X an ))(DU (f0 |f1 , . . . , fn ))
1
A0
((dtj ))1jr
M (j ) = (j )
M Matrr (A) E[det(M )-1 ] E/A0 A0 E/A0
A = (B , O)
/ Hn SingB1 (Ztr (X an ))(DU (f0 |f1 , . . . , fn ))
A A1 1 < 0
g
Jac(P1 , Pn )
0 > 1
B
CorB0 (V (B /B0 ), X (D,O) B0 )
Colim
/ E an
|k |, >1
CorB0 (V (B /B0 ), X (D,O) B0 )
an B0
RigDM(k)
E
RigDM (k)
eff
E = Spec(A0 [t1 , . . . , tn ]/(P1 , . . . , Pn )) A0
B1
RigDM(k)
/ CorB (B , X an D B ) = SingB1 (Ztr (X an ))(B ). ^
/ E an
E
/ CorB (V (B1 /B0 ), X (D,O) B0 ) 0
/ B0
>1
|(ti )| < 1 |(ti ) - ai |
A0
Pi A0 [t1 , . . . , tn ]
n i=1
DMeff (k)
fij dti ))1jr |fij - fij |
B1
an B0
DM(k)
64 2 0 ( & $ ! 7531)'%#"
a s uS g q Y
D FS w G I P D C !b##QR!AEEg
D V P q q o g D S h h s V F Xu4d(#reeii(t)D I D I V q P F q P s F P WQXr(hReURi4Rlg QRW4WpxWeU4uRP Er D I P h q D s I D D G G s q g q PS V h D F D q VS q ReX)##iU#4eS DV P q q o XR#pdg D q PS V hS I V F D I s (UQbXQ)X0e WD V DV P q q o XR#dg V q s qS P rR4U!R#
Z0 / Y. ^ Z = Z 0 U0 U /Z s s s0 Z
mS V s V P h V q s q D UXuQ0(itru#WQI h D p )RBs
D FS w G I P q D VFS D g I D F P G G F V D G q P !b##QR!#Qbi7eS Ertr#4l( DS h P F F UWXQRs S E4( V qS P G D F q V q DS V q P reRn7#WX#HrWeXrRWh
/ U0 / FltNis (z ) 0 (U, u)
z0
D y s V q s p g I D q s I D V VS S F q RRQru%Bs 7HW#AuQ!iQU#r4'X4eS s q P h I D V h q Pm q D V FS D S u44Rit#Xi4R6p#hXbWEecg q D FS w G I P Xb##QR!#
u0
g
D I DF P G G F V D G q P #rfr#4k!(u
/ ... ]
I D q s I D V VS g D IS s P I P h D I s G g T #A%x`lXX#ruS Ug q (dQUReeRQRWHUjRXR#
/
(U,u)FltNis (x)/(U0 ,u0 )
D D G P h D D s I D I V q P q ie#!Riie#rdxWQXr(do
[ ... /
^ (U,u)FltNis (x), ZU B Y
F I P s VS I h g F a c Y D D G P h D xReutUXWT Wf(XS g q Aie#!RWiEC g D D #rWQV T I Dm q V F D D y s S q P QW6p#th(uAUpRg #X#4fD I F S q VF D FS G D I G P I G V F #rXb##`Q#RQ#tD D FS w G I P !b##QR! D e VI P G G F D (4#p SF VD D q j)iwkg I F F DS q #X0)U44 D F D T I Dm F D S yS I F T V TS I s F P F F D 0QW6d#e(UQQieXR%p 4RX D D F q g V I P h I s e#dWD 'iX#RWQR4G D FI P 4(QxRe F D D V q s I V q PS q D q V F )XrRQQ4 (U4##4d)D D G PI y D C 4RQRg DV P q q Qu#RP T I D m D F q s F q D V q P h F V I P G G F s F D S q F D h q s q P G F D I I P h F D D G P I F P F D !QW6dU`4R#drXrRWdx(4#Rhe#4f)i4R#4RnXQX(ihin#(X(y 4RX!U D S yS I T V TS I s (bReXiQieXR%p D q #4 I P G #(Ag #Xe#4dW!QW6i#e(UQXWUQu%p 4(!## I F DS q D T I Dm D S yS I T V TS I s F P F D q V iD V q DS P rWeR
[ ... /
(U,u)FltNis (x)
y y yz y iWxbt0iRR
/0
VS I h g UXWT WF q D D w hS p D q FS D I s C g D V q s I V P xbr#XelhQ#40#RXrRQQ4Rih DS I P T V s h D q V F g eX(RyWQ%##!D djg D D #rHWQV D F FF D 4Q#us V iD F q s D s V T D S yS I T V TS I s 4R#peuxi#e(UQXWUQu%p D q 4# h D p (Bs F D G P h F V q P F F V D P F D V q P DS I P y T V s h s D q yS F T e##(it)DUtr(txiV (t)UtrR#heQRRXuWAbi#ReX D a g q PS VS q T s IS P pY F q P D G G s #rHueg q R(UQ44iAbteR(6i4(U4utI g #H)QbU I F F D F FS F D S yS F T V TS I s )e(UQIt)XWUQu%p F D w hS p q FS F P G P V D I P G F D I F I D V h q Pm D D pS V s p I D F q P h DS sm D q V q D I Pm F V qS P G F D h r g xeWD4bt`(nRQded4Rnl)X#tx#WQ4(6d#dReXuBQWX4(i!UeU!Rd#4drW!QR6lQrU(`)i7Weg q R q PS VS F P P I G s s G V D D q g q PS S F P P I G s D D p D I G s s I D p v w h s S h D g D S h h s V F ReXUrG(X#biIR4)iwf E(UQVX(nGRQ#bAAR#X4eQR#xRkW(#reeii(0D q WD D a w hS p D q FS nxeW4bt DS y P P G P s I P G D I D s I D V q eR(U((XVbk4Rn6Y0Q#4sed#r`xWQIXr(Pdg #XtX(4!RWh (4ridQbU q o I F D V h s G P S F s V D D F FS D S yS I T V TS I s bReQHXieQu%p D q g V qS P G q #4 7frU(4 V P g T h q P g D D D G h D USRf(i4(#qWT 'ie#!RPWe I s G s I D q yS F T q o R4xW4Re) dkg I P 4RnG D V q DS p q P h q P V #rrWeRiRAWD s qdg o D F DS G P #ie#RWh D D F F D d4QF# usTQ#4iU#Q#e I VS P I G D DV P q o Qu#q dg DF P G q rdo 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( y x v r z xy x wxb Rx xWb f(ArQUwtuQA4WH4 i4itWRW x | y y z ~ v x x z } y y iW y (
... /
(i1 ,...,in )I n
x
W (u
i
wx
a ((S g q Y % ) z ~ v y y y | v y rQUwtuQ iiUz Wi!b4~ A( ~ | y x nb 4WAi %xx x v %% f iRR W wx v iuwu
k B k
g r q g q g q QUReeRQRWp(itWd q g q g q (UQX(nRQ#e#pxbW#XelfXrReQu%f#eRXnc4RQ#RH4Rl4g H4XX)us D IS s P I P h q P F D V D q PS VS F P G P I G s D w hS p D q FS F D V q sS I s p F D qS P F D F q P I s F P D F FF D F D S yS I FmS V P F D F DS I P y T V s h F D q PS V h I V F q P h s I P G w hS p D q FS D DS y P P G P V s F q P I D FSS V F P q D S I T y s DS I V T P T )e(UQfUXu!H)ceX(RWQ%tReXi4QX4RWbH#(xbr#Xelh# heR(U((X0bf4RQWXeeeXf4(#R#rbX4WReueXQi!RRy q D D P g D sS pS I V q P q F s s I D F P G G D q g D q q DS T S w h I q P q D I P q D q I P G V D G P h F G I P h q !(f(UReeXQR#R4G xWX(n##XF #q RP Er##U#!U4xQRs (#!QR#4##(pie#!RW4QRi4 VS P UR 8 8 8 #j8 FE4df#l7 l k#nlA (lk
@ @ 9 8 53 " 2 " 1 $ & & $ " BA7'764#''%0)('%#!
k
RigDM(k)
Y
B
uB
^ ^ Ztr (Yi1 Y . . . Y Yin )
/ ...
/
(SmRig/k)/B
iI
Ztr (Yi )
(fi : Yi
/ Ztr (Y )
B
%%
^ (U, u) FltNis (x) Z U B Y Z CorB (U, W ) CorB (U, W ) ^ ^ Z Y W U B W CorB (U, W ) ^ U B Y / U B Y ^ T
ff
n
U
Y = ^0 Y Y x P(X)
k
^ Z = Y Y Z X k
Y = iI Yi ^n Y = Y B X ^ Y ) Ztr ( Y x
(U, u) FltNis (x) (U0 , u0 ) s Z Z ^ Y ) CorU (U, ^ Z ) CorB (U, Y Z ^ Z0 U0 B Y
[ ...
(Et/X)br Ztr (^ Y Y )
FltNis (u0 )
U0
Z0
hn : CorU (U, ^Z Z )
FltNis (x) u P(U ) x P(X)
Colim
Colim
Colim
T
n < 0
n
^n Y
Y
U
s0
CorB (U, ^Y Y )
CorB
7Y
CorZ (U, ^Y Y ) B
^ Z0 U0 U
U
/ CorU (U, ^n+1 Z ) Z
n
D D G P h D D ie#!RWU#r SFF XQ4us V q s q D V qS s D q ru4XreuAh4do
(U, ^Y Y )
CorZ (U, W ) B a CorB (U, W )
k
n
x
n
Ztr (^Y Y )
U0
/ . . . ].
(V, v) ~ k(x) U
s0
/ . . . ].
Y
Z
~ k(u)
W
^ V = Z 0 U0 U
Z
n
B
^ T U B W Z a CorB (U, W )
Y
Z0 = Y
(U, u)
/ Y )iI
S
Z0
q D D I s g WhQ#40bc( h D p )RBs I P G D S h h s V F #(f#rbUWWRkD a s )g q Y
a a s (Wg q Y
a c (#S g q Y
I F D pS V h D V P I G D I V h I V F s D D h q D V FS g I D p P I G V D q o g a 5 IS P p h DS q D w V P I D D q q DS T #X(UQ)WQ#!X#Q4QXpeAAW4WQbiD '`W(#RQ#iG d6 U AU(R6Y x)U#4W#QuQ`##UeWnus DS I P y T V s D q I P D s s p s P D D v I P T w D P h T D pS V h D V q D I V h I V F s D D h q D V FS g UQR(WQ%Wh ##p#RnG e#ReRBdur%R!WQR#XV e#(i#ReXi$eS Q#Q4QXb#iWXbWD kC 5D!AA97531)' 4 2C 8 B @ 8 6 4 2 0 ( ~ x z z ~ x y y z v iUi(AdRi W4x hwi x y x y ) 'W xW&v i4x y xUW# (~ X% WUw~ Uy Uz z z v v ~ ~ | z #x4x 0iAW UW (i Qv ~ d } iAxb#~ ! ~ z y y | v y y | X # u 4 QxH Rx 4R~ y zz (UeWx W!y R y ~ y z ~ v y y | v ~ y v #uuv ' i6# !x y x xWb ' dRAQUnl%x( Ab#~ !ewA( Wxy W z y y z ) ixW eh i4x y 0xUW#(~ x z z ~ x y y z v x y x y X% hiUi(A0Ri 'W4x nki WUw~ Uy Uz z z v y 4u 4QxH Rx #(~ y | v ~ ~ z e%v i Ai Ui R i Qv ~ A4 } iAxb#~ ! ~ z y y | v X y z z (UeWx y y Wu ' i 6# Ax y ~ y z ~ v y y | v v v y y z Rx QWb `(!irQUwtuQR i!b4~ A%w%A0iRWxy i z v iuwu X pp7 y y W i6# dx y ~ y z ~ v y y | v v y x y y z Rx xWb ' HR0iXUwtux0(Ab4~ A%Wb y WiiR!Wxy W z y v 4%%v
Cof inj PreStr(V , ) Compl(PreStr(V , )) W Compl(PreStr(V , )) Compl(PreStr(V , )) U X / Z Z ( ^ n Z ) ^ Z V k V / a (F ) Cof proj W Fibproj W Cof inj Fibinj
g s D h FS sm T I G D 7u)iQeRiQ# F D FS w G I P P F S F s F D V q P F F D S sm F D h q D s pS T F D V q P !b##QR!(bS XR4rcekrRX)U#eufW4Weu%e#rerR I F F D V q D s pS q DS T F D v P D #Q(UR%U#r)T Uee#dP)U` F D I V h I V F D D D F P G FS q o g FmS V h D V P I G D V q D s F F V D s h q P S Q#Xi4QX0#drXbdwQ)WQ#d`(!RXXdW#RdR#r D mS V h D V P I G V D P q V F g fQXX4liV (#`)D d ga V PV I P #RXd#(G D D V g gS 4(WQnU(D 6Y I s G T V q D F T I G D I F V I Dm F q s I V h D p s s D h FS sm T I G D u4dXrWXQ#WQdQXW64uxX`)RBu)iQeRiQ#U sI DV P q q P xWQu#pR I P G g F D P #(#e#! F D DS I P y T V s h s F q s F I D s p V dUQR(WXuW!bu#cQ4Weu%dtWD I F S q T T F P G G F V q D D VS hS G S s I D # E44ihX(n##XHrXUibU4e 0QF Fx6uxXkRB!u)iQeRiQ#!#BRQURQQrRiAA(UQ(!nRXkWHXU#Xeju`uW)U##! V I Dm F q s I V h D p s s D h FS sm T I G q D IS s I V q P h q PS V q D m s F V D D VS F s F q s g F D P D F V I Dm F q s I V h D p s s D h FS s jx6uxX(B0u)iQeRm F D DS I P y T V s h s D D R D V F D q D g h DS q D w V P I D D q q DS T s DS I P y T V s h D q V F D F D P eX(RWQ%`bfH!W)#jWBwxe4#XRX`tH#4WeUReX(RWQ%f4#)U4 D F VI DmF q sIV h D p x64RQQf)RBs F V q DS h D P h D s D q q s q D q x(UWii! R##R##do s D h FS sm T I G F D RiQbuiQ#`# DS I P y T V s h s C g mS V s V P h s I D F P G G q g D eQRRXuW7Q%Q#d!RWQr4#F RP '0#r
V X X V tr (X)(Y ) = CorB (Y, X) Z PreStr(V , )
g r q g q g q QUReeRQRWp#`(#WQ#e#hQ%b!!edW4W#r)4(ii#4f7W(XrWbWpReUW4US `#`#xu(trU(R#tD D IS s P I P h D p D I G s D D V sS T S D h q D T F q P h D q V F D S h D g D V q D S p T q PS F h q g D D w h s y V qS P V s q V F
atr : PreStr(V ) / Str (V )
s D h FS sm D DS h P F F uWXbu`eleiXX(s
I D V h q Pm D C g g E D IS s P I P h F V I Dm F q s I V h D p s s D h FS s #XR6`E4 dg q QUReeRQRWpHx6uxXlRBpu)iQeRm I #XF F V I Dm F q s I V h D p s s D h FS sm T I G q S I D V h q Pm D QXW64uxXt(Bu)iQeRiQ#40E4(H#Xi4R6`e
F
% y ~ y | v x v x Rx xWb ' HRAxb#~ !e%Wy y ~ b ' i( i4x A4W4nxb(Q x y x x x y z
.
wU g q g q g q qReXVUX(PnGRQ#G!ehDUeRsUg#E g q (PXUFr(PX#GbQDbSUeX#q(# PS S F P I s DS q VS P G I s s I F S V P D F I P q D V q #rQQ(U4E4XrWD q D g E WeSc gdg q DXeSusbU(PXI(PihfRsQ#qWT(yF4e#GhRsh Rspr q g q g q QeSubU(PX(PiDR4WQ#GpeiDXRs#Rs 7AeUSWRmVFk 5D!AA97531)' I I T F D I s I h p D I s I V G g D h s D 4 2C 8 B @ 8 6 4 2 0 ( Q y y W i6# X u x y y y xd i6# dx Rx x c ""fk ! X y | 4WR % ! b#~ x y y y !' i6# x x xib y
F
v z x z AdUh4Wi ~ x v euwu
F
g D sS pS I V V F eubreQXf)D D q PS V s s p s D F I P ReXu4eu%!e`#rQQ(U D sS pS I V q P q V F eubUQXp(#H)D D D I P q s D F I P #`!QR#!bd4(QxRe
k {ra, Nis} = Nis ,
DS y P P G P V s D eR(U((X0e#
D s I D F P G G F q o g w hS p D q FS D #rQr4#dExbW#Xbti D D P F D S F FS s F V q D D I p P h D I F D D D h F D D F F DS y P P G P V D F D D q g s I D q yS F T eUih (He#eQe!(tQrX#(i)XldeeWinW)UeW44teRReRnRQ#Wd## '!xW#(UX# D P F D D VS F s F q s U(0denRXU#XAed4Rd
U (P1 )an k ^ U k B
g F q s VF 4u4l)D F I P s D P q s D qS s P q V F Q(UR(R#A%#euA(#t)D S a pS #UY g V q DS V I s G G rUQR4#us F I P s Q(URr D D P q s VI D p P q VF lR4!uXWR4R4D SF V XciD V q s q D V I s G G s D F FS D S yS I T V TS I s ru4XXu4#RlXXbUdbReXtQieXR%p D q VF #4D S aSSS bee6Y g V q s q DVI s G G tru4XXu44us F#(#A%fQXW(#RH)Hu4G D P q s FVI D p P F D I s D D S F FS s V q D D I p P h D I q D V FS D S D F FS D S yS I T V TS I s e#UXQe!(frW!X#(iQ4Xei7eRXQeehe(UQ`XieQu%p D q VF ##HD S aSS be6Y gS q #4a #H4Q#pRiX#cU#Q##eRD 'd)Xe D F F F D s T I VS P I G g gSY V h D IS V US#Q#Hu4di44H)XelU#X4RitR4e#uxddUefQWV RH)l4RQhi!b##QR!RXbS 7H)QRbilet(UQrRih P I G I s G V D S q V h D IS VS P I G P h I s G D s V F V F D D g F V D P F D F F P V D D FS w G I P P F g F D F F s h F D V q DS V q P aS bY F D V q s pS F F T V TS I G P I G F D V q s S I T p D qS D G DS I P y T V s F P F D q XrRBe#XfXWUQ#(X#tecrunQW(#UU4tUQR(WQ%Wh 4RXi## V D D F FS D S yS I T V TS I s ihQbU`bReXhXWUQu%p D q ## VS P UR
V X V X B k B B Ob(V ) V SmRig/B B V (Xi )iI X X U
g q g q ReXiXAeW#4ReUiiiQ(Xn q PS V h D F s D T q q P D D h D D V q D I T S VrW!WQW(WeA q g q g q (UQX(nRQ#e00 (#WQ###X4UR`ruWX#RnW)U#eXXb!RdQrW!WQ#RWQl)0eR(U((Xe`RWh us q D D I v y T q PS VS F P G P I G s D D p D I G D q S F qS s V q s q D V P F D S F FS s F V q D D I p P h D I F D DS y P P G P V s D F s I D V G s g V q D D s y T V D G q P D sS pS I q P q V F XR#Rs f(!Weu(A(uReubUQXV (#D #!(UQ%4eu%bH#rQQ(E((URQWtU(ReRnRQhelk(Wuk4Q# ilXrWXQ#G D q PS V s s p s D F I P C g D s V T DS y P P G P V s D F s h s F F F D S h D T V q D F T I 9 8 $ D p D I G s I D V G s s V q D DS h sm V D G q o g DS s I p D I P h q D V F D g R4WQ#hekWX#u#(rW!WeUWRWnAdrReuxHX(i4kfU#E g q ReXUr(X#iet#lURQWHXrubXR%hC ) b 7%" ! ' q PS VS F P G P I G s D D s V T D V q sS I s p s
D FS w G I P D h D p s D VS P I D T F P G P h s F DS h P F F !Xe#4X(he`RBQ(X!hWX(n!Ri!QUWXQRs
DS q D h q s q P G F D I I P h D q S D FS w G I P P P g e#4li4R#4RnXQX(ii##E#r!b##QR!((#w '
1 $ & & $ " n%!r%#0)4
RigDM(k)
n (s id) : ^Z Z
^ Z Z (^ Z Z )
n
^n+1 Z . Z
wU D FS w G I P D F I D !b##QR!ehxWRp
D T VS V q D g D DS G P V P P w D q V q D F FS q I P #XUXreS e#RQu!R###irWXXb#Q#R6m
g D T I V q P T V F D q PS VS F P G P I G s C g RWQXr(t)(UQX(nRQ#'#q F D D V q D D V sS T S D S I T p q e#r0rXue)!eA4UXRdo
$ 1 & " " %)R k e
V
= Nis
a (F )
D D I V h I V F s F D VS q q g Q#Xi4QXQlU##d(P Er 9 8 $ DV P q q Qu#do ) e 7B#" ! '
(W, Cof proj , Fibproj ) (W, Cof inj , Fibinj ) Fibproj
atr
hn
k
CorU (U, ^Z Z )
V
PreStr(V , ) tr (X) Y Ob(V ) Str (V , ) a (F )
VS q q U##ddo
V a E g R E V HS s G H V g H G G X R c Xe g Q X 0##)%b)pU#|YH 8!!irddU#51 H Q { H g R Q V s S R X I V E V E I I R E H g H G G HS HS R X I H ` Ve q H H z D V Q E V HSS H ` H a ` ae R I E R I H ` ` H D R Q E I HS X Ge q X #q~%b5|5dX jd585)|q#5Q jbd#!fkUjd5ky|U%Y5rl t dyfUYbk#}YdbFUy#)y5~yyFF#dkfdBf%&BG ` V H e Q { ` H HS se Xh H D HS X ve Q { |#qTYf#UiY5YUi&U#qa t H Q `V H e Q ##w|v#q{ H %g H z D V Q E V H Q `e Ve E z D #rl (t 5dy##Bf|pd5rr)E b#fY5dYw#dE H R QS D E D R Q V HS se X U#Uih V H D HS X ve Q { 0F5fdCU#qa t V H g ` E 0%vqdyV H g V ` H G aS a V HS H Q { V E ` %vr)!YUY0fu#q5)ydE dUi5iY}0H HS X t E D ` V Ee V QS D S H Q { ` H G H )fy5UF5Ue u#qqY!fS t qYyb))!&)x 5 5!H X c a H R ` E G n H G G i jg6 1%d#y5X i U#fybFdX t b###5rl t dy0%!y)7h V HS se V V H D D V` Q s H g V H zD V Q EV V H g V a GR E ` YH H %g H g H v Q HR I V HS s G H V V Q E V V H 0f#1Yy5YH t 5dye0fS ` YH R X I X R H ce V a g V HS X D E d5mrY#)fy%WUiY%US t Y5YUi&U#qe%))|g#q~0#rl t 5dye0%U#1Yy5YH V H D HS X ve Q { a V H g ` E V e Q { V H z D V Q E V V H g HS s G H V t 5)|TUS V Q EV H XR H` E rYyd#F He R E c a ` X D XS H g ` H s E H G G E D H a R a ge V E Uy)dYb&UB%Y%dB!!dYy%p|5)FD H zD #rl US X H g V` H s V Q EV V H e #rF%)E t 5)|%g )H 7u H g V H z D V Q E V V H g HS s G H V %5rl t dyv%U#1Yy5YH (S H ` E HS se Xh H D HS X ve Q { a H g e ve Q V H G Ve z I R E G E yi5gdU#fdmF5fd&f#q0##Q 8U#|!yU5#y)!d# t !5Q 8B|q!dYUB!!dYbF0a V E G g H a V E I G E D XS H G G E D `e R D H Q { Ve Q I H G Ve z I R E G E E G Q ` V #qbU5#1!yU5#y)!d#)#|H H Q { R H V E I I Q V ` Q H I HS se Xh H D HS X ve Q { #qYy)##y%)f#Ui!Y5YUi&U#qa
Sub(f ) f Sub (f ) f W W R Sub (f ) Sub (f ) f : K f
t H D HS X ve Q { a H `e )E H Q ` V H HSS Q { V E R ` E G ` H HS X D E 5#T|UfH jqB5)ybqd!TFUi%fS t F5fd&f#q0##Q H Q { V ER` E #qB5)ybqd Ee `e V E I G E D XS V X dUyf|q!dY!Up5d#g H %g a ` ae R I E R I XS H g l V V E bFUy5db#Um%0|b)I H Q { Ve Q I HS X D E 5)bp##mfdY%US t F5fd&f#q0u##v|H H D HS X ve Q { a H Q ` V H Q { ` He v S 5)vqYUj% V X g ` E 5d#}qdyV ` YH H Q { `e Q g a g H V X s H g V ` H G H c X z E D R X #q}f#%0%dE |)5u%pbqY!d#d5rD t Fvd5I ` H V H ` X R `S t V H `e GeS E D R X I HS s X ` V ` V Fp0yqiryf5p0yf!fUdYvi5uU#ir}|H H G G E !!de V X g ` E 5d#mq)|V ` FH q Q X H D Ve Xh a R I H g HS se Xh H D HS X ve Q { a H Q ` V dE %dFbpiFb#!%BU#fdY5YUi&U#q!##m0H
W Cof inj
he ` X ` Q G G E D a R R X D Q X HS se Xh H D HS X ve Q { fyib%1)Fmybi5!)f5fdY5YUi&U#qa
LocR R
t H ##Q `e )E 7Cd#n dUyf|q)yR jg6 sf)d7dUyiypUiY%US t n Ee `e V E I E H g R Q H` D Eh H %W#yF5d7US V E` dydE Re E vu Ee ` X VeS X D E Ee V QS D S R E s g V E R ` E V H ve ` D H e V H R Q ` D Q R ` V V HS D H v X ` H G H ve V QS D q H X R HSSe X v X R ` H `e Q V XS V dUy5fY5Ue grdiX 8k5)ybqdd)fb0&f0y5yyb~0f0dC~qY!Y)fy5UF#FbfUUi&Cbb)E yf#|mpk5dX w H R Q ` D Q R ` V XS H g V HS se Xh V H D HS X ve Q { a V H g H V V XS D X b#yF5ybBpm%v0f#Uie0F5fdCU#qv%m|bdpFBpS H Q { R HR` E 5)vybqd!Gm%!bB%yd|pfdY%US t H g X R Q V S H a VeS X D E H` E yd#0)y|Ue%YH ` H`V q ` ve Q V H ve ` D H E R I ` H H e ` D H e V H R Q ` D Q R ` V V H g gS H t V Q E H g V Ee ` X VeS X D ES V H n H G l R E a qdXCU#yBdUy0ib#FTdvUy0&Umb#yF5yb%pY5|5dT#dUy&bpfdY%fUS 7F%5B!y)Y#z H `e ` H I H G HSS He ` H V V H ` V V H D HS X ve Q { a V H g ` E H V V XS D yy)UY!X i UiUS t FYUi&f5)0p%}q)|V V HS X D E Re E vu E z z D V Re g H G l R E a z HS R 6 uUddPbRd#5ryrf}H%u!YbdY5y`mUviX )yfyFv`qY!YUUYUyqYyy0v0H e Q { V HS s V V H D D 5)p0f#Ueyy0FY)X t b%5u%byRY7|5iry})CXT%i0FbUedFb#uH%0Yf5I)Fu%p!yU5#y)!d#)v%uyy)UYpS V ` Q s H g V ` Hh V X R ` D H v q Q X H D V Xh a R I g V H q HS G E D H g V H G Ve z I R E G E E G V H g H V V XS D X H` E yi5W H%0f5fyyV0FdX t VrF%)#UyR)dYb&D t 5dyVBp!yqYUi&U#q{fbd)YayiYbb`F##Q 8jd5f#bdYUu#qWird5U#}|#qFH g V HS se V H D D ` H s E V H g He E c a ` X V Q E XS H ` HS X ve Q a He R E c ` X D H `e H I H g S X e g R X D HS H Q { g X R c V QS I ` V H e Q { ` H G R Eh S H g ` V e Q Ee `e V I E !y)7BXp1%u|H#q{ )H e F #n dUyf|qE)yR 6 1V5i#!G)FD G E S X e g X D V R ES X g H R I V Ee ` X R s t E D R X d5f#bRdYP5QvrdUisYb#jY5)fb&r#5dFmi5I X g H HS X ` H V a R U#sirqYyb#I t 0vdvUyF&Ub#yF5Qybub}%fe#5G ` V H H e ` D H e H R Q ` D R ` V X V H g H Q H Q { ` 5)FH vyqYUi&U#qpfbd)Yb&YyfyYI H ` HS X ve Q { a He R E c a ` X D H `e ` H H g S X e g X D HS H Q g X R c V QS I V #5Q 8i5U%rRi1U#q{5db)f#`0H H Q { R H V E I I V g X R c ` H G G X Q V X e g X D #qBXrYy)##Qysg8irdq!dyVw%ySd5f#bRdYP5Q `e )E 78ggY#! 5 5 F Fri yYY}Fy g F ~YFbYpi fb ( )1fY&F~F f&& rFdFr&f f BF yp 5 1g p)y1Ffi f F U&& r)1YrCf U C C Yy qY}&( Y kFy re)ufYe7 dv (iYp& fF xW8~j B F ( Y F
homD(PreStr(V ,)) (B, K) Cor(V ) R Compl(PreStr(V , )) W (W , Cof inj , Fib -inj ) R
s T !p5& u # rF bf}& BU ) ji f YF 7 b( Ci r piFb) FFfi f F f&& rFd!Yb&f f i p u 5vfFrYC F Brp%#i! % H S X D E t FYUi&f5)0a H D HS X ve Q { Ur fdY%fS FB(FF7 y && r p&F(y)wFyU}& ) !p%# &B s# % dyU&B&# r p&F(y Fr& r p&F ig ~|8s f i vY(FF7 y && r p&F(yw)Pbf}& V ` R HhV X R ` D H v X q Q X H D Ve Xh a R I V H Yr|75dbbv0dCT%dYypiYy#v%g H HeS a s X He R E c a ` X D XS H ##fUYi~UbddyiYvp%g H a ve R a g He R E c a ` X D XS ` V YdUbY%~Uy)dy&}p|H H Q {e I E ` E G E z He R E c a ` X #qp#dbi!d5kfbd)YyiYD X y Ee `e t w H g V H V Q X V Ee ` X R s t E D R X I V HS s X ` H V a R I ` E V n H G G HS Q g V HS l g E G H g V He R E c a ` X D V i &)x #n udUyf#5Ya 0j6 v#j5Yyui85)fb&r#5dFd5gU#irqYyb#jqdypo #o #e!fm%UY%%!k%80fbd)YyiYj0H i h f d gef Ee `e V E I E 7&0f W)fby)dbR t H g V H q HS I G E D H g q Q X H D Ve Xh a R I V H g V He R E c a ` X D V HS R Q E I H G G E %T0Yf5)Fs#r%dFbpiFb#W%B0fbd)Yb&YWUT#)P!)FD H g H v Q H R I XS HS I G H q H R X I Re E vu V HS Q g E 6 #d#y#xUPf#!YwiTUdds0f##!G
H (A) n K
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
LocR (f ) W W
WR
Compl(PreStr(V , )) R
Ho(Compl(PreStr(V , )))
A
// H (B) n
// B
F
Hn (F )
g : K
f
K
L
Sub (f )
Sub (f )
Sub(f ) HOM(1, PreShv(V , ))
lK
lL
// L
// LocR (K)
// LocR (L),
W
lK
nZ
// M
// homD(PreStr(V ,)) (A, K)
// Sub (g) Sub (g)
LocR (f )
n n dd#(u
lL
f
W
D(PreStr(V , ))
Cell(
(W , Cof proj , Fib -proj )
f
Sub (f )
A
2
Sub (f )
(R )
R
3
// B
R WR = W WR W // L f : K R
f Sub (f ) Sub(f ) Sub (f )
n Z
(W Cof inj )) lK lL
Compl(PreStr(V , ))
W
R
M
H (F ) n
R
L
2
9qi&Ue#|VviXbf#Q|0yRUS1%5fyVb|RdYD%FDvF5fyab|dY`qY!GY#qp#dbi!d50ybi0f5de)YuU|H iD FgU S #)#fbp%g ` X v Q ` `S V a H X V He a ` X E ` H V He V ` R X D H H Q {e I E ` E G E z V a R R X D V HS R Q E I V X D HS ` V ` H T H V a Q c e ` Ve V H c Xe ` H V X 0fS)#ipyRbuV0fSui#g H#g a ae R I R y`FUb#dEb#IXp`)!YUSi)aB5fyadTUSUS S bf#%%y5dF1bfyYFur##b#1bfyYd5BU#ir1|WfUS S S H G H X c H te R v H V `e Q g E R I E D V `e ` H I ` H V `e Q g E R I V `e ` H I R X I HS s X ` V ` V H HS Vi#g X H g G Ve I R E E V H V X XS H g H Q { g H I a g H H ` V R H %H!|p#z#bd!Gdype 8gyyV)USYDuU#v#qYY%}5YbrY t Rbv%yFUb#db#X i Q E R H g a ` ae R I E R I g & rr dpFE Fi BF ri!i#~
K (Y )
Ff 0B ) x n dyU&5bf sbf}& P dy& f di#(u #}& C sYrpFY&& d (pi yi 5 I F 7 G HyvdwF gbygf! F& !Y&(ifF! 5vi & FYrpFdd1FFiC bE % }& 5 rF ) &q& U (FF7 y CiP r piFyg)wyfgvi
// Z K
ig DB ~|58P C A 9qdCU#ydUyf#5F#BUp5)f5iX @ ` X ve Q V Ee `e t a g XS V ES H I I
// X Z ^ ^ eX Z : Y X Z
H G Ve z I R E G H !yU#5y)US
H Q { S H ` V H ` D X I G E e V X Q { V H ge ce R V a ` ae R X #quYyTb)5!dFD t |)5qW0%U)fbWyYfbi&v t IV HR jyb7u
= Nis = ra
dUS#fyrYdUv0H H se V R H v e ` V H g H G Ve z I R E %!yU5#y)!G #0H Q `V
e 5x
e u
R#Q)EIe0Hb`|RHdv#Q)EW)feyrY!!fu#5%j|0yBfdbFv|H E V R H G Ge H Q I V H Ru HS X ` a ` V H ` R H v Q E Ee V R H G Ge H Q ` V )yyYd5ddU|rY!!fm#5v0H
#n )
90yqi&U#|p0yFUb#db#pUm5fyd8Ue #d5fbp%g t V H ` X ve Q V V a ` ae R I E R I V HS H te R a v S V a Q c e ` Ve
U
u
aRR X D Q `V bydY#v|H
x n )i#(u
V H ` D X I G E e V X Q { V H ge ce R V a ` ae R X v H g He V a ` R X D a R R X D 0y)5!dYD t |)5qp%pdUyp0yYfbi&u%YU|0yyiYmbydYv
Latr Latr
h f d gef H a v Q E R I ` V H Ee `e V E I E R I )Yd5db#0W)fby)db#X i dU#U|rY)fv|H HS se V R H v e ` V a `e g H G Ve z I R E G H he ` X v R H V E D ` V b#5Q 8u!yU#5y)uUS fy&&bYy5dYv0H R Q H ` D Eh HS H G G E #yd7mU!!deg a Q {eS I I #qpf#5iX #ybdU!yU5#y)!dyUe (qYUy#ddE H g H V R H v e H G Ve z I R E G E V S ` He ` s E a `e g H G Ve z I R E G Q yf##Q 8!yU5#y)!TiX H Q {eS I I X S e #qUU##dW)E g|V ` R X I H R ` Q HS se V R H v e ` V yiby%dX w dU#fyrYdU|H a `e Q E D H g H G Ve z I R E G H ` H G HS Xe ve R ` H ve R a g H b5#dY%!|p##bd!US qY!YUipfby)fbY%yV H Q { a E g ` X #q#5d%Wqdb` H g H `e E R g ` e E g %b)y#TqUdC)X S V E` dydE he ` X v R H V E D ` V fyiCbYy5)Fv0H H Q { ` H H z D Q X c HSSe Q 8 H g R Q H ` D Eh #q}F#r5d)YUfU#%}#yF5d7#Q ` |VH H Q { ` He v S HS X D E #qmqf)fdY%US t Y5YUi&U#q##0H H D HS X ve Q { a H Q ` V e V ` H G HS Q H V ` H e V H G Ve z I R E G E V e V X Q { Q ` V yu)!YU#YymYy!|p##bd!)|pe t yd5qP5|H `R X I HR` Q |d51by#iX w R Q H` D Eh #Yb5)7TH i 78ggY#! H)vfb0&f5dUyib55iYBUS dH e !yU#5y))#d!UW5d%dbYyb#p)#FB0H V e ` D H e V Ee ` X R s t E D V H V H G Ve z I R E G E E G V HS D E g H v R H V a R I S ` D X q H ` V % 5 7 5 % Frif# q 5 f 6d Fy mYd |d % YrYpi f U&& ru rF F~ $|d31~) 4 2 0
atr : Compl(PreStr(V , )) Compl(PreStr(V , )) D(Str (V , )) Str (V , ) D(Str (V , )) Str (V , ) // Compl(Str (V , ))
g
V R Q HSSe X R X I X V H aS Q c Xe R ` V He R E c a ` X D H g H D HS X ve Q { YU#)#ipyb~0fbd)Yb&Y#mY5YUi&U#qa r#fUfdvi5 H Q `e Q g a g H H Q {e E X 5#f#%0%sY8d#qp#d5iYD V He R E c a ` X D H g H D HS X ve Q { a H Q X 0fbd)YyiY%FYUi&f5)0#5BdE q0wdd5Bp%}%U7s}7u H c X I XS H g veu e V Ee `e g E D V HS ` X te R a v H e HS I He R E c a ` X V Q E V H R ` Q X H Q ` V 5dUyf%5)F0fqdbY)B#UYU#1UbddyiYD t 5)|wby#iw##10H e byY7|5dbb0dC%i0FbUdh t 0%T#5YUfS V ` R Hh V X R ` D H v X q Q X H D Ve X V H g H He t a s X He R E c a ` X D XS H dBfbd)YyiYpw%g H a ve R a g He R E c a ` X D XS ` V Y)fbY#Uy)dy&U|H H Q {e I E ` E G E z He R E c a ` X D #qp#dbi!d51Uy)dy&X i 5 % ' UF(BB)By r! F Yf !~ YC5 u YBp5&B& !!FdfF Y!(FF7 y && r p&FdTFqbfgvi dFBrp%#i!iFp & dYb&f f &1& $ W8v" % #
Compl(Str (V , )) Str (V , ) W Ho(Compl(Str (V , ))) V SmRig/B Fib = RLP(W Cof ) V (W, Cof , Fib) D(Str (V , ))
!dE U%5#ydy#m5#YUfdUbddyiY##v|H X D He g H z ` E R H g H HeS a s X He R E c a ` X D H Q ` V H Q { a E g ` X 5)5#d%}qdb` byY7|5dbb0dCP%dYypih t %!Uy)dYb&pb|pUfb%P)E pYY#dE V ` R Hh V X R ` D H v X q Q X H D Ve X V H g He R E c a ` X D XS X R H VeSe ` Q XS H D R Q H g qe E z D Q g V l R I V H aS Q c Xe R ` V H D HS X ve Q { a %)#r%mb0y50YU#d5ipybFYUi&f5)0u H ` X g H I a g e ` V H HSS Q { X R H R ` E G H aS Q c Xe R ` He R E c a ` X D H Q ` V V H a VeS X D E b)d#5%5U|fUYH jq}rYbyq)1&)YU#d5ipybkfbd)Yb&Y#5|H dY0YyUUiUS t 0y#b5by|%#qU5dyd!d#efbd)YyiYD V H R Q ` D Q R ` V V H g H Q {e I E ` E G E z He R E c a ` X X pS XR H` E V H zD rYbi#Fb0#rl #wf#1Yy5Y#qdCU#yU5dw%dUy&bpfdY%fud5!Y)y|5)FD H g HS s G H V H Q ` X ve Q V gS H t V Q E H g Ee ` X VeS X D ES R X I H a v R H V E ` X ` a a ` ae R I E R I H ` ` H D u V Ee ` X R s t E D R X I V HS s X ` H V a R I ` E V o n H G l R E a z ` Q g V H a VeS X D E qirFbFUy5db#1b|YF5dUyib#iFmivU#dbqYyb#}qdy!%%!y)Y#b%|pfdY%US t UY%%!#vUy)dy&0H i V HS l g E G H g V He R E c a ` X D V
f0 f R g Ho (Compl(PreStr(V , )))
h f d gef ` X `S Q V a R HS 0&bf#ybmfdE w %ubUiBUiB1p#byqdYF0edU#UiF5fdCU#qa t H g a `eS X Ge q X G XS `e g H R ` E D e D H HS se Xh H D HS X ve Q { H Q ` H G HS X c a ` V #5qY!fd)Ym|H H Q { `e Q g a g V HS se Xh V H D HS X ve Q { #q##%Y0f#Ui}Y5YUi&U#qa t V H g D E g ` H V HS X D E 0%5d%Y0fdY%US t Y5YUi&U#qa V H D HS X ve Q { V H g ` E 0%qdyV ` YH H Q { HSS H ` ` 5)fUYbvFH 5i5#Yyq)Fd5 U#U|bYYdX t %#%g V X g H Q H` E D E HS se V V H D D ` Q s H H #g H zD V Q EV H Q R H v Q ER` D E #rl (t 5dy15#Y)#dbymd%g ` Q H I #Y8 V X g HS X t E D ` V H n n u H g H ve ` D H E R I H `e GeS X n H G G 5i#1fd55iY|pid#(T#dUy0ib#b!Uf!pS i %51!H i 6 dX %g H H zD 5rl R t 5dyBUS H H z D V Q E V H Q He D E V V X HS X ` E e R E z H z D #rl t 5dy##uUF%|bdmUibq)Yy)#u#rl UvYb)%pqp5dUy5fY5UU))|0fdYpyyY)p#r0l UdE XS ` H V H ` H ge v a V Ee V QS D e V HS ` E V V HS X De ` R H v V H z D V HS V Q E V X %g
g0 = g f 0 g f0 f Sub (f ) g f0 g0 = f 0 g g = g f 0 = g g0 f 0
%
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
R
Latr
Rinc
inc
atr (f )
// id
RigDM(k)
atr
Latr tr a
k
atr
K (X)
B
V
k
atr
// X
// Y
// K (V )
// K (U )
e
Latr
f
K
K
^ U X Z =
D(PreStr(SmRigqc /B, )) 2 3 : id SmRigqc /B atr B
F
: // Rinc Latr Cof
% y && r piFb BYfFb i F
HS se Xh H D HS X ve Q { a H dU#UiFYUi&f5)0u##Q ` |VH #q{yqdE!G%feddX Vq H Q R H` H g ` c S H `e V `e ` H I HS R Q V S X D E yf|fyF}U#|gi%fS t #iYypiYy5%Yf5)F#Q q Q X H D Ve Xh a R I H g H q HS I G E D D H v X S X D E 0dCgiY%US t F5fd&f#q0T#Q H D HS X ve Q { a `e fdE H `e `e ` H Q yf|VfyFIiX H g Ee ` De R ` V H R X %)fbpyb0yTUS H ` E H V VeS H ge ce R a ` ae R X bi#sdE |bpf1%U)fbbFUydCv t H ##Q R Q #dE X D E SiY%US t `|H V H { R H E I I Q D E ` H I #QqvYyV)##yVd%g}%Q R X d5I H D XS I G H R S e V V X I ` H c X z D H n u H g V H R s G H F)U5y)E 8|)5qYd5i5ru#di#(w%0y5Y!G q Q %YH%gVH i iX%fS t r`|7VirypD0dCBq%i0FbVUdFb#IH#g%qFU#!dYW#Q S D E V R Hh X R` H v X Q X H D e Xh aR H HS I G E D D H v X HS X D E )CUiY%US t F5fd&f#q0#5Q H D HS X ve Q { a H `e fdE H g qe E z D Q V l R I H D HS X ve Q { a %T)#r%g b#mF5fdCU#q d5YY%g V X I g H I a H# H G % R5|j#F%qY!YUfy5y&|YU#bY%uFFH Q V e t a g ` H G HSS H R Q ` X ` V H R He R H g H D ` ae D E V V X Q X H D Ve X fYybdWdYypih &GH w t %gW`qY!H#qUe#5g5YIY%f)cdUd!Gd5dFrRY#z S `FgTUH Q H G Q { Q H a g He ES E E z E D H I $ H H Q { R HV E I I QV ` Q H #qw|q#5|W#YI 78ggY#! S m(FF F " ibfi!i# Y5b# j r piFbU F &5 fYrF& !
K X V
T% !p5&Bf ) vq eYF7 y( Ci1 b p&F(ygdqbfgvi &8 Bee~~ A x n i&%(u 7 5 p& F b& Fd&5 Yd CiE F!p5&BFduYrCf fk C sC Yy}YP&( Y FfY' 0!sd fYe 7 5 )P (pi f &&B d U Fy 1FdwU& U F f&& rw)TYrCf f} & fYe7 )w &Ypi f! B ~~ A H`VR H jy|b t Ryu%uyYfb#db#pbY)8fjyvqY!YU#Yy}F|0|pUiY%US t dUyFdy#mb#yF5by|pv#)qir#5t )H 7u Q E R H g a ` ae R I E R I XS H te R a v Se e V ` H G HS Q H V ` H e V H a VeS X D E H ve ` D H E R I H R Q ` D Q R ` V XS R Q E I ` X R s e ` XR s qir#5t t qdUyFib#|H ` H G H ve ` D H E R I ` V V ` R Hh V X R ` D H v X q Q X H D Ve Xh a R I H g H q HS I G E D Q { ` H g a D a R I H G l R E a z ` Q g `e Q g a g byY7|5dbb)CT%i0FbUdFb#%uYf5)F#Q jqpqY#F0y5!Ybd#yW%}f#%0%W
f K K Compl(PreStr(SmRigqc /B, )) // tr (X)/tr (U )[n] tr (Y )/tr (V )[n]
h d gfef H R S D E H g ` G R H Ee ` V E I E R I db#QfY5dYD%vFH!bYI)f#o #n)fbey)db#X i F0d5Yb#Uu%uUyyi5bYUy51pvd57sy|b t Ry%g H v Q H R I XS H g He ` R X I H R le G H R I XS R X Iu H ` V R H Q ER H a ` ae I E R XS te a yYfbR#)y#ImpH5fyRdv H HS X e Q { a H Q ` Ve X q Q X H D Ve Xh a R I H g V H q HS I G E D H g H G Ve z I R E G E V e V X Q { Q ` V `YfdYD%US t FDYUi&vf5)0v##eV0H pid8%dYypiYy##~%FU#!dY}%p!|p##bd!dype t |)5qw#|H H HS X E H HS I E D D v 0HdCX Ee ` X R s t E D H Q XS H D R Q E I `e Ve E z H Q R HR` E #q{yb))!G%g`!%yS(dYD%US t %qYf#!G)F#Q H QV S X E HS Xe ve R fdUUyb` t )fb&r#5iY##wUF5d}ypd#rD (dYD%fS t 0H S X E `V H#q5dbyqdEj f)dUdEdbwpS5QdIYHy`|VrRYH t QREyRH%g1bFUy#)y5wp5fydv Q { V ER` He c ES I E ` X R E a ` ae R I E R I XS H te R a H Q { HV E I I QV #qy)##ys ` X `S Q V a R HS ibf#|0ymUdE w d!yU#5y))|pe t yd5q#v|H H G Ve z I R E G E V e V X Q { Q ` V
K
H Q { ` He ` s E S X D E #qvqYUy%)W)E dY%US t
// [tr (U ) tr (X)]
V H q HS I G E D H g H G Ve z I R E %Yf#!)F%u!|p##bd!G HS H Q { ` He v S fW#q!qfp ` H G H R v Q E D H R Q g z D H H g H q HS I G E Q qY!Yb#dYb%r0e#W%FU#!dYD %g Q X H z I R E G E V e V X Q { ` V H n dw##bd!dyUe t |)5q!|dC%u ae D E V V X HS I Ge V H q HS I G E D HS H Q { H R ` E G ` H T H fY%|bdf#!U|%FU#!dFf#q1byqd!dE FgU S &U%%s)fby)db#wp}d5pfyb|dY%F|b5d5d#mY7xYUyyyi|H o n Ee `e V E I E R I XS R X I He V a ` R X D E D e V V Q X D E g ` Hu He V a ` R X D ` V
tr (-) tr (U ) // tr (X) (U X, Y X)
BYf YYgi f
q Q X H D Ve X %dFbpih t %bbiYuUS )&%(m0y)5!dYD t |)5qpyypfp0yYfbi&v t %g H g a R R X D H x n u V H ` D X I G E e V X Q { V H V VeS V a ` ae R X H e 78ggY#! abRbRiXYD#Qpa##)E%g`qdXb`Fa `FHgTUH H ` V R H t Q E R H g a ` a e R I E R I X S H t e R a v S S X D E t ` V S Yy|rY Ry%yFUb#db#U5fyd%Ue i%fS |H I P G dyU&5bwfu # rF H yv)BYrymf F&k v(YF y &&W r piFbe) bf}& A 8B 8~j w F
B K
h f d gef HSe D e g V QS I H R l Q c ` V V HS se V Ve G g X V ` H G H R v Q E D H )fUFwf%v5U#y#dv|H jp0f#Uyyp!%dpbqy#)F0yR V H g He c ES E I E ` XS H g V X D HS Ve X 0%f)dUddyPp%BdfwUd!G rUY5|pe UddUddbPU%B)YUwqY!Y#qp##bdbbW!y)Y#bWep%x %d!Ybd#y` z De v H V He c ES E I E ` XS H g V X D HS ` H G H Q {e Q H `e X R ` H G l R E a z ` H n H G l R E a z Q V H `e V V `e ` H I V HS R Q E I V ` R Hh V X R ` V X V Ee V R H v X V H g ` H G H ` Xe g a G Ge HS Q E D a g Ee `e V E I E R I i %g dH (e byubfyYUm#dbyY7|5iry}5dys)fyrYdWb%qY!y&p%0!U1U#)F#sdUyf|q)y5TX 78ggY#! % Y(F F7 y && b p&F(ygp)B !p5 B&Yp C % rF & rif dyfi#rTfu # I P rY G RbBdYbbBfm e YC Y mYfi f f&& r W Y&#) % u(YF y Ci r p&F(y})1bf}& dBBp5&B ig A B ~|i2 3~) A 0
f a a: K
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
SmRigqc /B
a Q c e ` Ve 5d#Uy|U#g t
[tr (V ) tr (Y )]
`V |H
H G G E HS X D E !!de)fdY%US t
H D HS X ve Q { a H Q ` V FYUi&f5)0u##v|H o n dd#(u
n q&#(u
B
K
// L
I
H0 (X, K )
Cone(K(X) K(U ))
K
K
f : K
V = SmRig/B
V
// homHo (Compl(PreStr(V ,))) (tr (X), K )
K|X
tr (V )
// L
K
L
// L
L
// tr (Y )
// Cone(K(Y ) K(V ))
H0 (-, K )
SmRig/B
K
L
K
L
SmRigqc /B
Et/X V
nZ
Ho (Compl(PreStr(V , )))
(Et/X, )
X
K
L
K
r : K|X V
r
// I
)i#(u n
f
V a ` ae R I E R I q Q H g V H D R X I V l R I H G Ve z I R E G E Ve H Q {e Q Q u a e G R H ` a g ` V yYfb#)y#W%Y%0Fvd5eb0y#m!yU#5y))|pm#qU5##!B#U!yyF#}0H V QS I 5f5uH w X g H E H g ` H G H c ES I HS ES Q g E G q##)~%)!Yd5dU#uf!U#%%!7u H ` H ge v a Ee V QS D S D H v X H ge D E D X g H E H g ` H G H c ES I HS Xe bqY%pYdU|5UFfe (})Cm#UY5(F)5#)~#qd#)f5UUv H g He R E c a ` X V Q E V H Q H G G E D H Q v %}Uy)dYb&D t 5dy##u!!dY#7u Ee ` De R ` V H R )fbpyb0y!X b!Uf)FW#iy#!dYD V H `e GeS E D q Q X H ` Q G G E R Q H` D Eh #yd7H i
atr atr f h(X,a) = hY,f (a) F atr (F ) = // K (B1 ) X F : (V )op X V (V ) (V ) Cor(V ) otr (H) I (X)
h f d gef VR H rY)v H g V HS Q g E %pU#%%!G t %}b%5dT%dFbpiFb#u%u!yU#5y)#Q 8g H g Vhe `e g g X q Q X H D Ve Xh a R I H g H G Ve z I R E G H E X Ya5#d%gBpS!y`qHfdXCUe#qaV0ufUf!dufUYbu##Q 8##d#BX i H S v Q { ` H HSSe G Xh HSS H ` H g H a E g X dE ` Q E` R Q E I H Q %db#d#q{ HSS UfHy` R Q #)EI V ` H G aS g HSSe G Xh H Q R H E g H V ` He v H bqfa jwfUf!d15#mY##)%B|qYUYbR V X 5d#g H g V`R HhV X R %mbyY7|5iry` D H v q Q H Ve h R )CX!%dXYDypiXYay#IH%H!yVU5#IyR)E!w#Y##)%H U#%%!G t #byY7|5dbbp)CwiYypiYy5T#Q g G e z G Q R H E g V HS Q g E H g V ` R Hh V X R ` D H v X Q X H D Ve Xh a R I `e )E 78ggY#! 5 n &)#(u &5pYYdU d!5 ) F Fvk YF7 b( && b p&F(yg}Fd&qi f! )TYp~ YT1& fE %T iYd dmFB&)
(X,a)(V )/F
FvYFf 0B
)Wi)&
7 U! )wr fiC
i i# FFygy% ! &d) r p&F(y &g
(X, a) b = f (a) F
VR H bdv
` H G aS Q ` V qfa 5w0H He R E c a ` X D X Uy)dy&}pS R Q V H R seS HS Q g E 5|b#UfU#%%!G t
H` E yd#d HS ` V Uw0H
H Q { S H 5)jy` H#zrD0l #} H ` FH D H v 0dCX V HS I Q E D V HS ` E V V ` H s E V HS ` E U##dYBUB))|BrF%)!Uwqd%g H g H q H E D H ` X V E I G E D H g V H ge ce R V a ` ae R X %e%F5#dYeyqdyq!dY5#Q j#U)fb0yYfbi&v t #0yp##bd!%U#|YH (S H g V H G Ve z I R E G V H g V HS s G H V H Q { S H` ` 5)YbBFH H Q { V ` H s E H G % G ` X X H ve `e g g X He R E c a ` X D XS ` V #qwbY%d!Y&Bqiq$ )fb##dUy)dYb&Uw|H dE
(V ) B a F (X) V // Mod()
H Re X a e bfd#UUS t fyf%#)v#Yb5)7#u!)FD he `e g g X R Q H ` D Eh Q H G G E Re E v ` Q H I Ud)%YT5 R Q V V HS Q g E #|p0f##!G t %fyf%#)p5YyF5d7W#Q H g he `e g g X R Q H ` D Eh `e H ` X ve Q V H R le X G XS H g `e Q R ` V E D ` V H S V ` R Hh V X R ` V H fdE dbqi&f5|yf#dBp#f#ybdF|8qYr|75dbbe%g `%)&X 85YyF5d7rYUUY#iT)E ){ Q E g R Q H ` D Eh X R HSS H I I X Q H z D Q X c ` e E g X Q ` H G g X R Q H ` D Eh H #r5iqeqUdC)T#F!%d#Yb5)7ve# R Q V Vhe `e g g X q Q X H D Ve Xh a R I V H g He R E c a ` X #yefyf%#)%dYypiYy#e#vfbd)YyiYD X pSVi#b#fd&T X g V R Q HS X v V ` R Hh V X R ` H g eS s Q g ` H ge v a R Q H ` D Eh g H V E I Ve g r|75dbb#U##)E jqY%pYm#yd75Q 8|qyp%P V S Q HSSe Q H g V Ee ` D E g X V H Q {S H Q { H g H ` VeS H Q V E Ve Xh Ee ` D Q R ` V E D XS H g H I X ` a H R le R H g XS R H V V X I H g ` X HUUey`%fUf58 %5dUyF5dC)}0#qU#q1%y|UU##}5dypidUyF5by|5dYwU1%1irFubYU#bY%wpwYyy)5%)d&v % u #}Bp#i!iYp r& ) s% r !Yf Y~Yr& f rif yYe !YC Yy YF7 y( Ci r piFyg)yU}& &8 jBW8" A
B
1
bF& f YF7 y( && r piFbvfp YrpddFiC
RigDMeff (B, ) X Meff (X)
H D H ge v a H Q ` V H ` X ve Q V H G G HS )F5%pqu##v|qi&U#|!!YUmH i % d% FYd B1i bF)r YTbF0yi) f j ( } (i!1 gB ~|8P A
B Compl(Str (V , ))
H ` HS X ve Q { a ` H G HS X c a ` V H H R Q ` D Q R ` V H ` ` H D H g H Q {e I E ` E G E z He R E c a ` X D mbqYUi&f5)0)!YUi)v0my#b5by|yyFYm#m5)p#)yi!)#UbddyiY!X i qCf#o #nP)febe#tFa%gWUS5i5gHa|VpfSUy#Q|VbdXpFD!Y%&wpqi&U#yU5|5d!%dUy&bpfdY%fTUT1UiqYa o E ` X V X e e ` H V S H G G XS ` X ve Q V gS H t V Q E H g Ee ` X VeS X D ES XS HS X c R Q 5|V H Q` Q ` H Q eV Q X bR#bD5bRy|V5#5yyViXW# V HR Q`D QR` 0y5yybV H X E USiD%fS t V He R E c a ` X D V HS R Q V V HS X D E 0fbd)YyiYUe5|eUiY%US t V 0H%g V H Q e I E` G #q{p#dbiE!dE5zvUbddyiYs%d!bqYUi&f5)0|H V He R E c a ` X D q Q X H ` HS X ve Q { a ` V He R E c a ` X D fbd)YyiYX i & F1Ffi f ri!i#m&qi f Yr&eYFddwY&&!FdTF)Bb (i!wFd&qi fm u 5 Fy (Tdu BFfgvyYf&5
RigDMeff (B, ) = Compl(PreStr(SmRig/B, ))[(WB1 - )-1 ]. = Nis B1
5 Fy F Y&# pYFf& f Fd 5 )xYb &f fB C fYe7 s) &Ypi f&5! Yd Fy vU Y
B1 (WB1 - , Cof proj , FibB1 - -proj )
i
T U& U 8B A
V HS se Xh V H D HS X ve Q { U#UiY5YUi&U#qa t
q Q X ` R E I I X R R X I R H VeS X D ES H ` Ve V E D H I X ` a H G le Ve E R ` #iv|)5ird5YypfdY%fuy|p|5)FuibYmfypdby1X i
I (Xn )
h f d gef ` X `S Q V a R HS ibf#ybufdE w V H %g H Q {e I E ` E G E z H `e GeS XS H z I R E G E V e V X Q { ` V #qp#dbi!d5yf!fU!U!m##bd!)|pe t yd5q|H H G % G V H g H Q {e I E ` E G E z H `e GeS XS H z I R E G E V e V X Q %#qp#dbi!d5yf!fU!Uw#5y))|pe t yd5q{ &H w `V |H S X D E t `V H G G E V ` R HhV X R ` D H v X q Q X H D Ve X !)e)Yr|75dbb)C%dFbpih t %Uy)dYb&uU5i#g V H g He R E c a ` X D XS V X X wq iY%US 0H H g V` D X I G E eV X Q { V`R H v Q E V H %bFd5!)FD t yd5qpbyY)#dp0%g D H v )CX H ` X V Ve E R D Ee Q H Q H G G E b)dybpdbFdU##u##m!!dYD `e R D a S X R a a c V X D HS R Q bF0W)E db#Y)pdYuU#dE n H G l R E a z ` Q g V R ES X HS Q E D a g ` X `S Q V a R x %d!!Ybd#y%prdUiuU#dY%v&r5|0yuH i f)dUddy1p#dwy|b t Rb%uyYfb#db#pbY)v He c ES E I E ` XS R Q E I H ` V R H Q E R H g a ` ae R I E R I XS H te R a rU#fybU!%)kbqY!Yb#dYb~0%vf)dUddbmUe#dyqdUbi&mbwd7qx #n )bUipf)y)F}U~iI H G % G V HS se V Ve G g X V ` H G H R v Q E D H R V H g He c ES E I E ` XS R Q E I H ` Xe R X v X V Q Eu n H Re XSS E R E D HS R X He c ES E I E ` XS R Q E &vH w UddUddbmp#)I H`VR H y|b t RyB%!yFUb#db#WU15bY)v Q E R H g a ` ae R I E R I XS H te R a n o n H G l R E a z ` HS R H ` D X I G E te V X Q { ` V H Q { gR E s g V EV E I I 5)Pb)dX jdy)##Q 0U%%B!Ybd#yUdX db)5!dFD |)5qu|H
j
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
X
I
tr (X)
(B1 , ) tr (B1 )[n] X
X Colimn tr (Xn )
RigDM(k)
X
(WB1 - , Cof inj , FibB1 - -inj ) Compl(PreStr(V , )) // tr (X)[n] nZ X Ob(V )
Ci
homPreShv(V ,) (F, otr (-))
f hom(V ) (Y, X)
1
RigDMeff (B, )
RigDM (B, )
eff
K
K (X)
V
Compl(PreStr(V , ))
K (Xn )
B RigDMeff (B, )
otr : PreStr(V , )
Colim
F
V
X = nN Xn
tr (X)
hom(V ) (X, Y ) X Y (V )/F f : (Y, b) (X, a)
K
V
HoB1 - (Compl(PreStr(V , ))) B1 (WB1 , Cof , FibB1 )
// PreShv (V , ) atr V
Xn
otr
7
H atr (F ) H f hom(V ) (Y, X)
h(X,a) H(X)
K
X
(X, a) (V )/F
V
PreShv (V , ) K (X)
n i)#(u
# A DB W8v"
tr (X)
K
F
FBrp%#i! fY' 0Bd |8P
HV E I |q ` H ` X R `S t a E g R E HS s G H V H Q ` V Yvqiryf5m##)%b)uf5Yy5#v0H t 5)|#Tf5Yy5H S H I ` H g V H V VeS V H R s l cS $ T%0|bUUu0y#dUiX t V Q E V V H g HS s G H V
(B, O)
R Q V`R H V R` H v Q X DV Xh a I #yVpbyY7h|5iXry}D)CXdHFbpeiFbR##Q `e fdE Q V H g G l E ` HS R H g e %#5t k jk FDbIdE H%H!YbRda#zymUvdX Hyd#xdE yypeUw%g #B&X t ` E HVV S H E #Q R#QdE 78ggY#! g % bu F
F X inc Rig : PreStr(Sm/B, ) B
5vi
d Fpf
(X an )
YB )Y | ` YH
(Sm/B)
H z D Q X c ` e E g X g H V E I Ve g H G % G #ri)vqf)Cd#Q 8uy)|p%dE !Y&uH w H Q { HSS H #qfUYb` H Q `V #50H H zD 5rl #v H ` H H V VeS X G a z D FwybUUBY#rbV t #Q D H v 0dCX He R E c a ` X D XS ` V UbddyiYUu|H ` FH V Q V V H te 5yy0%g e
Rig F =
(X,a)Cor(B,)/F
H R ` E G H a ` I X g X ` H G HS s X H v E byqd!vy%d#dqY!f#d5Y)dFD f#o #m!fw#vd#y#US n H G G HS Q g H v Q H R I X
// PreStr(SmRig/B, ) .
R Q V V ` R HhV X R ` D H v X Q X H D Ve Xh a R I Q ` V #yr|75dbb)C!dYypiYy#T#e|H
H i
HV E I y)wdE
R Q #|V
H z D Q X c ` e E g X Q H g l V V E #r5d)}qf)Cd#m%yb)I R Q H` D E #yF5d7h V ` R Hh V X R ` D H v X Q X H D Ve Xh a R I Q a E g ` X he ` X ` Q G G E D ` H G H Re XS D ` V byY7|5irydCwdYypiYy#W#}5#d%qir` fyib%1!dF)!YbfdUYp|H
Rig Cor(B)
Rig
e Q ##q{
aRR X D ybiH i He t H D X g E I V H R R E D XS He D E V V f55vY5i#d|0ybdYpmfY%|bdX q#QYH%gBHbRy`qYH H Re X ` H G aS a He t H D X g E I V H R R E bfdbqY!YUYBU#5F5d#5)yby)FD t H Q e Q { he `e g g X R Q H ` D Eh H #5W~5)vfyf%#)#Yb5)7!fS `qY!YUiqYvbi#B# H G HS X c a H ` E H ge ce R a ` ae R X %pdUybFUydCv t pUF%|bdX XS He D E V V e t H I ` H g H V VeS X G a z D #5$ v%vybUUmBY#rbV t #Q H Q { V EV E I I #qdy)##Q R Q H ` D Eh g H V E I Ve g Q { V ES H I I #yd71#Q jpy)yU%!dE jq5dUY#dX @ dyypU% #BiX t #k|H H V VeS H g E Q `V
B (Et/X, ) B k
h f d gfU V R Q H ` D HS q Q X a V Ve XS X R H V ` H ` Xe g a G Ge ` V H Ee ` X VeS X D E Yr#Yb0fT#im|bUdUrYy}F&p%0!Uv0W)fb&bUUi%fS t XS Ee ` X VeS X D E p)fb&bUUiUS t pm%mdibbd5uH i &r5|0yR XS H g H c X V V X I ` X `S Q V a HS UdE w H Q { ` He v S #q}qYU5 H `e V HS R Q V H R Q ` D Q R ` yf|U}#yuy5yybV t ##vqbd7h H Q ` H GR E D H v )CX V H `e 0yf|V V `e ` H I H ` H H bfyFV0fS YT S X )E &TH w H G % G S H R Q ` X H G Ve z I R E G E V y#b&5B!yU5#y)!dyUe #Q 8 g R Q Iu H E Ie g 5dE7wyV)f%)E f#o #mbUipf)y)FUviX n H Re XSS E R E D HS R V X g H ge ce R a ` ae R X 5i##UdUbuyFUbi&v t H ##Q `e H G Ve z I R E G E V e V X Q { Q ` V fdE )!|p##bd!dype t |)5q#}|H H Q { V ER` E S X D #q!5)yb))iEUS t eb##dX%dYypiYy#%%FU#!dY#Q Vh `e g g q Q X H D Ve Xh a R I H g H q HS I G E D D H v X HS X D E 0dCfdY%US t FYUi&f5)0##Q H D HS X ve Q { a H V ` R Hh X R ` H v q Q X D Ve Xh a R I H g H q HS I G E V ` H s E V HS H v R H V a R H Q { R HR #qYby` `e fdE SiUS t r|7ViryD0dCX %dHFbpiFb#%%FU#!dFD #Q X D E `e fdE g#iY%US t rF%)UsdbYyb#I q Q X D E t G H g XS D R Q I ` Q S V H X D )E!T%PUHF15dE!%yVY0fSdY%EUS t FDYUi&f5)00f1i5)fb&bUUi%fpW|bd5)fb5)CdyyFYW#q5)ybqd V H HS X ve Q { a V HS R X I Ee ` X VeS X D ES XS H V V X I Ee ` D E g X H ` ` H D H Q { V E R ` E D X H H S Q { Ve Q I V H G Ve z I R E G E V e V X Q { V HS ` H V H G Ve z I R E Ge I a V HS H v R H V a R `)#qFv`|VjUe )bp##~0!|p##bd!dype t |)5qvUvF0yp##bd!f5Y0fdbYyb#I ` H T FgUH S VYypfdY%fSrd510dUyFib#Vb#bD5by`|V q%d!qY!dUy&pYbrYUfU#8 %w0H H a VeS X D E E V H ve ` D H E R I H R Q ` Q R Q X ` H G H ve ` XS H R HSSe Q H g `V Ee ` D E g )fb5)CdX i 78ggY#! s Y yY u f1fuC58 fF Yrif BFfgvByF&
otr (K ) otr
Y FC Yy vU& U Ff& f f 5 d~
% Y & 7 5 f u F j1Y% (i# BYC5p5 F&( Yy g FC Yy gFy Fd Yrif5 ) vf8Yv Yf8 g F B ~|d31~) 4 A 2 0
atr : Compl(PreShv(V , ))
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
otr
// I
X Hi (K (X)) XV
atr
X B B ^ Z an X an B Y an
(atr , otr )
Latr : RigDAeff (B, )
B H-i (X, K ) P (B1 , )
K I
X an
V
Hi (I (X)) (V , )
// RigDMeff (B, )
// Compl(PreStr(V , )) (B1 , )
iZ H-i (X, otr (K)) Hi (K (X)) Hi (I (X))
Rig : Sm/B
Rig : Cor(B) B
` YH
V H V VeS V X G a z D 0|bUUdBY5ryV t
F
homPreStr(Sm/B,) (F, Rig (-)) (X, a) X f CorB (Y, X) Z
Rig : PreShv (Sm/B, )
Rig : PreStr(Sm/B, )
Sm/B
Sm/B
Cor(B, ) = Cor(B) Z B a F (X) b = f (a)
Colim
Rig
// SmCor(B)
// SmRig/B
// PreShv (SmRig/B, )
Z X B Y
tr (X an )
G
SmRig/B
tr (X an )
R X i5I
R X I a D XS I G H R ` V d5mF)U#!yv|H
inc : AfndCor(B) RigCor(B)
Cor(B, )
Cor(B, )/F (Y, b) (X, a)
inc
&5p5
rYC5}& U rif &1%
C W8v"
y n u H Q c ES X X HS Q G R Eh H Q R X I a E )E )&%(m#d)fd5iuU#bd7#5i5u#5d%g n ) #(u
He t H D X g E I V H R R E U#5F5d#5)yby)FD V HS I Q E D V HS ` E V V ` H s E V HS ` E 0f5#dYUqdybF#d0fqd%g H` HV aR I HR E yq|0y#y)FD y n )i#(u H Q #q{ e ` H H |V YT T S #(u n
(X, O) ColimM L (X) M
inc
(F )(X) = Colim F (Spec(M )). L (X) L (X) Sm/B // PreStr(SmAfnd/B, ) Rig (G) = G Rig // SmRig/B // RigCor(B)
M L (X)
(X, O) = X u : u Y
5
)
7
(
h f d gef n #o #1!!YUUvi5d#Y}0H n H G G HS HS R X I ` D X q H ` V ` V H R Q H ` D Eh H G le Ve E R ` |#Yb5)7B!YU|pdbyTH i
// Str (SmAfnd/B, ) .
H De D R H q H H H a V Ve XS ` V H H v Q H R I XS ` E g ` H H `e Q YUYbYYYyypipv0md#y#p))%}Fub5|V X pSV5dX#gWayUeUUey`%QVdX5IXrRYHyVWH#Ue)7hFHb%db#q{ qi&U#|&bf#ybwUYbyq)%T%!qYyyF!bYu5dUyib%p|5)F1!Y&0H i V S V E ` Q E` e Q ` X ve Q V ` X `S Q V a R HS R H R ` E G a g H g ` H ` ` H G R H I V Ee ` X R a ge V E D V H G % G V ` X `S Q V a R HS &r5|0yuUdE w R Q H` D Eh H t #Yb5)7ufS 5# S dUF%yy)i0FbUdv#yd7%|5Ui0Yfe jqyp##I Fd#FH ae D E V V X Q X H D Ve Xh R Q H ` D Eh Q g e V e X ` V H H S Q { Ve Q ` D X q R Q H ` D Eh Q g H `e E R g ` e E g X ` V H S Q { Ve Q I V H `e GeS q Q X H ` Q G G E D R Q H ` D Eh R He G H R I #Yb5)7%Wb)y#sqUdC)|fe jqbU#510yf!fU%iy%1)F!#Yb5)7f!y#WH i
StrNis (Sm/B, ) // PreStr(Sm/B, )
inc Rig
dE )x #n )!YbdY5yfmqibUUUy% S y H G l R E a z ` HS ` X VeSe ` Q
R H` E D g 5QYyFD5d7h#Q5)E%!XgjrU#|pe z De v H V Q X H D Ve Xh Q ` V H Q { `e E #qv)dv i0FbUdW5v0H H Q { ` X H ` e X G V E V E I I #quqi5yqfdBdy)##Q R#QyVVr`|R7V5dbRbu0HdCXPdHYyVpih t #u|H Hh X ` D v Q X D e X Q `V V l R I H Q {e E X D H G Ve z I R E G E Ve Q H ` Q G G E Y0y#m#qp#d5iYm!|p##bd!)|p#y%1)FD
Rig G PreStr(SmAfnd/B, ) G
otr
aRR X D H ybifS H#q{vrR#fSUfdX jgyb`))E!GdUy`&DfeyqY%gpHb|`Ye8 R#YHbD5)7huUSmdvCX Q V Q H Se H R Ee X t ` H e ` H Q ` E H D H ` X t ` H e H H R le X G H G % G XS H g H R ` E G a qifeyq%gUbYU#iB!Y&Bp%1yb))!Y%g H ` H s X X ` H H X c ` V he e g yV1FHBUS#ipSi&vqY!GfSd)Ya|HVb`#g#dXPq#iXYDypeiXYay5IuHUR#)IqYH#gaFD0ayR5IwH!GGfS#gH#Q)dUi5dX i 8 8&%x Q H V h R V S Q E ` H Q c ES X @ 9 $ " H G G HS Q g H v Q H R I XS H e G R H ` e D H )!f%md5Yb#!Um#U!yy8Ye h f d gef
M L (U )
#Q 5Q n )f)P)fby)db#pi5dE ybUUm%#B&X t Ee `e V E I E R I XS R X I X H V VeS H g E R Q E I Ve X H V VeS X G a z D #dpUd#dybUU!#rbV t D H )vCX E ` X S YHB)feb&5QUiX&vYalb#iXvHU#U|rY)f0H #qyb))!}%p#|gfe yf!fUdY!#iqy%1)FD V R I S se V R H v e ` V H Q { R H R ` E G H g ` Q V S V H `e GeS E D q Q X ` H ` Q G G E ` YH H!!GdEe%`D)X#qFH~`qYH!YbUeipFDe|VH #q{bU5Q#IvH!!YUB%vd#y5uUrYd#rdF0eqdU#U|rY)qU~|H #qyb))!k5dUUi~5)E G G H R XS ` H Q Ve G G HS Q g H v Q H R I XS X R H v l z D X e D H HS se V R H v e ` V H Q { R H R ` E G V ESS X V Q H S H R Q ` X `V H S Q { ` H V ` R HhV X R ` D H v X q Q X H D Ve Xh a R I H g H G Ve z I R E G Q `V 8Yb#yi5v0jUe jqvFpr|7iry}dCT#iYypiYy5u%uyp##bd!#0H V X 5i5g Ee ` D H V H Q ` H V Ve t a dUy0|u#5qYybU#F%g H#Qq{fe)EdvH%gv`&Xp%!!fp|g R e g a G Ge ` V H S Ee V QS D S H g `e Q g a dUy5fY5Ue %uf#%#g R Q E 5dI R X I a` HV aR I H d5yqYyb#yR H!G|pe#z#Ibd!UYH V R E G HS ` ` H H V VeS H g E Yv|bpf% 5!iX t
r : (F )
#Q
HS I Q E D f#5dFH i `e H ` X ve Q V H R le X G XS H fdE dyqdCU#y}bYU#iBmp}%g
ayRbRiXD}UYb%!)FqibUdh HS R H ` Q G G E D ` X Ve X ` FH H g H V V XS D XS HS X c a ` V #pybdpFupmpUiqY|H #qv|q5#|B% H Q { HV E I I QV q Q H g HR` #Y%BbyqYH ` H G aS a Q ` H V H V VeS V H g E )!YUY#FybUU}0% #B&X t
V R ES X H ` Ve q H S rdUi}bp%Fd e X ` R H D Q R Q E Uir|FT5#)I ` H G aS a Q Re t a g V ESS X V Q qY!fB#~U#5F#k5dUfd~5)E He t H D X g E I V H R R E D H Q H a E g V E V E I I U#5F5d#5)yby)FB##Y5#d%udy)##Q
M0 L (X)
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
(F )(X)
RigDM(k)
CorB (Y, X) Z B (a) (F )(Y ) a N0 L (Y ) 0 CorB (Spec(N0 ), Spec(M0 ))
Spec((Y, O))
(a0 ) 0
Q X H D Ve Xh a R I i0FbUdFb##Q HR` E `V byi5|H ) ` FH
H G Ve z I R E G Q `e Q R ` V E D !yU5#y)!5v#by|5)FdE R Q #yV V ` R Hh V X R ` D H v X Q X H D Ve Xh a R I Q a E g ` X byY7|5dbb}0dCiYypiYy555#d%}qdb` R QV V `R HhV XR` D H v 5|r|75dbbv0dCX H g `e Xh ` H Ee `e V E I G E D XS HS se ` X I G E D ` V H V He t V H D X g E I V H R R E D V H g Ee ` D X H ` ` H %vfiYTdUyfy))F!p1mU#fb&5!)F}|pU#5~Y5i#d|0ybdY%dUyFduyyFe ` H G aS qY!YUYa H g qe E z D V H g H ` X g H I a g e ` V %fd#r#pbqi#5%5Uk0H V X 5i5g H g H V V XS D XS H Q { ` H G H ` Xe g a G Ge H te R a v %|bdpFmpe5)~qY!y&p%0!UebY)1
(a)
X
r
inc Rig B
U B (Spec(M0 ), s0 ) Cor(B, )/F
SmAfnd/B
r(s)
Colim CorB (Spec(M ), X)
r
s (F )(U ) s0 F (Spec(M0 )) M0 L (U ) // Spec(M0 )an M0 (U, O) U Rig (F )(U ) = Colim(X,a)Cor(B,)/F CorB (U, X an ) Z F r F = tr (X) U B
Rig : Str (SmRig/B, )
PreStr(Sm/B, )
inc Rig
Spec(N0 )
inc Rig
F
CorB (Spec((U, O)), X)
Sm/B
(F )(Y )
otr
0
// inc Rig (F )
// PreShv (SmAfnd/B, )
// Spec((X, O))
// PreShv (Sm/B, )
// Spec(M0 ).
// StrNis (Sm/B, ) .
SmRig/B
inc Rig
CorB (U, X an ).
A 0 a0 B0 (F )
a a0 F (Spec(M0 ))
atr Rig (F )
H a V E I G E D XS ` V H a D E Yy))F!pp|uY5d5Ya (S H%gv#Qyd7mUS R H` D Eh H V He R E c a ` X D H g H D HS X ve Q { S ES Q g E V He t V H `e GeS q Q X H ` Q G G E UbddyiY%Y5YUi&U#qH (!U#%%5U#5p0yf!fUT%dy%1)FD H { R H R ` E G H g `e c V S V H `e GeS E D q Q X H ` Q G G E D S H z D Q X c ` e E g X Q ` V #Qq!ybqd!%w)iX YvY0yfUUdFr%ib%!dYvfe #r5d)Tqf)Cd#B|H H G G E !!de 78ggY#! bF Fe ee~~ R H Vb#Qy`FD5d7h0F%r|75dbbe5ibb)dUbi&p0%v)!YUi)m|qyU#)E %5yqYYU5qUdC)vFFD E V H D H g V ` R Hh V X R ` V X V V H ` Xe R X v V H g ` H G HS X c a H V E I Ve g Q g H ` H He ` e E g X ` H HR E D H H` E y)F5bi# Q X H D Ve X i0FbUdh t UpfYybdX HS He D E V V V ` R Hh V X R ` D H v X z De v H V byY7|5dbb0dCwrpY#yUe iYypi!5m#q#r5d)uqUdC)w#%yb)#yd7e Q X H D Ve Xh Q e Q { H z D Q X c ` e E g X Q H g l V V E I R Q H ` D Eh H
Rig
Rig
Rig : StrNis (Sm/B, )
// PreStr(SmAfnd/B, )
Str (SmAfnd/B, )
// Str (SmRig/B, )
inc Rig F
atr
atr Nis
Str (SmRig/B)
Rig
Yd
5 % pU& U &( Yy gey !fd! rd ' fF)mFfF(B1d 5 & y Y!)w i)& f & |v58P H g ` D H Re g R Q H ` D Xh H Q { Ve Q I ` X R s t E D ` H G H ve ` D H E R I ` H s E #}bU%v#Yb)m#qyp##pqir#5iYpqdUyFib#vY%d#Q ` |VH % a e b`#5QwdUy`FyVpSH%gBd)$ #)FU0U#fyb)BdU|57#dFm##BUe YV Ee D H X Q X E E D HS HS se V V E I Ee V Qh E D H Q X $ S Q E d7u H` E yd#w % 5 F E `e t w 0qo )fdUeyf#5Fa 5g6 &&!f&) !uFfY' 0!1d&qi BU&d !mb ( E % fY' 0B)1i)& 8 8" ` X ve Q V H G G HS HS X qi&U#|!!YUuU1 R X I a E g ` V H a `e Q ` H s E d5u##)%v0uyf##vF#dv
T MT (V , ) (WB1 - -st , Cof proj , FibB1 - -st-proj ) TB p Rig A1 A Nis
1
Ee `e t a g R X I X dUyf#5Y%vi5dE YH ` V `R HhV XR` D H v r|75dbbv0dCX q%QiX0HFDbUedXhFabR#IPq#QYH%gu0a##dE#1`qib` V V g X R ` H {e ` a V HS X g E E G He R E c a ` X D H Q ` V He R E c a ` X D UbddyiYX i dHbfedXbfe##Q`YH5Q)pbRyF!G$ Ui# 5d!1UbddyiYw##|H 5 5 fm 1f& f &( Y mb (BfY1 Y!fm f& f &( Y my 5 (!U 1U Y1 FBfm u&( Y 5 y (mU 7 Y rFd i)& f Frif1# q f 5 d
F & }Fy
5
FdF
%
n dn #(u % F0Ybp) F !YC5d ~|d31~) 2 0 qdn%%uHyUeiXpUSdbRdEFD%g}qYH!GHy`&pe%0!Uuf5dF0%p5dyb%g t YW)fby)db#X i n o n R S E Q ` X g a G Ge HS Q E D a g V Q E V V H e D Ee `e V E I E R I aR` E G a g `V H H G lR E a z` dbyq)%v0my)Y#bmH i R#yV Q H g Ee ` D H E R I XS ` V #WdUyFdy#p|H 0dCX D H v ` FH Ee ` X DeS Ie `S Q G )fb&UU#Uyf#WX i dUi%fS t HS X D E H D HS X ve Q { a H FYUi&f5)0B##Q
0p B1 B Rig : Compl(StrNis (Sm/B, )) X LRig (A1 )an X ^ X B1 (A1 )an k B X an k // (A1 )an X tr (X) p
h f d gef #0y5U#dbi!d5z t HVR H v S `V |rY)fe (|H H Q { HR` E G e D H #qbyq)8Ye5 H g V l R I He I E ` E G E H g a `e ` H g S H R ` H He I E ` E G E %Bbb)%pe !yb)1U#)yi!)#z t H Q `e Q g ##f#%5Ue
// tr (A1 ) X tr (X an ) B1 B1 // tr ((A1 )an ) X B1 0
`V |H H Q { R H R ` E G H g `e c V S V H G R H ` V H R ` Q g #qybqd!%!fddX fe !yy1by#iX 8r S H V VeS X G a z D yypUBY#rbV t ` Q E ` R Q E I HS X D E %db#dWUi%fS t H D HS X ve Q { a H Q R Q F5fd&f#q0##v5|V H G Ve z I R E G HS He E v !|p##bd!uUuf)dYH H Q { R H R ` E G H ` V H R S R H e G R H ` R Q #qvYbyq)!b0yfe #f!bYb#dE n n o n H Re XSS E R E D Q g HS Q E D a g e D H H G Ve z I R E G E V e V X Q { Q ` V qd%%uyUipUdbdF%U#dY%jYe#d!yU#5y))|pe t yd5q#v|H H Q { R H te R a v H g ` Q V S HS X D E #q5bY)1#m!%yUe UiY%US t F5fd&f#q0##m|H H D HS X ve Q { a H Q ` V H Q { R HR` E G R Q #qybqd!5dE bqir#5dFD V` XR s t E ` H G H ve ` D H E R qY!dUy0ib#I R Q V V ` R Hh V X R ` D H v X q Q X H D Ve Xh a R I H g V H q HS I G E D H R ` H HS X D E #ykbyY7|5irye0dC1%dYypiYy#}%~%Yf#!)FvbyqYpUiY%US t H D HS X ve Q { a H F5fd&f#q0v##Q `e fdE V H S X D E t Y5YUi&U#q%#y}Ui%fS t V H D HS X ve Q { a V H g R Q V V HS X D E V H D HS X ve Q { a V HS He E v FYUi&f5)0}0ff)dYH H z D Q X c a ve R a g R Q H ` D Eh HS H Q { R H R ` E 5r5iqB1)fbY%m#yd7U5)Ybyqd!G UiY%US V H R Q ` D Q R ` V q Q X ` H G H ve ` XS H R H z D Q X c b#b5by|#iTqY!Y)fb&pYb#r5d) fUU#%0H HSSe Q 8 H g ` V H Q { R HR` E G R Q #qTYbyq)W#dE H g ` Q V S HS X D E #!%yUe fdY%fS t YfdY%US t ` H HS X D E V`R HhV X R byY7|5iry` D H v q Q X H D Ve Xh a R I H g V H q HS I G E D H R ` H V Ee ` D H R Q V V HS ` H V H G Ve z I R E G E V e V X Q { V HS ` H G H Re XS D H v R H V a R )CX %i0FbUdFb#%B0Yf5)FbyqY!5)fb0#yw0fwY0yp##bd!dyUe t |)5qwUB)!YbfdUY)y|0y#I ` H T H FgU S V H a VeS X D ES E V H ve ` D H E R I V H R Q ` D Q R ` V q Q X ` H G H ve ` XS H R HSSe Q 8 H g R Q H ` D Eh Q ` V YypfdY%fd5e0dUyFib#p0y#b5by|T%dqdUyiUyYUfU#%v#yF5d7#|H R Q H` D Eh 5YyF5d7H i 78ggY#!
1U 1fi f fu f& f
F & }Fy
5
% Y !F& #pi F5 &5 ! 5 &Ydr Fed !fg C & Ybey 5 (!U Yy y 5 rBfF1 YBU f& f C Yyy rBfF1 YBU & Ybey 5 (!U f& f !fY YBfm uFC Yyy (mU 5 7 Y rFd i)& f Frif1# q f 5 d FdF %
Rig : Compl(PreStr(Sm/B, ))
F
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
(Rig , Rig ) : ShvNis (Sm/B)
// Shv (SmRig/B)
%
n 0 #(u % F~0Frp1) F F&#) W8~j B ` X ve Q V ` X `S Q V a R HS R H R ` E G a g ` X H ` e X G ` Q H I qi&f5|v&r5|0yuUpYbyqd!Y#qd5Yb)UiB}#YW ypr F Fef 0 C5bdBBrp%#i! ( iY)!r F})Wfg 8"
Rig
(A1 )an X
f: A
Rig
// B
PreStr(V , ) F G
RigDM (B, ) = HoB1 - -st (MT (SmRig/B, )) = MT (SmRig/B, )[(WB1 - -st )-1 ]. tr (idB ) Rig (f ) Rig (Rig , Rig ) : DMeff (B, ) Nis F tr G = LRig B
U Cor(V )/F, V Cor(V )/G
(Compl(PreStr(V , )), WB1 - , Cof proj , FibB1 - -proj )
k
Colim
tr (Gman ) B
SpectT (Compl(PreStr(V , ))) B1 // Compl(Str (SmRig/B, )) // RigDMeff (B, ) . ^ tr (U B V ). Nis (B1 , ) (B1 , )
0
// Compl(PreStr(SmRig/B, ))
Nis
tr (idB )
a Rig (f )
Sm/B
// tr (Gman ) B
a Rig (aNis f )
HSSe Q H g Ee ` D E g X H g H V E I Ve g YUfU#8 %dUyF5dC)##Q 8u|qyp%g R Q V HS X D E #yUiY%US t H R Q ` D Q R ` V XS H g H `e Q y5yyb!pu%yf#%g t %BdUyFdy#Bb#b5by|yw#g a g H ve ` D H E R I H R Q ` D Q R ` V X V H He R E c a ` X D XS V E V Ve Q Uy)dYb&Um5)|bp## 78ggY#! % fY&Fe f&& b F )Y!!p5&BU r&f % % TYF7 b( && r p&F(y1)sbf}& YYrpsddY&& ig A 8"
SingB (-)
ddY && U) (i!1fmg % YrpT!Y# Y % W &Bf&& 5 &) BY f (W F 5 Y % ( &( Yy &5 UF)5 7 & ( Yygvy rFBd TUvFWfYdrBf& f g}g rFBdw Yfv y Yrif!# q U 5 dF Fp C W8"
X Rig : MT (Sm/B, ) B A
1
`qY!G#q{U#5QuHyi5W)E Fibyd!fdy%g H H Q e ` E H D X ` R E I Ge E V H EV X dyUedbRB S jYUSfU#Q8 #gk5dEUyFD5dECg)~VYDYHbpUS%gBfe)|vHUkVr#QYbDU1q#iXbR|bUedUS!q%in #o #n}!GYbdEY5zy!%gYH0f#o #!)fby)db#I H Se H V e ` X H R ` Ve H E V S R H ` HS Q X H V V X H l R a ` Q ` y n Ee `e V E I E R X H H Q X pSB%g5zrD5iqcWYUSUfe#86#gubR#HyFD5dE7V0%guUS#iXrV5)feybRdvm0fSqYUy`#dEdE C)#x 5dUyfe|qI)yR j6 B`qibUUUy% S H S Q H V Q ` h H V H s ` V E V H V H ` He s o Ee ` V E E X VeSe ` Q 0fYrF&
Colim homRigDMeff (B) (T n A, T n B)
nN
)YB!p%#&Bu
(B1 , )
dYYd )
Y
&8
4 ~|d31~) 2 0
V R ES X ` Q H I V HS s X ` V V HS X D E rdUi#YgY0f#db|UiUS t
` X ve Q V ` X `S Q V a R HS Re H ` s E R Q E I H G l R E a )d&f#yv&r#ybmfU#y%)v#dgF# #x 5m!Ybd#z g6 uY#qpf5#iX R H Q {eS I I V H ve ` D H E R I V H R Q ` D Q R ` V q Q X ` H G H ve ` XS H R H z D Q X c HSSe Q H g H D HS X ve Q { a H Q ` V 0dUy0ib#pb#b5by|%dv)!YdUyiUym#ri)!fUf58 %F5fd&f#q0u##}|H
(- 1 ) : SpectT (Compl(PreStr(SmRig/B, )))
yYd
R Q H ` D Eh H y 5YyF5d7mUS i&#x 5 H!Ybda#z jg6 pRdX )#zr5iXq}!YUSUf#Q8 H%gkR#QHy`F5d71#k0H G lR E H D Q c H Se D Eh Q `V H Q { `e Xh Q g ` H H Q {e R s a cS X He R ` a G E a c H H ` X g E I V H R R E #q~d%~Ye#qpy5YdUifbyY!dY)1b)d#5)yby)FD a ` a R I E I S H g H I G H X HS E D g D H bFUey#)yR5!XU#fS5H%qFvRd5IU#QdYa%eFej V X 5i5g R Q V a `e ` H g S R X I H R l I #y1yfyq%Ue vi5ubYdE Ee ` X ` Q G R H dUyib%1yI X V Q HSS X US bR5YUfUeivRdX V5iX5yqdbb50yf!fUdYW%ib%!)FD g V H ` X R `S t V H `e GeS E D q Q X H ` Q G G E R Q H` D E #yF5d7h HS H Q `e g a UP5){f#Q%0%g} a` yfe##QdUy0|rpP%g Ee ` D H V XS H Q X E E Q X ` HS X ve Q { d)$ #)FD i)fd&f#q0a t `V |H ` H s Y%dE i % yYi f% )YYd !& Yrpv)ivY&& p &B # ~|8P FB&" ey 5( fjF H G lR E a 7 x 5x my)Y#z j6 !%U#)F#jfu!!dYu H g HS Q E D a g Se H G G E D
// RigDM (B) . Sus0 (T ) T
V X g HS se V R H v e ` H s E Q ` V 5d#U#U|rY)fvY%d#|H
V QS I 5f5uH w
S X g E E G aS Q c Xe R i5#d!mf5d#dUby` R Q H ` D Eh Q D E g X S X g E E G ` H G HS X c a ` V H S V H ve ` D H E R I V H R Q ` D Q R ` V q Q X ` H G H ve ` XS H R H z D Q X c HSSe Q 8 H g ` V #yF5d7#!d%xi# 5d!qY!YUiqY|8)fb0ib#mb#yF5by|%d)!YdUyiUy!#ri)xfUU#6%1|H R Q H` D Eh H #yF5d7e8 H Q {e R ` a G V H R ` D H I #qpybF!$ byF|V t fuUF%yy)X HS He D E V V V ` R Hh V X R ` D H v X q Q X H D Ve Xh a R I H g H q HS I G E D Q e Q r|75dbbm)C%dYypiYy#1%%FU#!dYs#T#q{
Sus0 : Compl(PreStr(V , )) T K T // MT (SmRig/B, ) (T
tr
!Yfi) bF&
HU#5%UedUe|5yV5yV%gv#Hy`Dd7h%gu|qEyp%gdH#q{pyRbF!$ VmfdX# E#)EGYUS#)c#ipeyb`}UbddyiY##v|H e t E V H I Q H R Q E Q H V I Ve Q e ` a G HS g H a Q X R He R E c a ` X D H Q ` V H#Qq{p)fv8)i&#x 5}H!ly)Ya#z g6 mi5I#qUebyYaG$ VUiX# 5dE!GvUyR)dcYb&XH#5Q}bRdEFDYp0H ` He S G R E R X H Q { R` HS g E He E a ` D H H `V He R E c a ` X D fbd)Yb&Y1X i 0&q i Y 7 F' f&& b r u Yr&
MT (V , ) = Nis
) gb&B&5x&q&
f
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
B
RigDM (B, )
RigDM(B, )
T B1 homRigDMeff (B) (T, -)
(123) 3
MT (V , )
RigDM(k)
RigDM(B, )
Sus0 : RigDMeff (B) T
A
B
B
T qc
RigDM (B, )
T
M(X)
Rig : DM(B, ) X an
tr
3
Rig
RigDMeff (B)
RigDMeff (B)
// MT (SmRig/B, )
X
// homRigDM (B) (Sus0 (A), Sus0 (B)) T T
// Spect (Compl(PreStr(SmRig/B, ))) T
// tr (B1 ) tr (idB ) B Compl(PreStr(SmRig/B, ))
T
// RigDM (B, )
n
tr K)nN
Sus0 (tr (X)) T
(B1 , )
k - {0}
` X ve Q V H G G HS HS X n n Ee `e t a g XS V X g `e Q g E R ` e a ` a 0)d&f#y!!YUU&5 %%dUyf#5Y%Bpp5i#vf#%%byqfubFX R Q H` D Eh #yd7H i He t e Ee V H I V Q V H g R Q H ` D Eh Q g a `eS a g t H e HS I XS H `e Q g X R U#5%UdU|5y5yu%}#yd7W%yff%m#UYU#pv5%dry` H e { Ee ` D t S I Ge V H g a ` ae R I E R I XS ` H H Q {e I E ` E G E z H R Q ` D Q R ` yVj#QqWdUyiXYfeU#!fyu%bFUy#)y5pF#qU#)yd)#}b#b5by|V t #5Q 8mFYy|pYH S #qp##rb~dUyf%5dYe0#UibyYYD t 5)|V H g H D H ` Ve q V H Q {e z D H ` V Ee `e g E D V H e X ` R H V Q E XrYHbyq)G R R` E ` YH V He R E c a ` X D V HS R He g Q ` a R Q E I n Ee ` D H V XS H g V ` X `S Q V a R V HS X R H VeSe ` Q UbddyiYUYU%5bFp#) #)fb0|!p#r&r#ybpU1b|pfUy#W H q t HS Xe ve R E X R H V E I I Q V Q { H He g a Ge z D R E Ee ` X QS X v H Q R Q E I ` HS I G E D V I R E D 5!)dUipqUby` t d#mrYy)##ywdE jqv##f%0!f#rriX t )#!dUy&fdC}##5dFU#!dY~5bdFW#Q fdE `e
I 1 G7 E E 1 $ $ B @ 9 4 7 1 4 4 1 ) $ # $ # F"2H!AFF2D5&%CA%&8&65!320&('&%!"!
SmRig/B
k
(p# , p ) : HOM(op , Compl(PreStr(SmRig/B, )))
1
(B , )
1
B
HOM(op , Compl(PreStr(SmRig/B, ))) Compl(PreStr(SmRig/B, )) k K
// Sing (K)
t
B1
Sus0 : RigDMeff (k, ) T
K
RigDM (k, )
B1
// Compl(PreStr(SmRig/B, ))
eff
RigDM(k, )
// RigDM(k, )
H Q { ` H G H ` Xe g a G Ge `e Q g a g #q}qY!Yb&p%!!fvf#%0%
H Q { V `e ` H I V QS I V a R c H g H He c ES E G E S R Q V V H G Ve z I R E G E Ve V H g `e Q g 5)}bfyF5U#bd%Yf)dUd!d5z #y0!|p##bd!dypp%f#%fe
(V, (n+1 SingB (tr (X an ))) -f ib )
H Ve I R E Gyp#z#bd!GmfS #qyfyYv5f5}UddUd!)#dY)fy5U%g H H Q { H `e ` H I V QS I He c ES E G E z E D Ee V H Ge H ` X %gB)db`Ya HbyVTfeyFIH i dUyV5YH!U%g%gHybUUST%g 5!iX t `e ` ` H Ee Ge H V Ve H E 5Q ` e)E V a Q { ER` V HV H 0#q5dby10|%g rbyYH7|V5iXrRy`D0HdvCXuq%QiX0FDbUdXFbR#0H%gHadUeya%~Hfbd)cYb&XY}XUV5i5g}H#Qqp#IdbiE!d5zebe!GUfS}U~!!dYe%FU#!dYeYyUbF%Y V`R h H Ve h a I V v R g eR E a` D S X u {e E ` G E H ` e XS H G G E D H q HS I G E D H D H Re R D a ` Q H I HS X D E )fdY%US t H Q { R H R ` E G H g `e c V S H V VeS X G a z D 5)Ybyqd!u%}fddX #)|bpf!BY5ryV t
#Q D H v X E R H a R c H g H a D XS )C!bY9y)Y%YuYdp#I Q X H D Ve Xh a R I H i0FbUdFb#uUS R Q E I a D E S H g He ` R X I H R le G H R I XS `eS s X ` a E e D #)uY5d5Ya (u%UyyibYU!Yb#!pvU#irFdF & R HeS Q De ` R X I V X D H D S X R a a 0YUf#YUb|d5pduFB8irY#dc dE V)Ys%PdUy%b#5|!YUyU)ybBX i dyypUBBY#rbV t X D Q g Ee ` D Q g a R H Q ` V H H G le Ve E R ` H V VeS X G a z D Q R X I HS s X ` H V a R I H R V ` R Hh V X R ` D H v X Q X H D Ve Xh a R I Q ` V #i5uU#dbqYyb#Ybvr|7iry)Ci0FbUdFb##0H RfU#Fpyyi)YmfpqYyfirybYU!Yb#w%%pH i Uyyi5epdbyw!y)Y#bw%d#y#1pyUU%WFY#x %%u!fmUeUd)7u HeS Q De ` R X I V X D HS ` H `e X R ` V H R le G H R I q Q H g V V He ` R X I Ve E R ` H H G l R E a z ` Q g H v Q H R I XS H Ve ve g o n H G G HS HS Re E v H Re E G a G R Q E I Ve G ` V H HS se Xh H D HS X ve Q { bf)!u#dup|BU#Ui1F5fd&f#q0a t H Q `V ##|H H Q { `e Xh #qfi!H i 78ggY#! % F(YF y Ci r piFygvdbf}& dBfi f f&& b1
^ Hom(tr (U k B), SingB (Rig (K))) Rig (K) K
igjbF0( & '
% s dFYd B1i % yfgvi fvYfFiFv fiC r !p%#&B! # % F!Yrp1d (YF y && r piFygp)!qbfgvi f pF y& f&q jeF u Y&#) 8~j w rF ` X ve Q V HS ` V H H z I X R c X R X I H D H g V ` X `S Q V a R V HS V Q E ` ` E R H V E I H R S H Q { HS R Q h HS D H G l R E a z ` qi&U#|U0}#5irdiriuFu%bibf#ybpU5dbqdbYy)Ybj#qU#|V FUFu!YbdY5yH i a R ` E G a g ` V H H G G HS dbyq)%v0u!fuH i
B1 B Bn B
h f d gef
R Q 5|V
H g He ` a z ` E G E z R X I Ee ` D %uUyY#yd)#iW)fbdX S
` H ge v a H G Ve z I R E G Q q%U!yU5#y)!#g R X I H c X G S g E IV HRR E D a D E S H d5mdiBUe T5)ybbdFuY5d#a u%g H G Ve z I R E G H V Ee ` X D te ` H ge V H D ES Q g E !yU#5y)1fS 5dUyiYfyqY#UYWU#%%!G 5U#1H w (d55}Y%ds#Y!%dX V QS I S X t ` H s E Q ` H G g H Q { Ve Q 5)bp##I X H G % G PdE !Y&H w
K
9byY7|5iry` V`R HhV X R D)CX q%i0HFDbUedXFb#I%gT%qFU#!dEYTH#YadUbY%gUy)EdYb`&DUBViX#H#qpe#d5iXYH!yVU5z#y)!dyU#X )E ri#5db|W&r5|0yR H v Q X V h a R H V H HS I G D V g H ve R a He R c a X XS g Q { E D G e I R E G E Ve Q g R X g X ` V ` X `S Q V a R #QiX sUFD%yy)XPU#I!fyPH%Yf#I!)FDfV0H ae E V V HS Ge V q HS G E HS ` H Q V ES I I #q{WdUYH#iX H q HS I G E Yf5)FD @ HS He E V UBfYD%|bVdX Q { ` R Hh X R D H v q Q H D Ve h a R H V q HS G D V g e R E a ` X X pe#qVbyY7|V5dbb`!)CX%dXFbpiXFb#IW%g0HYf5I)EFH%wHUbddcyiYDpSViX#gipFpf#!U|!F%)E S S Xe DeS I Ge V ` H s V E` dydE ` Q E` R Q E %db#)I ` H s E Q e Q { R Q H` D Eh H Y%dw5g#q#yF5d7}fS D H v 0dCX
H Q { S H #qgy` )H e
p p ()
R Q HS X v H 5YUi&#g
` X ` V E D ` H G HS Xe DeS I Ge V ` H s S He D E V V qib|5)FqfdUYUU#!U|pF%)E vUF%|bdX
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
M (n) =
nN M
^ hom(n k B, K rig
Lp# p (K)
Lp#
)
SingB (K)
1
Sing (K)
Lp# M .
B1
p K
B1
v
V E` 5dbiE b#vU#dbi!d#z t V l R I He I E ` E G E ` Q E ` R Q E I HS se Xh H D HS X ve Q { %dbB#dsU#UiY5YUi&U#qa t H Q `V 5#T|H
// M
H Q { R H R ` E G H g ` Q V S H R QS D E D R Q #qwyb))!s%W!%yUe y5fY5dYT#dE
// hom(Bn , K ) B
H V R H v e Q ` H G g v|rY)qUw#F!%)X
K
a: K
Bn B
// M (n)
B1
b : hom(Bn , K ) B
B1
HR` byqYH
// K
He I E ` E G E f5dyd!d#z t
H Q V R Q HSSe X R X I X 5#b#fUUii5
H Q { Re XS D ` V H S #qfdUYv08%
a `e ` H g S ` bb)%pe vFH H g HSS Q Ee ` D H V XS R X I `e Q g e H G Ve z I R E G H %uUf#WdUy0|!ppi5vf#%5Uyp##bd!mfS H Q { XR HR` E #qurYbyqd!G
ab
(tr (B1 ) tr -, hom(B1 , -)) B B
H G Ve z I R E G Q !yU#5y)%g
Ee ` D E g S Xe v H `e Q g a dUyF5dC)X Uub##%g
m
h : tr (B1 ) tr hom(Bn , K ) B B
hom(Bn , K ) B
k
hom(m,-)
// hom(B1 B Bn , K ) , ^ B B
b a = id
// hom(Bn , K ) B
Q
M
p ()(n) =
n
n N
h
B
%
%
D H v 0dCX
k
K
Rig (K)
// SingB1 (Rig (K))
B1
Sm/B
SingB (Rig (K))
Ho (Compl(PreStr(SmRig/B, )))
1
D(Str (SmRig/B, )),
1
% f}Yy
d
Y&#F1Fd (F& yYe BF
f}Fb
&8 % d bF ! Bf d U& U
B1 (B1 , )
Rig (K) U1 , . . . , Ur (A1 )an k
!p5&Bj
Fd
1
B1
// SingB1 (Rig (K))
an
H R Q ` D Q R ` V XS H g H Q {e I E ` E G E z He R E c a ` X D XS V X g S X D E b#yF5ybBp%5)p#)yi!)#UbddyiYwp5d#i%fS t
tr (X) X
`V |H
B
n SingB (tr (X an ))
HolimnN n SingB (tr (X an ))
1
1
d
1
Sing (tr (X ))
V
B
(SingB (Rig (K))) -f ib
// HolimnN (n SingB1 (tr (X an ))) -f ib // Hom(tr (U k B), (SingB1 (Rig (K))) -f ib ) ^ K
B
// (V, (n SingB1 (tr (X an ))) -f ib )
1
B
^ ^ U = U 1 k . . . k Ur
n-d
B1
d
SingB (Rig (K))
(Et/V, )
1
H Q { VS H #qvpYb` V ` R Hh V X R ` D H v X q Q X H D Ve Xh a R byY7|5iry0dCT%dFbpiFb#I H g V H q HS I G E D V HS ` E V V ` H s E V HS ` E %W%FU#!dYUWqdyWrF%)T0fqd%g H g H aS Q c Xe R ` He R E c a ` X V Q E X %PYU#d#dUbyUbddyiYD t 5)|V US V E` 5dbiE & i#r#Yb)m%g b#)#uf5|uUfYddUyF5%5Uu##Q V X g VR Q H` D Xh H H R s G E HS R Q V HSe D Xh Ee ` D Q g e H R X I n H G l R E a z ` Q g ` X H ` e X G HS Q E D a g a z D R H z D H R ` X `S Q V a R d51n #!YbdY5y%}qiYyqUiBuU#dY%u#rrY#r0yvibf#ybuH i H Q { S H` ` 5)YbFH ` X H` E qiYyq)FD #y|bUUS R Q V H V Ve H a G R Hh ` H G HS X D ES H ge ce R a ` ae R X V Q E V H Q H ` Ve q H Se E V X D Q X R H H G X R H V R Q E I n H G G HS HS X R H VeSe ` Q u Y!bY7qY!YUiU%U)fb}bFUyd&v t 5dyu##y|U%Y8fWddd#Y!ir|v#) U5 %!!YUU1b|pfUy#Wg H H %g HR s t X b#5pS D H v 0dCX D H v X H te ` H g dCvb)%pe V H `e GeS E D X y n H G G HS HS R yf!fUdYmUS do #n #}!fUkiX H V VeS H g E ybUUm%#B&X t g H Q {e R a a c ` e E I #Q 8W#qpy#Y)B)Udr#Q D H v dCX R Q E 5dI ` YH H Q { R H R ` E G H g ` Q V S H R QS D E D R Q #q!Ybyqd!W%!!%yfe b#UF5)F!#dE bqY)yddYb5|BqY!YU#Ui%i0FbUdFb#%g V ` H c R H v E D R Q V ` H G HS se Xh q Q X H D Ve Xh a R I V H ` E qdyV H Q { `e X V #q!dy)E )x dU%dUyfy)db#spdX )f#)yd)#i5qdUbi&f!FqdbYd)Fr#|!qY!YU#Ui1byY7h o n Ee `e V E I E R I XS R He I E ` E G E z R X I ` Xe R X v e ` H ` H c R H v E D R Q V ` H G HS se Xh V ` R H t 5iry0dCi0FbUdFb#I 5T0H V X R ` D H v X Q X H D Ve Xh a R Q ` V n H G l R E a z ` HS R R Q E I S Q ` #d#WFH R Q E #)I H z I R E G E Ve ` V s##bd!)|pW|H YU5 %!YbdY5yUTdX
v ^ k a (F ) T K ^ ColimW Flt(v) Hi (U k W, F ) = 0 ra v i = 0 r ^ ColimW Flt(v) Hi (U k W, F ) = 0 ra Gi : V SmAfnd/B mC,v = 0 i > 0 U D(PreStr(Sm/B, )) ^ Hi (U k V, F ) i > 0 Gi = G i ra Nis ^ ColimW Flt(v) H0 (U k W, F ) ra v M(V ) F ^ ^ Hi (U k k(v), Fv ) ^ ra Fv ^
H Q { R H R ` E G H l G X R V Q E V ` R Hh V X R ` D H v X q Q X H D Ve Xh a R I H g V H q HS I G E D H g V a Q c e ` Ve g V HS c Xe R #qYbyqd!u#Y!dbp5)# byY7|5iry})CT%i0FbUdFb#u%p%FU#!dY%5d#Uy|U#Ud#dUby` V0fw5i#g HS V X H %g a ` ae R I E R I XS ` X VeSe ` bFUy5db#UW)dypUfb%Q UUFdY5Yby5Fb#}dUiUS t F5fdCU#qs##w|!dx #(x#qB5)ybqd HSe D Xh H D H R R Q D a R H HS X D E H D HS X ve Q { a H Q ` V H nu H Q { V E R ` E V HS X D E Y0fdY%US t H D HS X ve Q { a V HS H v R H V a R I HS X D E FYUi&f5)0UBdbYyb#WUi%fS t b#yF5by|spyd#dbi5WYU#FpiT#qp#dbi!d#b!UfsU#q!fi%qd5Yb)UiBG H R Q ` D Q R ` V XS ` R E I I X R R X I H aS Q DS X D H Q {e I E ` E G E z H `e GeS XS H Q { `e Xh Q g ` X H ` e X HS Q E D a g H a `e X z Q E V Ee ` D Q g a R U#dY%syfi55dy)fb5#bX i V H 0%g fdY%fS t b#yF5by|UTyd#dbWi5I HS X D E H R Q ` D Q R ` V XS ` R E I I X R R X H aS Q DS X H Q {e I E ` E G E z H `e GeS XS ` V YU#YUdYD #qp#dbi!d5ByfUUTpu0H H Q { D E g `e Q g a g #q!5)%5%% H %g ` XR s qir#5t t qY!Fdp#!Yb#|H ` H G H D XS I G H R Q ` V H Q { H e X R ` H e D H HS X D E 5)#UiryqYF0e8)fdY%US t F5fd&f#q01#5|H H D HS X ve Q { a H Q ` V
Sing (tr (X ))
H Q { ` H G H ` Xe g a G Ge `e Q g a g #q}qY!Yb&p%!!fvf#%0%
H Q { V `e ` H I V QS I V a R c H g H He c ES E G E S R Q V V H G Ve z I R E G E Ve V H g `e Q g 5)}bfyF5U#bd%Yf)dUd!d5z #y0!|p##bd!dypp%f#%fe
^ (V, (hom(U k B, n+1 SingB (tr (X an )))) -f ib )
He ` R X I H R le G H R I XS V X g H G G E D `e Q g a g Uyyi5yf!Yb#}U5i5e!!dFkf#%#dE ` X Q {R X G HR qi)riBy S #qyfyY5U#)fy5U%p%p|bUU~% 5!iX t H Q { H `e ` H I V QS I Ee V H Ge g H g H V VeS H g E
// hom(U k B, (n SingB1 (tr (X an ))) -f ib ) ^
2 $ ' ' $ " &!q&%#105
RigDM(k)
SingB (tr (X an ))
1
V a Q { E R ` V H V H g H Q {e I E ` E G E z H `e Ge XS ` H G HS X c a ` V 0#q5dbyTy#P#qp#dbi!d#b!UfS UqY!YUiqY0H
HbR#Qy`FD5QbRy`|VpSuH%gmH#QqUe5IdEy`dE!Gd#fbd)Yb&Y!p5d#g X { E z He R E c a ` X D XS V X H Q { ` He v S HS X D E #qqYUm)fdY%US t
n SingB (tr (X an ))
V H q HS I G E D H g H G Ve z I R E G H 0%FU#!dF#m!yU5#y)!ufS H Q R H te a v H ` Q S Q { E R ` E 5){Y5fbRYd%g!%|V8Ue )V5dbyq)% R Q E I V R Q E Q E ` H G l R E a z ` Q g He ` R X I H g E D H V XS H V V X I #)er#d&)y}y)Y#bT#Uyyi5%5dYy1p|bdT & n ` n V H G l R E a z ` V H g V R ES X HS Q E D a g ` X `S Q V a R %YU5 %YHy 5 %v0!YbdY5y#rdUiuU#dY%vibf#|0yuH i dU#dbi!d#d5bqipbi&fqdyV He I E ` E G E z R X I V ` Xe R X v e ` E H R Q ` D Q R ` V XS H g H Q {e I E ` E G E z He R E c a ` X D XS V X He c E E G E E D H q Q X H Ve Xh a I V S H { H Re g ` Q X ve Q { a e D H HS X D E f)dUSd!d#z)F%gx%dFDbpiFbR#HU#QqbU%Pw%d&f#q0YeCfdY%US t y5yybpW%#qp#dbi!d#Tfbd)YyiYp1i#g Vr#diQdy7iX%fS t R Q E E`u S D E `V 0H He c ES E G E g Q X H D Ve Xh a R I HS H Q { R H R ` E G V R ES X H l G X R V Q E HSe D Xh Ee Ud)f)!d#z 8Bi0FbUdFb#f}#qYbyqd!mkrdUiv#!irk5d5vfUF)!dUy` t D5%fH#}dV#)#febp%kV0f)#iXpybVUV5dX#g %g yFUb#db#mpbY)%iY%US t Q g e a Q c ` Ve g HS c e R ` HS H a ` ae R I E R I XS H te R a v S X D E H R ` g a ` ae R I E R I X ybF% 8yYfb#db#mpS V R Q H S S e X R S X D E t `V |H r#fUfdiX iUS H R ` g a ` ae R I E R I ybF% 8yFUb#db#X i H Q R HR E H D #q{~Yby`qd!G%gp5)E%g!%|&Y0#qU#)yd)#k0yfUUki5U#ir|%dY%US t ` Q V S V H Q {e I E ` E G E z V H `e GeS R X I HS s X ` V ` V H S X D E HS X D E UiUS t H D HS X ve Q { a H Q ` V F5fdCU#q#5|H
$ 2 ' " " &0%d
1
Hi (-, F )
B1
nN V B 1 1 ^ ^ hom(U k B, n SingB (tr (X an ))) n hom(U k B, SingB (tr (X an )))
#Q
^ HoLimnN hom(U k B, n SingB (tr (X an )))
`e )E
` Q E ` R Q E I HS X D E #dy#)fdY%US t
^ hom(U k B, n SingB (tr (X an )))
B
1
1
1
1
B1
// HoLimnN (hom(U k B, n SingB1 (tr (X an )))) -f ib ^
// (V, (hom(U k B, n SingB1 (tr (X an )))) -f ib ) ^
K = tr (X)
n-d
1
H G G !% uH w H aS Q DS X H Q {e I E ` E G E z H `e GeS XS ` V f#YUdYD 5)p#)yi!)#!yfUUTp|H
V H HS X D E 0%g UiUS t b#b5by|!Uv|)5iri5I H R Q ` D Q R ` V XS ` R E I I X R R X V H GR H` V HR` Q g H S X D E t F5fdCU#q##u|H H D HS X ve Q { a H Q ` V 0!yyby%dX j S )fdY%fS
H Q { H v Q H R I XS H #qud#y51pu%g H Q #q{ H D HS X ve Q { a H Q ` V F5fdCU#qp#5|H n )x #(u
B1
an
HolimnN n SingB (tr (X an ))
F = Hn (SingB (tr (X an ))
^ hom(U k B, F )
^ hom(U k B, n SingB (tr (X an )))
1
^ hom(U k B, (n SingB (tr (X an ))) -f ib )
HoLimnN (n SingB (tr (X an ))) -f ib 1 ^ hom(U k B, (SingB (tr (X an ))) -f ib )
1
ae D E V V X Q X H D Ve X uUF%yy)dFbpih t
HS H Q { H Re g ` He v H R e D H Uu#quyU%!v)fyjYe
D H v X HS X D E 0dCufdY%US t
H D HS X ve Q { a H Q ` V FYUi&f5)0u##v|H
F ZV
^ ColimW Flt(v) F (U k W ) B V
SingB LRig (K)
1
B1
Ho (Compl(PreStr(SmRig/B, )))
V X g S X D E i#jdY%fS t
`V 0H
^ hom(U k B, SingB (tr (X an )))
// hom(U k B, (F ) -f ib ) ^ // HolimnN (n SingB1 (tr (X an ))) -f ib
1
1
1
2 3 1 F = Hn (SingB ((X an )))
Ho (Compl(PreStr(SmRig/B, )))
n SingB (tr (X an ))
B1
d
2
1
3
H He s E D X H De R ` X G X H V VeS ` V ##f5dF)|kFpby&BupS |bpf0H
Y = Spf
Ee ` X ` H V a R I H Q V E V Ve Ve E z )fb&r)|0y5u##5dybUyU)#e ` Q E` R Q E %db#)I H Q { HSS H #quUfy` D E g H `Ve q 5d%bp%FH S 8 R Q V HS s X R X I a V e ` H G HS X ` E ` ` H He t ` V 5|!f5iri5|5UuqY!YUiriyuYBf55m0H VR E b)fS H Q { HSS H ` HS s X R X I a V He t Ee V H ` q H H 5)wUUYy1U#iri5|wU#5PdUy5YbqY!##Q `e fdE H X S H GR Eh X G a z DV Q `V d5!ijY!bd71BY#rb#v0H H Q { R H V E I I Q V ` Q H I V H V VeS ` H V H X ` E #qpYy)##yp#YTdE yypfY0#B&))|V V HS H Q { a E g ` X 0f#qm##d#pqir` ` FH D H v 0dCX V HS R X Upi5I H a R H c H H aS Q c e R ` He R E c a ` X V Q E V XS V X g ` V Yb%g5d#uYU#d5fbyUy)dYb&D t 5dywp5d#|H H Q { ` H G H ` Xe g a G Ge `e Q g a g #qqyiU%0!Uf#%0%dE o n E e ` e V E I E R I X S V H U%%dUyf|q)y5BU1H w Y0#B&X VS H R Eh V G a z D V V H g R X pY!Gbd7k)XBY#rbe0%ed5I H g `R H v Q E e t ` H G HR v Q E D HR %|d#)#~qY!Yb#dYbw#Q `e fdE R Q V H V Ve #yybUUS H g S H G R Eh HS l g E G #Y!bd7}f%%W#Q `e fdE R X iI H g H c X G S V X g `V #m)iBUe p5i5}0H H5)RYbyqdE!%g15d#g!%|5 Q { HR` G H D E ` QV S H g ` D H Re g R Q H ` D Xh Q ` V %Fbf##y)#}|H H Q { HR` E G e D H #qbyq)F0e8 VR E rdUS `qY%peYm!|p##bd!%mYy)|iry1US yd#W)%pdUyyYfbi&v t H g v a H G Ve z I R E G Q g H a V E I V X R ` X H ` E H ge ce R a ` ae R X H Q H G G E D H a R a ge V E 5#m!)FmYbY%p|dFD H ge ce %pdUyR a ae R X X y`Yfbi&vmpS VdydE )%gpdUyRpyYfbi&v t #q)dbYdUyF5%0y#5d~F!%dX E` H e ce a ` ae R X H Q { ` X ` H Ee ` D Q g a R H E s ` H G g H Q { HSS H 5)vUUYy` HS s X R X I a V He f5iri5|vU#5t E V H` H H Q H`V q H S H q H dUe|YyqY##y|peY#&d%Y##dEYD ` X V E I I Q V H a `eS X R a a c XS V X I ` e H R ` V H R H qib)##yBYpyffdb#Y)Uk)5)UYby|b#!q R X d5I H g H c X G S V X g `V %dd!Ue (ki#|H H Q { ER` E G ` E `D #qV5dbyqd!pFHWdUeyQ%gabm##dp)!YUffb)ydqir|F!%)}yFd5!)FD t yd5qm%pdUyyFUbi&v t R H E s ` H G HSS He ` H ` E I ` X ` ` H G g X H ` D X I G E e V X Q { H ge ce R a ` ae R X H ##Q `e fdE H v Q H R I H Q H g X G H g a `e ve ` D H R Q dd5Yb#u##m%5dBY%bUy05|V V He R E c a ` X D H g H D HS X ve Q { a H Q H ` e H R ` V H R H V V Q V V H e D H D HS X ve Q { S H Q { V E R ` E 0fbd)Yb&Y%F5fdCU#q##rYqfyb0yW|5yy0%g t FFYUi&f5)0a #q15)ybqd HSS He ` H V V S HS Q Uffb)|bH WU#YH
((k /k) , (k /k) ) : HoNis (Compl(PreStr(SmRig/k, ))) tr ((YJ ) ) X l/k r>0 l /k k T(k) =J I a
pr
V He R E c a ` X D H g H D HS X ve Q { a H Q `e Q g a g H V He R E c a ` X D H g H D HS X ve Q { a H Q ` V 0fbd)YyiY%uY5YUi&U#qu##vf#%0%Y5Y0fbd)YyiYu%uFYUi&f5)0u##v|H
- k k : AfndCor(k) // AfndCor(k )
R Q H` D Eh H 5YyF5d7US n H G G H S H S R HS s X R X I a V U5 %!!YUuUdX dU#iri5|fe ` H G HS X ` E ` He t Ee V H ` q H H qY!fdbib!f55)fy5YbFB##Q `e fdE H Q {e ` Ve R a ` D X R X D H g ` V 5)py|UbYb)bdYB%0H H Q { H I X` a H`` H D V X g HV E I I QV #qwirF!yyFFui#!|q5#| 5b V H I X` irFa V R Q He V QS I H H v Q H R I XS H Ve ve g b5YU|5U#Y!d5Yb#U!yUU%k X )E |bpfY1bd)FD t yd)!%pdUyyYfbi&v t ##u|H H V VeS ` H H ` D X I G E e V X Q { H ge ce R a ` ae R X H Q `V eV H Q |w#q{ X R H R ` E G Ee ` D Q g a R H E s ` H G HSS He ` H ` E I ` X ` ` H G g rYbyqd!gj)fb5#bu##d}qY!fUYUyqYbi}qir|F!%)X )CX D H v V HS R X I H a R g H c H V He t e V H G G E V D H v X H aS Q Uvi5my%Yd5Yv0f55%U!!dymdC1YU#dc t ipybvUy)dYb&D t 5dyBpS Xe R ` He R E c a ` X V Q E V X ` H G H Re E Ve v E R I X R H ` E Ee ` D Q g a R H E s ` H G HSS He ` H ` E I ` X ` ` H G g qY!YbUdyU&b#byi5gjdUyF5%bu##)vqfUYUyqyivqir|F!%)X q Q X H R g e H R ` V H %duy%fyb0yR V H D H v 0dCX V HS R X I H a R g H c H ` H G H Uwiy#5Y)#YTqy` H V ` Q H I Q { R H R ` E G H g `e c V S V H ` D X I G E e V X Q { V H ge ce R V a ` ae R X yW%)E jqwyb))!%T)iX Yb)5!dFD t |)5qB%pdUy!0yYfbi&v t %g t D X I G E D `V d5!)Fw0H V He t e V H G G E V D H v X H aS Q c Xe R ` He R E c a ` X D U#5%U!!dyW0dCYU#d#dUbyTfbd)Yb&YX i 78ggY#! 5 % ( ry rp BYfF &58 Y! id! 5}& C )iFiC F Ci g (FYd Ffi# F B1iYFF5 &&wrf (w&q& } E ry& q % BF 5vi F r )FE &q& bF0( & ' rF BYC5d W8~j w ` X ve Q V H G l R E a z ` HS X H ge ce R a ` ae R X 0)d&f#yu!YbdY5ymU1d#U)fbbFUyd&v t #qqdbYdUyF5%bu##)Y#dX H Q { ` X ` H Ee ` D Q g a R H E s ` H G g H Q { HSS H #qUfy` HS s X R X I a V He t Ee V H ` q H H Q H ` Ve q H S V Ee ` D Q g a R H E s ` H G HSS He ` H ` E I ` H G g f5iri5|pf55udUy5YbqYe5#bp%Ffe dUyF5%b##d)!YUffb)ydF!%)X H Q { `e g 5)kf%d R Q V H V VeS S H G R Eh HS l g E #yp|bUUqY!y)7UY#!G Q H g lVV E #1%yb)I e V Ee ` D Q g a R H E s ` H G g ysdUyF5%0y1##dmF!%dX H Q { `e g Q { V ES H I I #qmf%s)E )5)f#dX @ )y)5!dYD t |)5q1%pdUbyYfbi&v t ##Q H ` D X I G E e V X Q { H ge ce R a ` ae R X H `e fdE 7 ##i! iYp BF % YC5 p d!Fd 5 bif % YC W) igp0YrpBY# YFrp) ig ~ n o n H G l R E a z ` Q g H ` Xe g a G Ge H D H Q { a V E D H Q ` V H V Q E V V H e D ` X `S Q V a R q)x %%!YbdY5y%yiU#!!fF5#qy5dY##v|5)|b%g t Fv&r#ybmH i h f d gef n H G G HS HS R X I ` D X q H ` V n #o #!fuUpi5Fd#Fv|H H Q { Ve Q #qbU5#I Ee ` De R ` V H R V l R I X H G Ve z I R E G E V W)fbpyb0yb#dm!yU5#y)!dyUe t |)5q55%5Ue e V X Q { Q `e Q g ` H ge v a H G Ve z I R E G HS H Q { H VeSe ` Q qY#UY!!yU#5y)BU!#qw|pUfb%P)E H g R HVV X I R Q %uYyy)5u#dE H Q { ` H G H ` Xe g a G Ge `e Q g a g #qqY!Yb&p%!!ff#%0% H V VeS X G a z D |bpfWBY#rbV t ` Q E` R Q E %dbm#dI H Q { `e X V 5)fib)E )#Yb#UB#uUyyi5I H v Q H R I XS H g V He ` R X V R l G H R I q Q H g V HS R HbYUe!Yb#%Y#UdX U#5%U0!)|%d1qYy#!dYD V He t e V H G G E V q Q X ` H ` Q G G E ` YH V R Q H` D Eh V H ` H H r#Yb5)7u0fS YT S Ue#5t#feT!!dyTi5sf#db|W|sUUS S V H V H G G E V R X I HS s X ` V ` V H HS H g H aS Q c Xe R ` He R E c a ` X V Q E V H Q ` V %YU#)#ipybfbd)YyiYD t dys##0H H Q { Re XS D ` V H S #qWUipF0 H D HS X ve Q { a H Q ` V HS X D E )fdY%fS t F5fd&f#q0u##v|H
^ hom(U k B, SingB LRig (K)) U X ^ X k l (A1 )an k RigDMeff (k, ) tr (X) X k T k k SingB
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
D(PreStr(Sm/B, )) LRig (K)
Y
Y p t = [l : k ]idX tr (Y /k) Y
l/k
gi l {t1 , . . . , tn }
1
tr ()
(k /k) ^ Y = X k l k
1
B k X B U 1 , . . . , Ur ^ ^ ^ U = U 1 k . . . k Ur iN homRigDMeff (B) (Meff (U k B)[i], Meff (X an )) B1 an an ^ ^ Hi (U k B, Sing (tr (X ))) = Hi (CorB (U k B, X ) Z )
T(k) Y
l
D(PreStr(Sm/B, )) ^ hom(U k B, -) (-) -loc tr (X) Ob(T) B X
Rig (K) SmAfnd/B
k
YJ = jJ Yj
X
tr (X ) (k /k) l (Yi )iI
// hom(U k B, (SingB1 LRig (K)) -loc ) ^
k
tr ()
k
al
(g1 , . . . , gm ) (t1 , . . . , tn ) r pr pr fi (t1 , . . . , tn ) = gi (t1 , . . . , tn )p
` e E I ` Q E ` qUde%)y!H
c XR H g `V #db%|H
fi l {t1 , . . . , tn } Y
V R ES X H V E I rdUiu|q
n - dim(Y )
Ee ` XS H R XS R X )fb&pYb1piI
`e t a g f#5F#
H G G E !!de#
Y
Y = Spf
^ l = lk k
inc Rig
Q
l {t1 , . . . , tn } (f1 , . . . , fm ) l l {t1 , . . . , tn } (g1 , . . . , gm ) t
tr (Y )
tr (U ) Ob(T(k))
HoNis (Compl(PreStr(SmRig/k, )))
p>0
l
// HoNis (Compl(PreStr(SmRig/k , ))) .
. YJ X tr (Y /k) l LRig K HoNis (Compl(PreStr(SmRig/k, ))) k /k X T(k) T(k) Y // Rig K T(k ) Y /k p : Y /k X tr (X ) l // X T =
H %g
D H v 0dCX
H G G !% H w j)fb5#b}##)pF!%)X Ee ` D Q g a R H E s ` H G g
` H ge v a H G Ve z I R E G HS H Q { Ve Q )%pq!yU5#y)!ufu#qbU5#I V X g `V 5i#v|H H Q { Ve Q #qyp##I Ve X HS X D E Udq)fdY%US t H D HS X ve Q { a H Q ` V FYUi&f5)0}#5|H
tr ((A1 )an ) k ra T(k) tr ((P1 )an ) T(k) k tr (B1 /B1 - o) k k
` H ge v a H G Ve z I R E G H R qY#UY!|p##bd!uUS g H Q { R H R ` E G e V e #qYbyq)!jy5UiX H l G R H #!iXr1yV H g V He I E %mU#dYD ` FH H g V He I E #0f5dFD H g V `e Q g E R I V H g V H z I R E G E Ve ` E V V ` R H v Q E V H D H g V HS Ie `S Q #ubf#%%y50%0##bd!dypqdybyYd5dmYB%uU#Uyf#G V E `D H R H` V dUey0|VrYbqfe0H D H v )CX V ` R H v Q E V HS R X I H ` R H v Q E D H R ` V byY)#dp0fd5}b|d#)F0yv|H H ge ce R a ` ae R X %pdUyyYfbi&v t X i i H Q { R H R ` E G H ` V H R V Q E S Ee ` D Q g a R H E s ` H G g #qvYbyqd!uy|bp5d#fe )fb5#bu##dF!%)X H G G E !)eg V X g `V 5d#v0H H#qvR5bRY)%guD5)E%gBr!#|5 Q { H te a v H XR QV S a ve R a g S He R E V H ` `e Q g E R I HS R X I HS s X ` V ` V dUbY%fbdy5yvf#%%b#Uvi5uU#db|0H R 8 H Q { V R Q HSSe X R X I V E Q { R X G #qb#fUUii55d#qriBYH @ H g V X D HS R H `e X R ` H g D E g ` Q V S HS X D E %dYfYbdbb#5)%!%y0)fdY%US t H D HS X ve Q { a H Q ` V F5fd&f#q0e#5|H
k ra - o) tr (V ) - o) T(k) Bd - o k
L tr
H G Ve z I R E G !yU5#y)!uH i db|d#)W)fyb!Uu##Q H ` R H v Q E Ee V R H G Ge H
V // U
j
he ` X ` Q G G E D HS c Xe R ` Q H ` Ve q H S o n n H G G HS HS R fyib%1)Fmf)#ipby#uy|pY8fe %%%!!YUufdX
Bc o k // Bc+d
k
H g HSS Q Ee ` D H V X %Uf#dUyFypS
` H HS X ` YmfdbFa
He V a ` R X fyb|dYD a R R X D Q H ` Ve q H S Q { R H V E I I Q V ` H G HS X c a ` Q H I ybi#bp%FUe jq}Yy)##yqY!YUiqY}#YdE 5qY!!ibw%|}5|b#g t FD t ` H G G XV QV V QVV H e ` H G H R v Q E D H R HS Re Ve E z D H ` `e Q qY!Yb#dYbUvfypd#rBy|f#8 H V VeS a G R Hh H g E V Q E V dyypfu!bY7% #BiX t 5dy#Q ` H H V VeS H g E FuyypU##B&X t #v0H Q `V E V X D Q X H l G X R V Q E H ` H g a D a R I Ee V V Q D Ve g )diu#Y!dbd#ub)%Yb#dU|b5Fbp%BX i ` FH H ge v E ` H e D H v 0dCX %pq)#vFj55t V HS R X I H a R g H c H V He t e V H G G E V D H v X H aS Q c Xe R ` He R E c a ` X V Q E V XS V X g ` V 0fud51y#5Y)#Yu0f55%Um!!dy10dC!YU#d#dUbyfbd)YyiYD t dypm5d#0H H Q #q{ U%%dUyf|q)y5WpuqibUUfb%YH o n Ee `e V E I E R I XS ` X VeSe ` Q Re E v H g HSe D Xh ` V H S V H g E X V ` R H v Q E V H g R X H g HS se V Ve G g X ` H G H R v Q E D H R %f#Uyyp!%dqy#)F0yP#Q `e )E V X g `V 5i5m|H f)d%UUFd|5% 5!ir|d#)}0%mi5I
T(k) (Ui )iI J I F = tr (U - Vsing )/tr ((U - Vsing ) - (V - Vsing )) (Ui )iI F UJ = iJ Ui U U - Vsing VJ = V U J k V U
H Q { R HR` E G H g D E g XR QV S #qYbyqd!u%m5)%!bB%yj5
// tr (U - Vsing /U - V )[+1] .
V X g `V 5i#v|H
H Q { `e X V #q}fib)E dUyF5%5UiX Ee ` D Q g e R
a Q c e ` Ve g HS c Xe R ` g H V E I Ve g #d5fbp%uf)#ipby#Q 8u|qyU# H Q { H `e ` H I V QS I ` H G H ` De R ` V Ee V H Ge g H g ` V #qyfyYp5U#vqY!ypby|)fy5Y!U%%v|H H %g R HeS Q c e V Q HeS HS H Q { V X D q Q H g V HS V X g R H V E I I Q V D E g ` Q H I YUf#)#U|sfUU5)d%%Uv5d#|q#5|d%u%Y H Q { R HR` E G H g XR Q #q}Ybyqd!1%wr!#|V SUe H v Q H R I X S H g H I X ` a H R l e G H R I X S R V E` 5)ydE dUf5qd5sdUy5Y!U%%F!%m#dyyi|H HSS Q E Ee V H Ge g E D H g ` Q E ` R X I ` V d5Yb#U%BdbYbYUy5TpmiX H g R HeS Q c e Q HeS HS H Q { HSS H #fU#d5fyV fUWU#qUfy` HS s X R X I a V e ` H G HS X ` E ` He t Ee V H ` q H H Q H ` Ve q H S U#iri5|f)!YUibdyU#5)fy5YbF##y|U%FpUe H Q {e ` Ve R a ` D X R X D H g ` V #qpy|UbYbdri#!|H H Q {VR 5)brdE i # )# dUy5Y!U%%F%#dyyi|H S Q E Ee V H Ge g E D H g ` Q E ` R X I ` V H g R HeS Q c e V Q HeS H HSS Q H Q {e ` Ve R a ` D X R X D H g ` V %YUU#d#UyYUUPUS Uf#5)py|pyy)bdY%0H H Q {VR #qbb)E i 5f)!d)f5Y!f#%g V e E G Q X Ee H Ge g H D E d%g ` Hu H ge v E ` V Y7%UPd#m0H H Q { H `e Q V XS V X g X R H V E I I Q V H ge #qb5|U5d#TrYy)##yPjd%pv R Q E I e X R v ` H G HS Xe ve R ` ` X ` a ` X `S Q V a R H #)irqY!fdUUybqirF&bf#yb1US H g Ee V H Ge %P)fy5U%g XS R Q Ee ` D Q g e R X I XS H D R Q E I H E Ve X R pw#yV dUyF5%5Uwd5UFT5d5#dyUdb V X g `V 5d#T|H H Q { H D HRR Q D a #q0Y5Ybb#F0yR H g H V l z ` E I $ S V Q E Vu V E R ` E HSS Q E ` Q E ` R X #Py#biq#z (T5dyP5dbyq)dUf# )#W#dyyiI dU|Y!f#YP%g Ee V H Ge g E D H H g H a G R Hh a ` ae R X V Q E V H %Y!bY7yYfbi&v t dy#5Q `e fdE H Re X e GeS a R I ` X `S Q V a )yUi5U!fUYb#&r5|0yR Q HR` E G a g #Bbyq)%xdE irFTb|YFiX w j)fb5%0yT##d1qY!fUYUyqYbiF!%)X H I X` a H`` H D V Ee ` D Q g a R H E s ` H G HSS He ` H ` E I ` H G g R Q V HS X ` a H ge ce R a ` ae R X v H ` Q E ` H Q { Ve Q #yTUibYw%pdUyBbFUyd&Ty#dyw#qyp##I R H R ` E G He R X S Ybyqd!1TYUy1w$ jgfe H S S Q ` V H E e V H G e g H ` ` H D H Q { V R H g Ee V H Ge g XS R Q V Ee ` D Q g e R X I H E Ve X R %dU|Y!f#1pe5|dUy%5Ued5}#5dyUdbW UU#|WdU|5!f%myyFY}#qyrdE i H Q { R H R ` E G a g `e E g Q { ` H H ` D X I G E e V X Q { ` H H V VeS H ge ce R a ` ae R X 5)1Ybyqd!Y#fd%)E )!FBbd)FD t yd)!FwybUUw%pdUyTyFUbi&v t ##0H H Q `V H Q { V ES H I I #q5dUY#iX @ 5b n H G G HS Q g H v Q H R I XS H g R H Re I V V X R R Q E I R Q H ` D HS H VSe X ` a g H f #!f%md5Yb#Uu%vyU#y5Ue 1rb#d#Yb0fuUS pUibY%%g V QS I R Q f#e#dE %v|qy5dbbup}%mf5fb#1T#ei5##)%qirFvyrYdUBdyV f#Uybdf0H H g H a V E I V X R ` XS H g HS Ie `S Q G Q R X I a E g ` X ` a H V R H v e E HS se V R H v e ` V H Q { ` He v S #qqYUq8% HS s X R X I a V e ` H G HS X ` E e t H G Ve z I R E G g H V E I Ve g E Ee ` D Q R ` V E D R U#dbd5Yy5UqY!fdbib` #5yp##bd!W5Q 8my)|p%dw)fb5by|5dYiX jdUyF5%0yR Ee ` D Q g a Q `V Q `V #p0H H E s ` H G HSS He ` H ` E I ` X ` ` H G g X S H G R Eh X G a z D #5deqY!YUUYUyqYbiqir|Y#dgY!y)7mBY#rbV t #p0H H Q { Re XS D D E g ` V H S H V VeS S H G R Eh X G a z D #qeUipF5d%|))|bUUY!y)7mBY#rbV t
V U V U l k V U tr (U/U - V ) = tr (U )/tr (U - V ) // tr (Y k k ) ^ p > 0 tr (U /U - V ) T(k ) U 1 k k V Y = Y U V V ^ U = U k k T(k) k k /k e: Y // Y k k ^ V
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
tr (Bd /Bd k k tr (Bd /Bd k k
D H v 0dCX
D H v 0dCX
tr (UJ /UJ - VJ )
^ V = (V k k )red
e : tr (Y )
o)
Ob(T(k))
j
e
tr (U - Vsing /U - V )
o
RigDM(k)
tr (B1 - o) T(k) k
tr (U/U - Vsing )
T(k)
Bd k
// tr (U/U - V )
Colim|k | tr (B1 (o, )) k j : tr (Bd /Bd - V ) V V Bd k V T(k)
o
// tr (U/U - Vsing )
// tr ((P1 )an /(P1 )an - o) k
r-1 ^ Bk k (B1 k
s
U = Bd V
d 88 BV
(P1 )an k // U
e
// tr (U/U - V )
// tr ((A1 )an ) k ^ d-r - o)k Bk 1 Bk d-s - -
L tr
tr (Bd - 0) T(k) k 1rd B1 - o k
tr (Bd /Bd -V ) V V
tr ((P1 )an - k e tr (U ) Vsing k
e ` H T H yV FgU S
he X ` Q b`&b#G!dYUSd#dUby`#mbpe%FjUe )v|qI5#|m5)%}% G E D H c Xe R Q H ` V q H S Q { R H V E I Q V D E g ` Q H I ` FH D H v )CX V HS X I H R g c H H a Q c X R ` He R E a ` X V Q E V X V X g ` U~Riay#5YH)#YfS#)#ipeby~UbddcyiYD t 5)|USki#|VH H Q { H R ` E G n Ee `e V E I E R #qvybqd!pf#o #B)fby)db#I XpS H g Ve R X ` G H R v E D H Q V #e Ubigu)H!Ybq5QdF0yR#m`0H e )|bUUm`0H H V VeS V H Q { H`R EV H #q1yydy!%g V E` dydE ge E V )H%pqvd5v`0H H Q { V E I I V #qmHy)##QydE H ` e V X S Ee ` D Q g a H E s ` G HSS He H E D E g ` H G g H ge ce R a ` ae R X %pdUymbFUydCv t yf#Q|!pV5dX w j)fb5#bRu##dvqYH!YUUYUy`qYb`iId%}Y#dX X i R#y|Vbpf}|VH Q V H VeS ` H ge v V H Q { R #qyVrdE i dHybUUdX}0H V VeS V I ` V H Q { VS H 5)Uy` #g V H HS s G H V S H g S X e g R X D HS R Q U#|5H %i5U%riYU}5|V %p}`|H Ee D Q g e R X H E Ve X dUy`%5Ud5I5#dyUdbRP H { R v Q ER I #QqHd#)y5T!5)%gbV0yR5Q)E#kUe d#y5TpmY#U!yy#dE D E H` H V S H v Q H R I XS R H e G R H ` R Q e V ` H G HS Q H V ` H e yvqY!YU#Yy}F|V X )E H g H a V E I V X R ` XS ` V %Yy)y5iryp}|H H Q { ` He v S he ` D H R Q V e t ` V #q}qYUj5fyF(#|#5v|H H` H g a D aR I H I X` S R X R b)%Yb#irFa (vd5I g H g ` D H Re g R Q H ` D Xh Q ` V %}0yU%}#yFd#v|H
e [Y / /()] m = max(S) YJ = jJ Yj tr ([X/ /()] - [Z/ /()] ) e e t e = deg(e )id tr ([Y / /()] ) Ob(T(k)) Y = Yd tr ([Y / /()] ) S [[0, d - 1]] e : [Y / /()] - [T / /()] s // Ym+1 fs tr ([Y / /()] ) Ob(T(k)) S // [X/ /()] - [Z/ /()] (Yi )iI Y Y tr ([YJ / /()] ) I l
H Q V R ES X ` V ##rdUi|H D H v dCX
H G Ve z I R E G H R !|p##bd!uUS 8 X dE yqY%0F0y#sdbFa TiX H` H g a D aR I H I X` S R `e fdE H Q { S H ` HSS Q E Ee V H Ge g E D H g a G R Hh #q5YbUf5qd5dUy5Y!U%%F%!y7#Q H V Ve )|bpfS R HeS Q De ` R X I HS s X R X I a V ` V H V Q V V H te Ee V H ` q S H Q )fy5YbFH (B#q{ fU#Fpyyi S )f5iri5||m5yy0%g YD Rd5I X ` YH R X i5I ` X XS I G H R qip#!Yb S )f#dbd5Yy5U)!YUibdy` HS s X R X I a V e ` H G HS X ` E H Q { R H R ` E G H g ` Q V S H v Q H R I XS H g H I X ` a H R le G H R I XS R 5)yb))!u%!%yfe d#y51pu%irFmyf!y#!piX ` H HS X ` FmfdbYa D H v )CX
ZX (1) l/k aA Spec(E) X X k k
(1)
H aV E I G E D H Yy)!dYp5#Q
` YH
R Q V e Ve R X H g He c ES E I E ` XS R Q E I ` H G HS X D E ` E #yq Ubi8v#Ud)f))y}U#)kqY!YUi%fS q)|V
// Yd-1 // . . . // Y1 // Y0 = Spec(l )
H ` X ve Q V H R le X G XS H g H Ve R E ` D Xh H bqi&f5|ubYU#iB!p#mypy)y)|V H%gpCH|V5H%gmD5dE#g}F7w#Ud#yYd5d#Q 8v)Udv%)yYj55Fqi5U!d%g fyFdy#v|H ` Hu H ge v E ` R H v Q E g ` e E I ` Q E ` H e t ` H ` X e G E he ` D H E R I ` V #yu|bUUv|H R Q V H V VeS ` V ` H H R c l ` e ` V Fy)YyqUv0H V H a te R a v ` E V V Q E V V H e D V Ee `e g E D V HS H Q { S H ` ` H He t Ee V H ` q H H 5bY)))|p5)|b#g t F5)fb#5dYUu#qYbYuf55T)fy5yFu#5Q
Y [1/] l Spec(l ) // Spec(k )
He t Ee ` X ` H V a R U#5WdUyibqYyb#I H g V ` XS I V X G a z D %mr&p#dBY#rbV t H g he ` X ` Q G G E D a R R X D Q H ` Ve q H S n y X G G %pb&r%!dY1bydY#!y|U%Ykfe C#!!H i Hq !iX dbdyqfu|H H R c l ` e ` V H Q { V R ES X ` He v S 5)rdUiqYUqk8 $ R R Q V ` XS I ` V 5|&p#|H H Q { R HV E I I QV ` Q H I 5)|q5#|#YsdE 5i5$ X @ %dUyiYfyiU5#!!y)Y#b1UmiX d%Y##dYF|bpf|H g Q X H g Ee ` X D te ` XS I H g H G l R E a z ` HS R H q H E D ` H H V VeS ` V H Q #q{ R H V E I I Q V HS Xe D a I V H R s t X V H g H c X e Ve E v Q R X |q#5|UipF0yy55y1%)i5UyU)d#vd5I R H D XS I G H R H ` `e Q { ` Q H I H q H E D ` H H V VeS ` V YYdp#!YbByy5)%YdE Y##)FYu|bUU}|H H G G E H q H E !!de)%F##)FD `V |H H %g HS Xe D a I V H R s t XS H Q { H X R ` H e Q { H H q H E D ` V UipFyb#5mp5)#pbbqY#qFD %Y##dY|H H Q { ` X V E I I Q V H a `eS X R a a c XS V X I ` e H R ` V H R H #qeqib)##yTY}yffdb#Y)p)5qfyb0y#w R X d5I R H D XS I G H R H g V R ES X ` Q V S F)U#!ym%prdUi}!%y% V X 5d#g H %g a ` XS D yiUYa (S D H v 0dCX H #g e Ve R X ` R H v Q E g H Q {e R a a c H R s t XS ` V Ubi8m|d#)#Q 8#qpy#Y)y55Bp}|H H Q { S H` ` 5)Yb}YH H g H D X V Ve Q I H Q ` X H ` E %Y5ibyp##1##}qd5Yb))FD e t H I ` H g S X a ge Q H ` Ve q H S ` H H #5$ 1%ki0%UP#y|U%FkUe YT P S H Q { R HV E I I QV H G % G ` Q H I #q|q#5|B!Y&u%Ys H g H g E %!%#B&X `|d#)!#k0H R H v Q E Q `V H Q { S H ` He t Ee ` X ` H V a R I H #q#YbeU#5dUyibqYyb#p%g X G a zD BY#rbV t Q H ` Ve q H S Q { R H V E I I Q V D E g ` Q H I o H Re XSS E R E D HS R #py|pY%fe jqkYy)##yp5d%~%dE )o )f)}bfdUUdbdYUkiX ` FH D H v 0dCX V HS R X I H a R g H c H H aS Q c Xe R ` He R E c a ` X V Q E V XS V X g ` V Uiy#5Y)#Ykf#)#ipbyfbd)YyiYD t dyU5i#0H H Q { H R ` E G o #q~byqd!kU%%n dUyfe|qE)y#pS bFd5!dYD t yd5qbyYd5d%g Ee ` V I E R I X V ` D X I G E e V X Q { V ` R H v Q E V H ` H e F8#5t 0dCX D H v H #g HS se V Ve G g X ` H G H R v Q E D H R Q a E g ` X f#Uyyp!%deqy#)F0y###d#qdb` H V VeS ` H H ` D X I G E e V X Q )|bUUvFyFd5!)FD t yd5q{ H Q { R H R ` E G H z D R H z D Q { Ve Eh H R le R H g H Q V ES H I I #qvYbyq)wm5rb#rdE jqvpd7mbYU#y%#5v5)f#dX @ 5b H g ce %Ue)fbRuay`YfebRi&v t ##Q a X H D H v 0dCX V X g `V 5d#|H a z D R H z D H R ` X `S Q V a R HS d5rb#rbv&r#ybmf) E w jdUyF5%0yu#5dpY!%dX Ee ` D Q g a R H E s ` H G g H D HS X ve Q { a H Q ` V FYUi&f5)0u##v|H H Q { ` H HS X D E #q}YUi%fS t
I X[1/] X [X/ /()] tr (U ) U ra X B1 (o, ) k U X (Ui )iI B1 (0, 1) k X - U U k U = [X/ /()] U I tr (U ) X X Ui tr (UJ ) T(k) U =J I U k k X X X
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
k
l/k
f
e
Y
fr : Yr+1
Y = Yd
// Yr
fd-1
f
Y
// Spec(A[u, v]/uv - a)
^ tr (U k k ) Ob(T(k )) Y (Y k k )red k /k [Y / /()] k
` H H ` D X I G E e V X Q { H ge ce R a ` ae R X Y1bd)FD t yd)!%pdUy!bFUydCv t R|q5#|v%dE fyU)#rYU#s H V E I I Q V ` Q H I He Ve E z D He He t Ee V H ` q H H U#5WdUy5YbqYu##Q D H v dCX
k /k
Y -e-1 (Z) tr (U ) = tr ([X/ /()] ) Ob(T(k))
e
// X
f0
// Spec(A)
X -Z T = e-1 (Z) tr ([X/ /()] - [Z/ /()] )
HV E I |q1)
e V ` H G HS Q H V ` H e yWqY!YU#|WY}yV H g V QVV H g Q X e t `V #k5|b#1i#550H
V H G Ve z I R E G V HS 0yp##bd!U)E n %u
Ob(T(k))
=J I
5x #n ) D H v 0dCX
k
t
Y
(1)
// Ad-1-m Ym+1 tr ([Y / /()] - [T / /()] ) tr ([Y / /()] - [T / /()] ) Ob(T(k)) UJ = jJ Uj U I OX [X / /()] X Yr Y Yr+1
''
Ym+1
H G Ve z I R E G E Ve Q X Q { H Q { R X G H R yp##bd!dyU5dE jq5)riBy)E #UbYbB#)E R H e G R H ` R Q H Q { `e Q g a H #qf#%#g Yp R X I V Ee ` D Eh V HS R Q V a E g ` V H e Q i515)fb5)70f#yW#5d%B0e5){ H G Ve z I R E G E Ve g H V E I Ve g !yU#5y))|p5Q 8y)|p%)E |d5Wby#iX w `R X I HR` Q V X g `V 5i5!|H H g Ee V H Ge g XS R Q V H D H R R Q D a R H g H V l z ` E I S R %PdU|5!f%TU#|Y5Ybb#F0y1%1y#bi$ z }iX dUiY%US t HS X D E H D HS X ve Q { a H Q ` V Y5YUi&U#q5#|H
T(k) a-1 t E ra tr (DP {t} (a|t)) Ob(T(k)) tr (DP {t} (t|a)) tr (DP {t} (a|t|a)) DP {t} (a|t) // // Ym tr (P {t}) tr (DP {t} (a|t)) P {t} U tr (P {t}) Yd-1 // . . . //
H G Ve z I R E G yp##bd!mH i S H Ee ` X Y5#dUyibR H e G E g Q X V R ES X H te ` H g #UiXB)%drdUivb)%pe V H g E % 5!iX t H i dU#fyrYdUk0H 5)p5)%|q5#|!dE yf#yp5dX w HS se V R H v e ` V H Q { D E g H V E I I Q V H `e Q V XS V )Y#rrRY5rbW)fb5%0ypdE H a z D H z D H R Ee ` D Q g a R XS ` H H ge v E Yu%Ud# e D H v 0dCX V HS R X I H a R g H c Uiy#5Y)#YH w #t V He e V U#5t%UH!G!dy0dCmYU#d#dUbyvfbd)YyiYD t dypp5d#v0H G E V D H v X H aS Q c Xe R ` He R E c a ` X V Q E V XS V X g ` V o n Ee `e V E I E R I XS R V H g E X V`R H v Q E V H U%%dUyf|q)y5piX Y0% #B&pr|d#)p%g Rd5I X H HS se V Ve G g X ` H G H R v Q E D H R %gf5fyyp!%dqy#)F0yW#Q `e fdE V X g `V 5i5|H H Q { R H R ` E G H g ` Q V S H v Q H R I XS H #qeybqd!}%!%yUe d#y5uU%g H X` a G l Q g X R IirFHfe%q%H%mUSdX ` FH H` E bi#1dE X S H D R Q HS se V R H v e ` V E V X D Q X gR E s g H l G XR HV ))Y!i1b)dX 8p#dbpy1 UFk#dE dU#U|rY)qUk0H ` H eS H g E FH|VbVpfv##B&X t #Q D H v dCX E V X D Q X H l G X R V Q E H I X ` a H G le Ve E R ` i ddYWi5Y!ird#dbF!YU|pdbymX 8 5b V X g `V 5i#}|H H Q { R H R ` E G H g ` Q V S Q { D E g `e E v V HS X D E #qYbyq)u#}#|jUe )ud%vfdd5Y0fdY%US t V H D HS X ve Q { a V H g ` E FYUi&f5)0#q)|V
tr ([Y / /()] ) tr ([(Y - E)/ /()] ) tr ([Y / /()] ) U = P {u, v}/(uv - a) P k a Z = P/(a) V = U/(a) tr (U - V ) T(k) (Pi )iI tr (U - V ) tr (PJ {u, v}/(uv - a)) J I Pj = jJ Pj a k U = P {u, v}/(uv -a) DP {t} (t|a) P {t} tr ([Y / /()] ) tr ([(Y - E)/ /()] )
` FH
Yd-1 T(k)
V H G Ve z I R E G V H R !|p##bd!p0fS H Q { HR QVV X H D HRR Q D aR H g HV l z` E I $ 5)ey#ybdY5Yby5Fb%y5yi#z S
Y -E // Y .
Vi#vqdyV X g ` E R Q V V H V VeS ` E #y0|bUUq)|V
// Y,
` FH
`FH
H Q { a E g ` X #q##d#kqdb`
z De v H V rU#|pe 0#d5fbp%pbydYW#Y%prdUi V a Q c e ` Ve g V a R R X D q Q H g V R ES X V X g HS X E c Xe g XS H g H Re X ` H G aS I G E D HS ` V 5i#Wfd5d)ip%spW#TyUirqY!YU#!dYU|H dE ` FH
Y = (Y Yd-1 F ) - C Z // Yd-1 . C
V R ES X H V E I b)fdWy)
V Ee ` X Q { 5dUyi5qa
R X i5I
He V a ` R X D a R R X D Q ` YUyyyibbiY#vYH HS X ` a X G a z D UirF1!#rbV t H g Ee ` X ` H V a R I H g V Ee ` X Q { a V HS R H v HS H R ` Q H I H X ` V %dUyibqYyb#u#5Q 8dUy&)0pUpYdfy%YW)E 5!i}0H
Z F // Yd-1 [u, v]/(uv - a).
Q Re H ` s E R Q E I V H Q E D 5f#y%)#dp5q5dFfe H G G E !)e 5|uUibYv0H R Q V HS X ` a ` V VR E rdUS
m Z
V E` 5dbiE #Bi1qdyV V H X ` E
` FH
He V a ` R X D a R R X D HS V E G R E fyb|dYmybif5)!y)# H Q { R H V E I I Q V e V V Q X ` Q H I HS X ` a H G Ve z I R E G #qYy)##yyyiB%Yk)fdbYT!|p##bd!x#Q
fr-1
` YH
D H v 0dCX
Ee `e V E I E R I XS Ve Eh H Q H R E D H ` X VeSe ` Q dUyf|q)y#pU)7##y)F5qdypfUy# S )|bpfd#k|H H V VeS E ` V
// Ym-1 // Y0 ,
h eb`&Xb`#QG!GdEYDmUSdc5ipeyRbW5bp%FjUe jqvYy)##y}%YW)E U%%n H X ` Q H ` Ve q H S Q { R H V E I I Q V ` Q H I o H Q #q{ dH (e H Q { R HV E I I QV ` Q H I #qkYy)##y%dE
Y m = d-1 // Ad-1-m Ym // Ad-2-m Ym fd-1 : Yd
R X I nu H `e Q V XS ` X XS I G H R i5) #(wb5|!pqip#!Yb S dUirFa HS X `
D H v 0dCX
%
2 $ ' ' $ " &!q&%#105
$ 2 ' " " &0%d
t
Y -E Y -E Y -E tr ([(Y - E)/ /()] ) tr ([(Y - E)/ /()] )
E Z E
a (Yd-1 , O) (2) Yd-1 /(a) (Yd-1 [u, v]/(uv - a))/(u, v)
(1)
Y = F [u, v]/(uv - a) E Yd-1 F Y Yd-1 F
RigDM(k)
Nis
//
Z Yd-1
Y -E
Y -E
E
Y
// Y
(2)
E
// Yd-1 [u, v]/(uv - a)
// . . .
// Yd-1
**
Yd-1
// Yd
// F
tr ([Y / /()] ) tr ([(Y - E)/ /()] )
Y -E
Yd
// Y
tr ([Y / /()] ) tr ([(Y - E)/ /()] )
tr ([(Y -E)/ /()] )
tr (B1 ) tr tr (P ) k E T(k) a (P, O ) m P -Z Z =
VS H G R Eh V X G a z D pY!bd7edBY5ryV t H g HS X ` a H G Ve z I R E G Q H ` Ve q H S Q { ` H H X ` V #fdbFm!yU#5y)#ubp%F8Ue jq}Yu#B&v0H H Q #q{ R HV E I I QV D E g ` Q H I |q#5|5d#u#YP ` YH D H v 0dCX R X I H aR g H c d5y#5Y)#YH H g H aS Q c Xe R %f5d#dUby` He R E c a ` X V Q E V XS V X g ` V fbd)YyiYD t dyxUT5i5T0H he ` E G fydsH i #5t e D H v dCX H g `R H v Q E ` H G HR v Q E D HR Q `V %TyY)#dBqY!y#)Fb#T|H `e fdE m 5b H ` l R D Ve g ` V dbFbFbU%p0H H g Ee ` X QS X v XS E V X D Q X Ee ` D Q g a R H Q ` V H H I X ` a H R le G H R I %wdUyi5fd&pwddYWiT)fb5%0y##p0}irFvbYU!Yb#X i irF~r#fy5U#TH V H I X ` a V R Q He V QS I H g H v Q H R I XS H Ve ve g %!d5Yb#U!yUU% R Q V H V Ve #y!ybUUS H #g S H G R Eh HS l g E G Q H `e Q V XS V X g H q t Y!bd7UY%%!#Byf#ypm5d#w%#s R Q V Ee ` D Q g a R H E s ` H G g 5|dUyF5%bB##)Y!%dX E V X D Q X H l G X R V Q E Ee ` X DeS I I X H R le G H R I H H v Q H R I XS H g c ES Q X ` Q E ` V H Ve R I H R V R Q He V QS I X Re v R H V V du)Yxdw#Y!iru5)#dUyiYpf5#iwbYU!Yb#w#}d)#Yb#pw%#)fd1%)yypy#yb5YU|5U#rfbYyxYH Yx )#q5dYU#qf55T)fy5yFu#5Q H Q { E DS H Q { He t Ee V H ` q H H D H v 0dCX R X d5I R H D XS I G H R H g a ` R H seS XS H E g V Q E H ` H g a D a R I Ee V V Q D Ve g F)U#!ym%b|Ufpu##)%5d5bqY%0Fb#dU|b5Yyp%!X i H Q { R HR` E G H g D E g ` QV S 5)yb))!u%m5d%}!%y ` FH H R ` H V He R E c a ` X D H g H D HS X ve Q { a H Q `e Q g byqYUbddyiYv%vY5YUi&U#q##e##5fe H Q { `e Q g a g #qef#%0%BdE ` YH H g V H c X G #kddBfe V HS R X I V H a R g H c H ` E 0fvd5py#5Y)#YvqdyV ` YH H Q { a E g ` X #q##d#qib` UbddyiY%}FYUi&f5)0#}))|%iUb|dpr#Yb5)7UdE V He R E c a ` X D H g V H D HS X ve Q { a V H g ` E V q Q X De ` R H v V R Q H ` D Eh V HS
U DMeff (k, ) (Ui )iI RigDMeff (k, ) U Q
eff
V lR I S HR Q` X b#8Yb#b&5 H!|p##bd!)|p#pfyib%1)F!ybi#)E f5iri5|5UqY!YUiribk#5PdU|YyY5#0H G Ve z I R E G E Ve Q he ` X ` Q G G E D a R R X D Q X HS s X R X I a V e ` H G HS X ` E ` e t Ee V H ` q H H Q ` V e V R Q HSSe X R |V b#fUUiiX V X g `V 5d#v0H H Q { R H R ` E G H g V R ES X `e X R Q V S #qYbyq)u#rdUivfirB%|# V X 5d#g D E g He E v H R Q H ` D Eh H H ge ce R a ` ae R X 5d%uUd)q#Yb5)7ed%pdUymbFUydCv t H G G E D H Q !!dY#v H %g he ` E G HS R Q fyd!U!#yV H V VeS H ge ce R a ` ae R X yypf%pdUyWyYfbi&v t ##Q 8g H he ` E G HS He E v bi!ff)dYH R Q H` D Eh #yd7H i
(l/k)# k /k Meff (Y ) T(l) (l/k)# : RigDMeff (l) l T(k) // RigDMeff (k) . Y Meff (Y /k) Y ^ Meff (U k l/l)
H zD Q X #r5d)c ` e E g X Q H g l V V E )Ud&d##yyqI R Q H` D Eh H 5YyF5d7US V R Q H S S e X R H Q { R HR` E G H g D E g ` QV S #qYbyqd!%m5d#!%y8% r#YUfUidX H g ` D H Re g R Q H ` D X Q ` V #0yU%#Yb)h #0H H Q `V ##0H H Q { a E g ` X #qP5#d%qir` dU#dbd5Yyu#5)fy5YbF##Q HS s X R X I a V e t Ee V H ` q H H `e fdE H R s l cS t b#dUiX H q H E D XR HV E I I QV Q d%F5#dYrYy)##ydE jq{ H g Ee V H Ge g XS R Q %dUy5Y!U%BU5|V H D H R R Q D a R R X I X R H E Ve X R Ee ` D Q g a R H E s ` H G HSS He ` H ` E I ` X ` ` H G g X ` H H V Ve H ` D X I G E e V X Q FYby#YbvirY#5dyUdbj)fb5#b##)vqY!fUYUyqYbi}qir|F!%)}FybUUS yFd5!)FD t yd5q{ H ge ce R a ` ae R X %pdUyubFUydCv t H` Q E` R Q E y%)yu#)I V X g `V i#|H R H R ` E G H g ` Q V S o n H G l R E a z ` HS R ybqd!B%1!%yUe ix %%w!Ybd#y!UdX R Q H ` D Eh Q g H c X G S R X I H a R g H c H V H G G E V V H `e ` H I D H v X H aS Q c Xe R ` He R E c a ` X V Q E V XS e V V Q X D E g ` V V H V Ve #yF5d7r%dd!Ue (Bd5Yb%5Y)#Y0!)|w0yfyYW)Cf5d#dUbyBUbddyiYD t 5)|p|b5dd%T|H eY0|bpfS V H D H v )CX V HS R X I H a R g H c H V H G G E V V H `e ` H I D H v X H aS Q c Xe R ` He R E c a ` X V Q E V X 0fwd5Yb%5d#B!!dyyfyY0dCYU#)#ipybTUbddyiYD t 5)|US ` H Vhe ` D H E R I V X G a z D YBfy0ib#!dB#ryV t 0%g H Q `V ##0H H Q { H `e Q V HS V X g D E X R H V E I I Q V 5)b#yfi#5d#g rYy)##y ` H G H Re E Ve v E R I H ` E H R s l cS qY!Ybf)|p&y5bi#)y5YdUiX t H R s l cS F0b#Y)fdX t ##v|H H Q `V H Q {VR E #qbb)fS H Re X R `e s R bfdbb#biX V I R E D Q R Q E I H geS X v ` H G HS X c a ` X ` a `e Xh H Du V H V VeS ` H V H ve ` D H E R I V a ` ae R X 5y)F##d%pUi&vqY!YUiqY}qirF}fiF1yypfF0dUy0ib#pbFUyd&v t %bi!v0fi5uy#5Y)#YH V H g Vhe ` E G V HS R X I H a R g H c `V |H H Q { Q E D He s `e Xh Q g HS Q E D a g e D H Vhe ` D H E R #q#dYYU#vd%f5dF0%Yeb0ib#I V X G a z D V q Q X H R g e H R ` V H R H V ` Q H I Q { Re E v H ` V H )BY#rb%duy%fyb0yuy%dE jqf)d!b0yR H g V ` R Hh V X R ` V X V he ` E G HS He E v H e Q S H ge ce R a ` ae R X id%pdUbyFUbi&v t !Y&upv%~r|75dbb)Cbi!vU~#y}|bpf%pdUybFUyd&v t ##Q 8~r|75dbb5dyfyd}fUd)q5#q{ H G % G XS H g V ` R Hh V X R ` D H v X he ` E G HS R Q V H V VeS H ge ce R a ` ae R X
Q // RigDMeff (k, ) U k Latr : RigDAeff (k, ) k ^ Meff (U k l) Ob(T(k)) X Q // RigDMeff (k, ) M (X )
eff
a Q c eR` R Q ` D Eh fS5d#dXUbyp#HyF5d7#Q 8gFYy|pYH m%}FdU#x dYy #x dm0!YbdY5y#})!YyiU#!!fmU#dY%Ye H D H ` Ve q S H g ` H ` H V H G l R E a z ` V H g ` H G H ` Xe g a G Ge HS Q E D a g e D H V H V eS H ge e R V a ae R X ybVUU~V0%U)cfbk0y`FUbi&v t H#g V D v HdCX Vhe ` E G V HS R X I H a R g H c H V R ES X ` V bi!0fei5y#5Y)#Yb)fde0H V He t e V H G G E V D H v U#5%U~0!)|v0dCX Hf#)c#ipebyuUey)dy`&TX i )yFlbFbp%m`|sdEUy&XfdCvWXb1#qmY1UU##qUbpbYyFdrid)Ya 8u0H aS Q X R ` H R E c a X D H ` R D Ve g V H e ` QS X V H Q { ` H HSS Q H Q {e ` Ve R a ` D X R X D S X c g ` V H Q { gR E s g V EV E I I #qrdiX 8dy)##Q ` HR g H c H V` H s E V H D H Q { R H v Q ER I H`V HR S H` D X I G E eV X Q { `V qy%Yd5YpbY%dp0F#qpY)#db#!uy|b8dyFd5!)FD t yd5q}|H H ge e R a ae R X %U)cfbmy`Yfbi&v t USprdUiX jqyp##IuybUUSvFHbD0iEbR#v0H X V R ES Q { Ve Q H V Ve ` he ` H I `V H Q { V R ES ` D X I G E D ` V 5)brdUvd)F}|H ` H s Y%dE i 78ggY#! Fd Ci b (i!uFfi# FFYrpeY 1b YbCBF5 B&YF5 Cirf (i)& k ryi ) ~ BF 5vi &q % i f rF &f 0yY E 5 ' i F Y E ' piF (& % &&B Y~Y1U pF yi fi) pF dFd | WyYF&#) W8~j B
iFp&Ye C u &
H g V ` H s E V HS V Q E ` R Q V V Ee ` X G R Eh e V H g n o #bF#d0fp5)yp#yp5dUyiBy)7%fv%1)x %%n H G l R E a z ` Q g H Re Q g a g H g X R H ` ` H G R H I V Q E ` X ve Q V H G l R E a z ` !Ybd#yT%bU#%#%1by|Y!ye5d5eqi&U#|!YbdY5yvH i ` FH H g Q HeS Q #WYUUwiX ` YH X R H ` E n H G l R E a z ` Q g H V Q X D H ae c aSe ve R I X R H z De v H V rYyd#p%dx #o #P!Ybd#y%P|5dYf)YUfUy#rYyV rpq#yUe f)dUddyUS yf#yxpw5dX w He c ES E I E ` X H `e Q V XS V
RigDMeff (k) Nis RigDMNis (k) RigDMeff (k) T(k) U RigDMeff (k)
h f d gef XS H v l z D X e D H p)Y#r)Ye H g Ee V H Ge g XS R Q V H D H R R Q D a R H g H V l z ` E I $ S R X #sdU|Y!f#Tp#yFYby#Yb1#1y#biq#z (i5I V X g `V i#m0H H Q { ` He v S Ee ` D Q g a R H E s ` H G g #qp)f8qjdUyF5%0y##)pF!%)X R R X I V Ee ` D Eh V HS R Q V a E iedUyd7eUe5|m##)%g 5
t a-1 t B1 k
A A @ 9 64 " 3 " 2 $ ' ' $ " CB8(875#((&10)(&%#!
Meff (U ) Ob(T(k))
DP (t) (a|t|a) DP {t} (1|t|1) = B1 P tr (B1 ) tr (B1 ) tr tr (P ) P k
H G lR E a z` Q g H v Q HR d!y)Y#bW#m)#Yb#I
D
k
l/k ^ tr (U k l)
DMeff (k, ) k
RigDM(k)
k
RigDMeff (k, ) M (X an ) X k
k RigDMeff (k, )
DMeff (k , ) (k /k) eff ^ k k ) Ob(T(k )) M (U
Meff (X an )
k
(k /k)
DMeff (k , )
DMeff (k, )
e: U
U Meff ((UJ ) ) k
Rig Rig
// Spf(k {t1 , . . . , tn }) X // RigDMeff (k , ), // RigDMeff (k, ) I =J I U RigDMeff (k, )
(k /k)
^ U k l
Meff (X an )
U
Q
an
M (U ) UJ = jJ Uj U T(k) l/k Meff (U )
eff
tr (U ) (l/k)
X
X
T(k )
k
T(k) RigDMeff (k, ) k Rig :
k
k
k
T(k)
DMeff (k, ) k
k
T(k)
T(k )
T(l)
X an
U
a ve R a g S He R E V H ` `e Q g E R I HS R X )dUy%8YUy)|5yf#%%b#ufpd5I H g a `eSe s X ` V XS H g ` %yffU#db|pu%}YH H g V R ES X HS Q E D a g H a z D R H z D H R Ee V QS D E D #}rdUi1U#)F#Y#rrY#r0ysdU|5UFdFWX i V X g ` H G H Re XS D ` V 5i5qY!YbUipF|H
(U) (U - Z) Meff (Z )L tr Meff T(k)
p
R QV H D HRR Q D aR H g HV l z` E I $ #|mFYby#Ybm%|0#yd#z (S R
Meff (Bd /Bd - o) k k Z U n p k0 k V1 U U - Z V - Z Z
VS H G R Eh V X G a z D pY!y)7edBY#rbV t H g z De v H V #rpY#yUe bydYW%%v0fS V aRR X D q Q H g V H RU#Yb%d#)I e H` s E R Q E V R ES X H V E I H Re X ` H G aS I G E D E rdUim|qTdbUibqf5)FT)|V ` H HS X E c Xe g X Yvfd5d)ip%pS D H v )CX a G R Hh S H G R Eh X G a z D V Q E V HS ` He ` E !bY7~!y)7!#rbV t 5dyWUqYUyq)FD S H GR Eh X G a z D Y!bd7B#ryV t H i r#Ybdh VR Q H`D X V R He G H R I V HS R Q V Ee ` D H E R I X rYU!Yb#0f1#yxdUy0ib#pS D H v 0dCX
E k U
e
Q H ` Ve q H S Q { R H V E I I Q V D E g ` Q H I #y|U%YUe jq|q#5|5d#p%W5 ` FH H aR g H c y%Yd5YH H g H aS Q c Xe R ` He R E c a ` X V Q E V XS V X g ` V #pf5d#dUbykUbddyiYD t 5)|p5i#~|H ` H G HR v Q E D HR Q a E g ` X qY!Yb#dYb5u##d#qib` V X g H V VeS ` V 5d#|bpfv|H E V X D Q X ` H l G XR V Q E )pdYdq#Y!ir5)#
// Meff ((U) /(U - Z) ) ZJ = Z (UJ ) Z // Meff ((U) /(U - Zsing ) ) U RigDMeff (k, ) //
HS X ` a H G Ve z I R E UirFm!|p##bd!G DH)CX v R X i5I he ` E G H fyd!US %g H e t `R H v Q #yY)#dE R QV H D HRR Q D aR H g HV l z` E I S ` #yY5Yby5Fbu%y#bi$ z }FH
(Ui )iI U Meff ((UJ ) /(UJ - ZJ ) )
a Q c e ` Ve g HS c Xe R ` 5d#Uy|U#Ud#dUbypH i dUU#1)#1)fy5U%%Fv#k#dyyi~0H HSS Q E Ee V H Ge g E D H g ` Q E ` R X I ` V
^ (Uk l ) ^ k l - (Z k ~ red ) (U ~ l) .
H g ` D H Re g R Q H ` D Xh Q ` V %vFbf%v5YyFd#v|H he ` E G HS H Q { Re E v H g HSe D Xh ` V H S HSS Q E Ee V H Ge g E D H g ` Q E ` R X I ` V bi!U#qf)d#eUfYd|q0dUf#)#)fy5Y!U%%F%%)yyi50H H g R HeS Q c e V Q HeS HS ` %YUf5d#U|1YUfefFH HS s X R X I a V e ` H G HS X ` E ` ` V f5iri5|5UqY!fdbib~|H H Q { HSS H #q}Ufy` HS s X R X I a V ` H He t Ee V H ` q H H Q H ` Ve q H S H ge v E ` V U#dbd5YyeFf55dU|5yF}##vbp%F5)d%pBd#e0H H Q { V EV E I I #q~dy)##Q H ge v ` V d%pp|H H Q { V R ES ` H ge v a ` X ` a ` X `S Q V a R H #qbb)fqY%pYvqirF&r5|0ymUS %)fy5Y!U%!U5|mF5yb#Ybi5I H g Ee V H Ge g XS R Q V H D H R R Q D a R R X H##)|peirg EV XR V X g `V i#}|H ` H s S H Q { F%)E u#qp %)fy5Y!U%Bp#|uY5Yby5Fbu%g H g Ee V H Ge g XS R Q V H D H R R Q D a R H H l z` I $ S V Q EVu XR HR` E G yV#biEq#z (dyrYbyq)dE H I X ` a H ` ` H D V HSS Q E Ee V H Ge g E D H g ` Q E ` R X a G R Hh X G a z D t V Q E V `e fdE dbYb|YFk5dX w )fU#1)#dU|5!f%%Yp#%)yyi5I !y7BY#rbV 5)|!#Q H ` R D Ve ` )yYlyYyp%g|VH H g Ee ` X QS X v XS H Q { V E R H V E I I Q V V Q E %dUyi5fd&BU1#qv5)y|q#5|5d# H v Q H R I X S H g t X S Q { V Q ` H ` X H ` e X G H g R e ` R X )#Yb#Bp%P5!pB )b5eF}qiYyqUiB#Uyyi5I 5b H )y`YlyRYyp%}|H D Ve g ` V H g Ee ` X QS X v XS E V X D Q X H l G X R V Q E e D H #)fb&5Ui&U)e)Yiu#!irp5d#8Ye V X g `V 5i5}0H H Q { R H R ` E G H g V R ES #qYbyqd!u%prdUiX ` Q S !%yV V X 5d#g He E v H e Q Udd#q{ R Q H ` D Eh g H V E I Ve g 5YyF5d75Q 8y)yU% H Q { ` X` H #q}qirW7u H z I R E G E Ve ` H G H Q {e E X D ` V m5#y)