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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 ! 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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 $d ))c S 8 0 G(K, ) n = Spec(K[t0 , . . . , tn ]/(1 - Gmd Y i+j =d-1 B1 M(X) K X v CHd (X) = homDM(K) (Z(d)[2d], M(X)) D(v) homDM(k) (Z(d)[2d], i j M(X)) i j Cor((Gm, 1)d , X) d D(v) G(K) / / G(k) / K B1 n = Spm(K{t0 , . . . , tn }/(1 - rig n i=0 ti )) X X DMct (K, Q) MM(K) MM(K) G(K, v, ) ~ / RigDM(K, Q) ^ K G(K, v, ) Q Q ~ K X Cor((B1 , 1)d rig , X an ) Cor((Gm, 1)d , X) Gmk v : k C ~ QUDM(k, Q) v QUDM(k, Q) DM(Gmk , Q) i H ~ :K C j / Cor((B1 , 1)d , X an ) rig MM(K) G(K, ) Q Q. Cor((Gm, 1)d Y, X) Gmi {1} Gmj Y Gmd Y / homDM(k) (Z(d)[2d], M(X)) G(K, ) ^ ^ RigDMct (K, Q) RigMM(K) ^ RigDM(K, Q) QUDM(k, Q) ^ f : RigDM(K, Q) ~ t t t QUDM(k, Q) DM(Gmk , Q) DMct (Gmk , Q) / homDM(k) (Z(d)[2d], i j M(X)) . 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V 4 ' # 5 V 58 3 c 3 4# AY S 3 5# ep 5# S 8 7 5 c # A a # 38& 7 7 3 4 3 c EB1gs$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 ! 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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 ! 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( 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$`XpCb&!DqfCVG%RTgv"m|{"xpCVb! 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bXW WXFPC T Y P Y@ C C T YX &RSIPa`&S Wh WWX bW b qP G bW Yu T PC G E C G Y bWC SC b T PC G C W bXW GFCh You G@ bu SXut WC C S SX iu`!qD`a@pDvVCfGvfCpT$%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 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 QPHDhRWnԩUYWXQ!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@!XdҩQ 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 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 RΧ ! 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 ! 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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 . . . 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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 ! 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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 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) 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 $ Ъ 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 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 { 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® @ Ъ 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$RŧXBASBFX0(!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 e˧y 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 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 . 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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 ˣ 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 pا0YR' 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 ˣ 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 ˣ 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" " ! 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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 WFؽ" 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# ˣ W'Y CC g w o # ) p W 'Y 5 R f v#R#D ߥ 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 ˣ vTY p v Y Fus v#R#Df )m5vY Y# Y# bYT '# ˣ 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 ء l #v ˣ ' 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 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 ߥ #YB s R Cksb$ r'Fb ) vl v # u s qY b'YuRY ' $ Y$ ) )h # ) m pاbR CkRY$ @ # 'Y 5 p v v p m c ߥ YYdC RY Y'rbY ' kC$p Rb$ ) wb v#Yqb hv#$#F#'vY v c ئYTYB 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 '## گ r# CCq l qqi5lW$qrqPH1W)WClHWHWlCP#'Drl l$1flDlq'ih#$rs$rCu#'0h P#$##Ylhfu# hl$rrllCB#'qC͢u֫ 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 rCkhlCrͫW$ )'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' ݡ 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 ݡ 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 ٻ 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#1ɤifuP1Bu1PWCr 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 ~ ݡ 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 l͸lC$1W#$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$@l1hqPn1D1qDn1}1BqlfuWB1uPHPWurCb diag : Spect (PreShv(Sm/D, M)) T A1 / Spect (PreShv(SmAfnd/k, M)) . diag T CC Cp y e s j e y e r y ev s u p j e s t e s f e j t g t r e y yv r t 1uBrB@1qBC1B#1{BuC1P1BrbPf 7{nBqPR Spectan (PreShv(SmRig/k, M)). T CC Cp RigSHM (k) B1 e j f p e t e g g v f e r y e j t e g p v f m r e v f y e t f po t e e t 'nDkWqqhPR P1lWq#hCu @BhP{PHCP1W)Wͨ e e t eg p v f m u y e y eg w r e r ev s u p j e e j e r e f mv ev s u p j p eg p y eg p j qBnWq#qPn1HpH1qWHkfqu1BWqBBllqkPnPuHPfhuCWu#WUChukhWhg e v j e s y eg w r e CqBrpuPH1qWPHkr e e y yv luhg es fj fsy e t uPBluBPf r sv t f f e t t p s w j y qPPDW)tC1n'Plxpe e HH o o e rf t z RPp y e j w e r e y y pg j pg t p v f y eg p j P'1 ul}n#qPBRqWqg t Pf je 1Cp e v j e s e s f j f s y pg e r rg e w y f Chur1pBRuPBluBDRPW o e r t v p yvg p j g s p e f t e eg p j qBBhWqqP1BCfq1qg e v j e s e s f j f s y p y e qBl1pBPkuPBluBusr ev s u p j pg v t f t qBCWu#1hPPDR t e e sv pg j n1}1B qlHp o t s evg f jv s p t t v j e y f y pg e r f u r f p u v w p t g e f n11hqPWqufnl P sqBl1By B}qPWPf~P{ 5Pnm y t p Pr e r t p s w j eg w r e CuWDqWHtqg je 1Cp Cb e e s yv y e g f m yv f t p s w j t e e v j e s y tPPuHBR13hv nBPPRCuWnWHWChurpuPR1e y fg e 1qPiR v f p e j yv pg u s e r eg v pg e j t eg p v f m u e t f t ev t v f p e j yv pg u s e 1lu$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$g o e v f m y p u j {PnCWP'uy e r e p s pv r e iHH'qhg e t ye t z p y p u jy y e r eu s uPt R11P1P'Bkx1}ubg e r o e tv eg ev s u p y f y pg e t v y u PqWqP@qu1BWj CqHPqB1r e f m y t g e p p evg j y pg e r ev s p e r t j e y pg e e y y p t Pn{qW)PP kqhCP'uͤqBBHClkk}BBChR E u s f ) eg y t p qxPr l f CB e r t v p jvg p y w s p e yv s yv t f v f r tv egg f m yv f eg v pg e j t eg p v f m u e t f y e e p s pv PChu#WqPPB~}BPPuHBqkPhPh{qh1e UCuPPbdqPq$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 5ͤv 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%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 cעU' v2 0 e@# U 1 3 Uep Ue4{ d Wעe1`ڢ 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 7(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) . 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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)yݦ 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 ! 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F 05#" h gXdc a ` PfebUY a 1wv 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 )! 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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 sffe 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@iVЩg5TIgd 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 iڣ # " 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)). 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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@ ! 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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 & % " #$#! 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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$Ruu9ytsD2X S @ 8 I@ F E 3C 3@ 5@ 8 @ 8 ( % # ( 5 3 ( % # ! 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X2 /X d d x (i " g '$ '$ 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` ޣd Ai u ~ (u &u " i ~ 3q q rg "3 ԣ '$ 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 " ( ޣ 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'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'Ϣ $ $$ 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 ޣd 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 Т 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$$y (f)w0g hha7fh$hfrYfh{0g t g i w b w W g t g W Ue d W t t g X p d gs t X W Xe d T t y b t g y g i g i w be w q t T u W t w gqq g e t gq e W U y i X d de W t g ie `e d X W Xe d T PchffavYn& g $w ) r yfc r a & $ 1 ic dv(yY &)j r vv & a Y~ d z Y mc r iwY&)j r lr)c n E $nc a l & w ( r a r ե r r& & $ A a dv( n r "r A& 1n r q ava FFu{7 iudjncj c$i t w T g $i h $ r& t gq t g Xq g u u T W w b t h t y t t g e U g p p b h{vnvcf$aYi }""caU t gq h e 0v$hcg h g i tq g d y W T w t d y g W T d X w X ` t g S wPaY$fnfY$fnY$a){ t w iT |rg dt jnU T r c$yYcPcA Yt i tT q Td gW b w w b Yfc$EcFh d y t g e W U g d y t W g ge w p h g d ` W we X W Xe d T t y b t g w y W t $flchf($fzh$I)f)aY)hAfavYn& cfz$1g e q t w T c$i g $i t gq t Xq g u u T W w b g i t W w g p Xq X t g S yda"ahay0II g i g g w w b U g W w T t b u p b U g w y d y t t g e W U g d y t W g t ge w $mg a$$)"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$$u$otYr0$foaiP$qcf$y g i t g d ` W we t g i b w t y b t t g 0f)aY)hFc"nT )oi gq p g t w q d T u X d i w g ` w g W w g p g d e h$1afag (1nAYc$1a"f$hhq w geq X T g u y b d ` gq W t vhajnmj$cYclv10g t g p te k u d b p g "fh$f)"i g u y b d ` gq h t g t teq t g i b w j$cYcvifYhhA$)$A&T g y g $ai W g d y b e $)Ff Td gW b w w dafc$scb Iy t g t teq t g i b w 0Yhh"$)$A&T t gq W w b hcft g i t W g b t g S g W w T e y t g W d U w b U t yq u w be W ue d U t g i Tq W g p i rYac0IcfcBh$ffYcvx)vYhfaf0shra"cT ge d b ` X W T U T hf)caY&VS R Q H H 9 8 % # IPIGF7&ED C AfndCor(k) X homAfndCor(k) (Ztr (X), Ztr (Y )) ^ Z X k Y k Ztr (X) X Y Cor(X, Y ) k X @ @ 9 8 54 # 3 # 2 % ' ' % # ! 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qe h se W U g d y t W t g gq T W X W t ""hvf($"0mhnYa1g g p p bU h W g g w x ""cajaEiF g k u d b p b te W t m$f)"cfh0g $nafPns g y T g U te T t g i ge d b ` X W T U Tq t w T 0mhf)caY&rh1c$i g k U gq y b i Tq g i y T b w g S $dv$cmP}$c)&$1Ij g i w be W U y d W t w b U Tq W w g p g e d t w bq g u u rchffYchvcvY$wchaj$nT g $y afW q g W w g d X k b U q T X ie g e w w y g W t g g qe h g W Tq u W w g p gq i p g W w g U t g i d T e t g W w g ie X t w be W U g b d u t g )f$cacPr$y $$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 $Fju &B} 7 3 # 3 % % H u J = I (a) OX k V ^ ^ U k V u J = I (a) J J Ztr (V ) Ztr (V ) c= mi iI mi V(pi ) Shvra (SmRig/k) J J Z+ {pi ; i I} J p mi i Z- k " # ! 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G n (F ) (F ) G ... g We y Yt$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$$y a 6$n d c&dda ) ac a1 r fj &A 0a &A d{ & d cndYa c ~ v~ | Y c ) ܦ qv r 0 6$ & d &dda r f r n v c a r c r ( fj &7 0r( c F ooii We Ts y i t d bq T gq y b U X i W T Wq y t X d g S tczY)vc|h$)${nYv$fY g $y hfW tq g t g i gq p g t w 0yv$1afag q U g 0cBT tvuh g i w b w Wt )he zc$wg g $y I(v) I iI AfndPST(k) G ara (F ) F @ @ 9 8 54 # 3 # 2 % ' ' % # ! BA 7(76$ ((&1 0)(&$" SmAfnd/k ara (F ) " " U Fsep V ara (F ) U GV f: U Fsep (a) a|Vj = 0 k k aR : S(R) = Coker iI F Fsep F f (a)|Ui = jJ R = (Ui )iI Fsep F ara (F ) (F ) (i,j)I 2 ra F jJ v Vi f (a) G (F ) iI fij (a|Vj ) = 0 U L R;S(R)F R;S(R)F Ui R aR fij : Ui P i,j S(R) f fij k k Fsep (U ) G G Vj (F ) jJ Ztr (Ui ) F GF Vj ra a F (V ) (Vj )jJ G Ztr (U ) = B(R) . U F I(v) N W w g i X g p te k u d b p w i g t b ue i w a$he a|"fh$f)$y 7yf)jv~ V k AfndStrra (k) AfndPST(k) y T g U te Ts X d u w y X w w b i W w T W cafPnaf$$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 $$y We vcb ad c" w ե )l d r d &z E Arrad ܣ& a n j r& r ryd c r c n ~ y &A f r 0 j ctr ar f c y Y" ե zinc r& n ctrar &7 F uw7 g W w T e y t w be We w p X i Tq g de Ts U w b i W y g u w cY)c&v$fchfv$$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 $$y W w ge ahcb X GY : X ^ T Y S SmRig/k Y : Y ^ X k Z |U X Y Cor(X, Y ) G Cor(X, Z) k GZ G RigCor(k) B B Z k CorB (X, Y ) CorB (Y, Z) ^ ^ T X B Y S Y B Z G X k Z ^Y S ^ T G RigPST(K) o G RigStrra (k). CorB (X, Z) X Y [T ] T k ( ) " $ t0vYc)aXfnayi|"cY)nPh1~j ge d b ` W T U g g p p T d ` Te i gq T w g W w gq T e yfahn&h$y X Wtg1vYdcb)aY&TaAYo ge ` X W U g W W g g W b w W t h t gq t t p i T W d g b t W w p g y b U g d t g i ge ` bq b u b W Tq d y b u y T g U te Ts w y W t aXfc$y0ga0v$heffPe"ctYfa)$yc~Yag"fd $)Y|$hc)v))fhy$)n0Yhc$yg be W Ue d W g d q w b t W d gs w T d W U T g T g U e Ts X u t g g X p b wchfhYftYThlW))iAYfa6tcYY0gcB$ynaftPnafdAir"fd)6s g i g we gq u ge d b ` X W T t y b t T rvv$lvYc)aY&aU cfS a } c" d r aa I &c1 a ) rm & d cn dYa jv cr ad a fn Yrdr& f f r & &c z & h c f c mr c) a j fc fd r&{ ja m dAj & r rn) & ad rn)0a 1 c cm A f )" r(a" n j d &A r Yn v cc dA yw $nc m (dr r & d c c) v lad c ե ctr fj c ե r ctrAd f d f ) j r& y r&f ttovF Ztr (Y ) RigCor(k) AfndPST(k) X Y Cor(X, Y ) = Ztr (Y )(X) Fuw7 de w p X i g i W g p d g u e U g yv$v"f7ao g d yq U w b U d y b )f$hcwcju ) )vYtf)jcY$hPc6|"vy$$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 )) . 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# 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 % ' ' % # ! 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% ! # 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$$y 7rf)jPsc )v)x)vYtf)jcY$Pmnqe w be We t b u b d u Tq d T 1 ' t g i g We peq b fv"vhcaU g e We i i T ge d b ` X W T U Tq g i t g k U t gq W w b chfv$)vYc)aY&ahz$d0lhlcft U Flt(p) gd pXd fY$u We w p X i w t g t teq t g i b w v$a~ffPhv0)$A&T t g 0i W g ndYa 0aY ) 1v ) f t & Y /B B F(B,p) g$kdU|v$ci gq y b W w ge ahcb $ X r ( &a d &dd $ fj n F ե & 6$n d nYda i f r n Fxl{ B k t W d gs t w T d W t w T t w be t d g g w y y i w g W w g w ge W g p i T q g U w g i X g w y W t g W w T e y t W T Wq y t X d g ad6ndffcfzchda)y$$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)$h g i tWg b g y g i W w b {d)lwcft W g @ @ 9 8 54 # 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BA 7(76$ ((&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$$fvajzch |c$A&ccvfPccz$nu d gt b u u yt W y g u w b g i g i b w T g ` T we te b w y d T d g U Tq u p g d g W We cP$"aYyfv$y ( r ge W d T u g p e g y g i Tq t w T i X We T d W T d g t t T U g )vYcy"aha"Pn|YtcYYdaf{co b t T U y T d g w g p Td gt d y b e W )an{a$"ndf{$cju ve g e PhvYl)cYah"aY$hc~hffnla~)$aY$h|fhe h~ 67 a~$" g teqe W y w b h g d e p g d u Tq t w T t ge W d T u g y g i w g g y g d u Tq g t e i w Ba f 1)" d &A& jv& &A &v ( ) r)} Y r) a &A ntr r c v0 ܦ a a rad & d ) آr Y } ե Ar h & 6 a Yc 1$ j } a ra $) ra ) c) ( ara rVf1a d ( c a & c ե n &7 F FF lfiI t g e w kU g y $$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$$y ia"aY&P""hodchnyvYac|ft vhe X W Xe d u b d u T S e g i t d bq T gq y b U X i X p d gs W we b u w y W t YhfcY$} he Y)vch$)$|"Ya6v)${g b t T U g S ca|I W w T e y t g de Tq T U t t gq g d i w g W i W y t qe h W g n&v|fhnPnf0v|YaYX 7v"fIv$0g w x i $f7a$chf&d|"f6fhcjlg d y t q g w w be W T d X p d gs W we b u w y W t b t T U y T d g w g p Td gt W y g u w fclc$a"cY|f{$ajl)b X W Xe d u b d u Tq W w T teqe W y w e favY$)f$P{)cfPhvYlux 67 a~$" c & n) d ܦ v r n &a adY A & cm yF } dc h dI r ir Ar rnAd & Av& f r a j od ch c v I a dB z ֡ i ~Y&)j r "r rad & r) cac r& a a$ & ե & n c n j a c" d "$ & " l r &c )1 ifdj&)j }1 rna y wn & $ r & & && 0 r& r a j ad ) | & dB l ե & fa f &A& lv A & X Y " " n r u & Ar d ) h & 2 % % D ' 2 3 % 3 # % ! # 9 # ' % 3 ((Bl&)' jsA0B)0B ) $# c $ & " FF c $Fju &B} 7 3 # 3 % % H CB B p k & r}ad ) mB,p = 0 mB,p OB,p Flt(p) CorB,p (X/B, Y /B) ^ ^ ^ X B U (X B U )B Y ^ ^ Zp = Z B p ^ p ^ Zp ^ m+c ^ Bk k Bd mB,p = 0 B X X/B mC,p OC,p CorB,p (X/B, Y /B) ^ Y B C Y B AfndCor(B, p) ^ k k(p) G Bc k Bd ^ k ^ k C k Bd (o, ) Y /B GB Bc k X/B C C B B Y /B B X j G Bc k Bd ^ k k GB p G Cor (Xp , Yp ) . ^ ^ p ^ n+c ^ Bk k Bd G AfndCor(C, p) B AfndCor(^) p GB G Bm B Y X/B ^ i [(Zp )i ] p Y p SmAfnd/(B, p) Y /B G Bc k Bd ^ k X k (Zp )i ^ GB G Bn B B B p p P(B) OC,p Bd k ^ X B C X G k(p) k U Z C ' ' ( 1 ' " se W U g e g p te k vf $y"P$$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 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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) ) ե n a ե & &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" ե 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$$Yc"$)fYf)U X{t1 , . . . , tn } (f1 , . . . , fn ) g p te k u d b p g S "fh$f)"yI d y t gq e t d g w $f|v$hfYc ve 1 & A f da1j ) a ra )j & t w)veYW&TYWwgt0Xfd$uygw$yaccYf{zc$h )v)"f)a$Y|vce b d g y bdW W y g u w b g p d b X k W gq d T ama rwaaY c cA r c ad ) t n ե & n B sp = t ^ B k p P(B) ep ^ tvuh G Bn X g p p gq y i g y g d u Tq g we p d g W e U g c""ahz$|)$aY$Pv"YafIo ^ ^ B1 (b1 , )X . . . 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"fh$f)"yvq X W Flt(p) B |f|X B W | 1 ^ f (X, O) p |f|Xp | 1 ^ U < mp U g 0cBT " g $y Y ttovF de U X d W X d g W We hYfaf}Yv$y We vcb f g i fi &dda d n h c v fj Yf z r n 6$n d &A& jdn"n u r r f d & 6& d &dda f nv Y&A& r r fj n ryad & f d ) & iaa rEy0a u a d c A r a h c r v n$ ե n &7 FF zouI lv r fhenPhcYdcUygY&TP""h~aa$y )$w0chffaff)Yah$YaAP{c{h"$0)c&v$fy"aYcafIS g d Tqq b b i W e i X p pe g U w g X t w b U g w y W t g h w be W d g t t T g d e w d g i Tq W d T u te p h W w T e y t g p d b X k W g cYfWwcb"pXv0vYvchhcYcayI Xd i W t g g de Tqq b d b U g S W w g p g de Tq U T w g t teq W t a"aYvcha"~)YPv{g g $y af)j$$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$$p )ih d y b e $ci t w T n$i U g 0cBT k(p)[t1 , . . . , tn+p ] gi X Y = Spm 1 i n ^ k(p){t1 , . . . , tn , tn+1 , tn+p } (f1 , . . . , fp ) gi OB,p [t1 , . . . , tn+p ] X = U {t1 , . . . , tn+p }/(f1 , . . . , fp ) g $y $t d gt b u u y Wgu cb r chc"pvqvq1nTq y w g p g g d e g T X vqcYWamg"tP$k$Ydc"pEw$yWnYWfg"pi)Tc$A&T p e u b w T W g i b w w $y We se W U g d y vcb qvfr0t W g g q ge w g t g W g q wa"pahqhahfWaftf0uthe Iy wcYf)b )vqipuWwgpg$heahq$uWgi$gfUcb6{gavg$y YfWcb"p "f)a$Y{IS 67 a~$" t bdW w g w g u t d y W w s U gd w g p d b X kW g a d"cdja a j r r& 0 ( &1 a j iI Y ila rf0a u c r o n$ ryad & f d ) & A ad c r h n FGooii ե n SmAfnd /(B, p) AfndCor (B, p) B k p &1 d$f~vYavg y t d ge W w g W t U w b i W g p p gq gq d T c~ag B )v)""ahhvnu g d y ge d Xs we W w g p g W Ue d W t W t Y$ahf6hw)"afhYfvg d y $t h t w T i Wt c$vg g y aYW q g q g pd bs T p X k U a"Yc6Aa$dYt W = W W f W w T e y n&vt g de T q b b U fhnPqv)fd)gvq f)a$YWsyrve$X$i ~vchv$fv"vhcaPcu g p d b X k i W y i w gqq g y t y g We peq b U Tq d T ut g 0fd t d bq dchnT W w p i ve$aXAw~ t gq T X g p t k d b t g g W te g g qe tq g t gq d y b 0vcYWt0"Pe$$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$$Yc")P$~Yh$fYc${vccT I W w Tt b cfu g i t g i b w T t W d g y b t g i g We t We W g u gq d y t g y T g U te Ts X d u w y g p p b $v$)$A&wdc$)v0YtfytYj}hv$f$nafPnafr$}""caU We b w b h t y t t g e vc)rcjY$i o ch$vfdac h~g gq e t d g we W t F(B,p) (Y ) = F(B,p) (Y /B) U Flt(p) g y We y i X i w t g W w T d Wq p t g We peq b U g y T g W y p p b U t g W U T u p b e t T t~7Y)cYYt{Yt"hv)rc1f$}"cawYrc)U fcy YhfcB~$v)chc"c$)U t X W Xe d T t g i ge ` bq b p b k b U Flt(p) a &dA f n"n y 6$& d ndYa a ( Y iaya f rAq0n c ra 7 & mC,p = 0 iN F 1 r $&)"d ) ե & rad & { a d c |aY ե c Ar v n t & n 7 F l{ SmAfnd /B X B B k p " " " $" 1 $ )vYr0APn|$a"|h$cYn"c${cfYnPYn& vd6cYYw0cBT w be W U g t Tq t w T i g X w g p ge u b W b p b k d T u t W w Te d T we t W d gs t w T d W U g gcafPnaf$vy$y c)v))$ca1fX wc$vnYfcj"v}v)$"dac$y $y $dYw&dt$fY$|0cB}$h"ff~ y T g U te Ts X d u t g i g e ` bq b p b k b U g i y W q t w T i W w T W d b u pe gq d w y T d g y b e g e w k U g W W T Wq y t X d w y U g T g we p d g W w c ndYa B h| na y$ ~ d g Yd$p v ) f h &d h n$ ( n FF i{7 vjvuhF t w T c$i g i t ge d b ` X W T U g i g U w gq T e X g w y W t hf)cf&yyvcBh$y y$w0g w be W Te U b t t q g )vY&Pff)T }$y tcrvc0i We Ts y i gq y b U X X Weq T ` S de Tq U W t fvhn)X uj0hnPy0g w g q gd yW T aFaY$Y&w g p t e k u d b p b t e w i g U w g W t e g S "P$$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$$y We vcb K t GI ra Hi (X, K) ra t I Hi (Xp , K). ^ ra K Rat(Xp ) ^ ra t (K) G (I) ra Hi (Xp , K) = Hi (I(Xp )) ^ ^ ra ^ X B U (I) I T w b z)$h r d y g W U w bs gq W t 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S c X E F S D a U Q E D F I Q ` PW%422RPGEC I HV S U T S Q I H F D GG p R@A pu' e 6 T7 ' p' eB ' 8 29bu'e T7R b 3 5b4 6'e T2 peu 0 1 R "g$wg G p R )6V6)hp ( ' peu 6g' epr |&%V# $ $ !"e'b`6GyD""gDfb6pc$sf$I$pfTeS h D60ES$1$bpBp&V1&ffRpH1TRf$p$60g60p"$TW F I c a UE x x d Dd E I Ed D I Q` F c U Y D E D I` c x Q E D E Dd` P c E W F W` S Q S E D U Yd D U Y E I Q x U E " R0B s d D I pQ ` F c h D U Y E D S I g 6g)' u "g g 0 g6$G fTeS $$ $ 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$yR"&0c D U Y t I Q` F c U 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CB )86$ ))'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$$yGg6f$z$H$Y e! 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CB )86$ ))'2 10)'&$" j B k e B X w 0 X X{T } = X(B,O) (B, O){T } s0 1 PicB (X) X s 1 GG PicX (X) , B j : PicX (X) j:XY 1 G PicB1 (Y ) X E D U Q S E Dd S U u$0T0H1bs$TE X ^ Z X = B B X B c ([Z]) B F k{-1 t1 , . . . , -1 tn } (I (Z alg ), U, 1) I (V, O) = A F = nN I -n k(A) Z a1 , . . . , an xF FbpQ ` 6 T w0 B t I OX U I (Z alg ) OX X Z X j I = I GX B k ^ Z X = B B X U X an - X Z V j B1 I (Z alg ) OX 1 G X an X an X ZV Spec(F ) Spec(F )an c Z X B B U X -X (U, I (Z alg )) = (U, O) ZV = Z V (I , U , t ) X X t B I -1 k(A) t CorB (B, X) an x B B Z alg k V Spec(A) mi1 ...in ai1 . . . ain }, n 1 Spm(F ) Z = B |k | Z B alg (I , U, t) Z alg = Spec((Z, O)) I (Z alg )|X X an - X X X A Spec(F )an (a1 , . . . , an ) 1 = I (Z) Z U U U U U X X Z alg B1 = Spm(k{T }) k I|U = OU GX X an - X F X X F u i $& i 0bT0up$fe$w66R& i I Q` F` E Q Q S cd E I c ` I { W D x E` S Q x U W F`d c S U F c I D D F I c a` U E W F W` S Q S cd c I T$$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 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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 ߨ 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 ݤ 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 % ( ( % # ! 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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 /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 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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#(nR$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 ~ Rpnؤizix r xuWynآdxi 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 Φ ! 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 k6֩ 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$j#Br#%W!(#( '#r%eEr'nnn d CorB0 (Spec(A(n) ), X0 ) CorB0 (B, X0 ) dn,i = 0 0 i n ti = 0 E0 A(n) A0 CorB0 (Spec(E0 ), X0 ) = (Spec(E0 /ti ) Spec(E0 )) = 0 dn,i : Spec(A(n-1) ) = A0 [t0 , . . . , tn ]/(1 - B1 n i=0 ti ) A0 (n) A0 (n) (n) (Spec(A(n) ) Spec(E0 )) 0in fi 0in A fi A(n) (n) |fi - fi | (n) A0 = (Spec(E0 ) Spec(E0 )) A0 (n) A0 [s1 , . . . , sr ] (n) E0 E0 E0 /E 0 k V n (B/B0 ) ^ ^ B1 (n) (f1 , )A(n) . . . A(n) B1 (n) (fr , ) A A (f1 ,...,fr ) Spm(A(n) ) Spm(A(n) ) (n) (n) Spec(A0 ) U U A(n) A(n) (f1 ,...,fr ) 0 (f1 ,...,fr ) Spec(A0 ) CorB0 (V (B/B0 ), X0 ) u (n) / (Spec(A(n) [s1 , . . . , sr ]))an , / Spec(A(n) [s1 , . . . , sr ]). 0 / Spec(A(n) ) 2 Spec(E0 )an / Spec(E ) 0 / (Spec(A(n) [s1 , . . . , sr ]))an . Spec(A(n) ) (Spec(E0 /ti ) Spec(E0 )) = 0 /1 (Spec(E ))an |k | U E0 CorB0 (U, X0 ) U E0 /U A (n) SingB (Ztr (X an )) 1 E0 / A(n) U DV q I P WQp#H#RG DQ4l4#jxReurWeX#RAdu)b(F VS F D q F I P s V q DS V P q o g s T S q P DV P q q P V Qu#A(WD y q s I D D I S V F #ux`dQ#eUf)D F D pS V s D I F D DS V q D I T S F D D P (UQ%bWQk)UeWeXrXn f`U4 A0 I =p A0 /A G -1 F D P )U##! D DVI Ph DVh s XX#(iXiR#iD qReXuQrWXQ#l4#t##(lXdg PS V s V q D F T I G D q D q q P D F q o D UHXPFRsWhuscVrqDDUeSih(mtD#v!uxlXdo 5D!AA97531)' s q sI DF q 4 2C 8 B @ 8 6 4 2 0 ( % y | v )Uz W!!b4~ Ax x e E p : A0 [t1 , . . . , tn ] (n+1) Spec(A0 ) z X 0uQ!A0 % W x v x ~ } y yz f #QdR!WQbi )( z v y X x wiy Uz b(% y | v (W#WRi #6Ww x n4 %Wy x y x v wx wx ߧ v WRw )( z ~ } y yz b4k 4Q'RpWxbt )( z x E A0 E A E k A0 { x v z Rx n4 Ei 0%xURr dn%RWB4R)f x v y v ~ ~ xy k6pp7 g S hS D p D I G s F q P FS P I G D I F P q I D V h D T VS P P h s I P G g D T IS V V F D V q s pS F D D D ibhR#X#bc4RXe4X#XfR##)#X)UpiQ!!RWbH#(U j AXeXHHru%e#i!!WehEC dn+1,0 = wU D h q P V D T VS pS V h D V q g D p P I G S h D eR!ipXUreXi$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#QѢWe4r 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)4ѩ 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#! 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' 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)4ѩ 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%WŢUiY%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!Ť)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#ȩdUy&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)ũf#`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 ȤF ~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)(&%#! 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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%ŢYf5)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)##yVȩd%g}%Qũ 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 ...