Coursehero >> Illinois >> University of Illinois, Ur... >> MATH 0765

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

V y V V r r i Y q q m v v z eu i YRegbexYehht9 YChyhnrtrsngnr enCFr rsngCrrbbi rsrtr Gg1CrsxgiVsVrnrtr gh Vrnrtr hxix shCvrsnCdrsgFs HgnrqVs#Vrnrtr dCgn fhrqY Vrnrtr Vngt n2fhCgHrFYnxr!CdnCCxCxnCr nghs91xi CrsngHnrCtr rsnC YCrCHnhxnxHqrd1ChriyYVrnrtr esgnnC nxCxngrCrtYy dCgn Rih 2hxix YCrCgrsnC rHn sVh dCgn bhqh V esgnnC ehHnqxhCnC nCnHnxegCsr hCgCCrFYnxrnCg2rsnyHtr hsxi hCgCCrFYnxrns nfxn rsnCxny CxhrVrHgCxixrTgh1 0CT0!1 m i u e w u i s o s i u y xqqxqeeqiqgxQQ xeeQqo o s y p y ou ~ iu w y g s y wu y s qfYGhxxr!xxqsyrxYkqxY' eeqoxqeqG qYTxer qxu ghe!qxtxHA Tqo!qxbFeyexHqxqqq0yoe u s y o s y y i j wu w v v l W u y wu s y wu w ~ w m u i wu y w v v l W u i wu y ~ o s W mw u keQegqHfxn9qxsGHA !sQqfYy9g qqeeh!tehW 0 eu eyqixu qfxkeQxsf}qfxfqxqsxq"qQeqxeq9q y y y e s yuu y e wu ~ y s y wu s eQHxeqe}eQeqxfqxyYQ9rnQxRyosTtRbxrQqqfq0 ~ s ~ g wu wu w u u y A m u y i i w m s neeqexeeQ qxgRx9qxu !!sxY"ekqVqRxiQeqx9YCxxxxyq y i wu w y i o yu o y u wu wu wu u s euu y ~ u o o su y | m { m i w o s z m i w w v v l W u y o i j y w y w g teqqexs}"qyqtqyiCtxGHA 2s#egYkftq! s u w y g w wu wu y i e p o g lm l j i e QesQqxfxu !CeenYrqdeqTY9 eeexqQxqhkFnbiCeYkqe hg fdeq"qx9"qqxvyYQRexQ1qxvtI VqgigeYb` y yu e wu wu wu w u yu y wu s r d p h f dc a X WI U S D E PI B D E D B YQVRTR2Q!HGF!CA % @ % 8 7 6 !25922' % 4 % $ % % $ 51"3210&)('"&##"! m w s y s p y Qeq1xqshxx s y wu u g w ou s y i u e w e su w y g w i u e w w q0xeqx0YqReenYexeQxqg0RQesQqxxu !!eenY}xqg1oYq u y wu ~ y eu u wu w m w s y ~ s wu y s ~ y s kQRqCsbqsxexkeQxeRYTq0CQeqeg RYxGqxqsqxeqes!QesQqxu g w u s o s wu u g w u s y g y i u e w o o y u w uI g e w iqqeqqeqxYqeqx9Qsxq"eYqsqQbdq9m eeq ~ u i u e w u wu s w o o y u uu i u e w wu u wu u dxxqYRqx!qxqreQx}eqs YTQqxkqxxxxfxqRYTxq0! Chxtfeqd}xihr'ntqC#nb#h} r D Tr r vgqh s q i Q gCsCsxq RisdCryxG qu shCYdhH1GiGxhdeCDrxx RYrqh ygnrqVs ynr rFgGxrQyghtgCCd9 gryx}frsrff FgHf gReYnb ' q#xgnyr resgnnCd !gCxe qt YYnCxrt#sF Q ngCr D g D T r pg"g &n hns nhr#xghtrxg qh nrC'rsx s q i Q i"dCgn Yrqh r nD hr2b D xhn CYv r rsnhnCn rxCCd}gnrC' brxrdefCgrCg 'nrC' Y W D hCgHrFYnxrnCqg esgxrnCd# bg Cn '#nh ngCrCxr qQ Yxsssn"rsnYxybrdsnd Q 'xnh srnCtV D nrCfgrhn"fi e YCrCrFnhxnnC2sxCr"xrQC r#rix#f D eC# Cxhn#feqxH 2s1CrssgCsCsx sCfrsCrxbhribdCgn bhrqY Rg#xrh 1 !rrsngnrxh qQ ( 2 d 0 " dH&&cF b E xnCnxe tVngt qd xrxi1 r9ngh nrxih D shCnrCbYdhH'!r nrhf !rYi D hhC hqh nhHrxrh yYnsnCs0}rsxx Q Y W s qa `Yrqh 2ynr XhsrF2tsGsqYrqh H ygnriysVngt xg s }xhdeHgn }hqh VhFxxh2xrngTn A V HgF #h 7 idCrxx hqh brvnrxih D Vxgh9st 6 rsrFrx Vn s YngxsnqnYC D r E U E rt YfVrrsrt D Yrqh hngsxrnEtrxYH'rrsrFrxb qQ Yt x #TRrbhe nCtb S P hC gtb D hrit#Tts1s trrd9 D g gsxrC sng S A irsngxynhxnCg1nrbhe E n !r nhn }nCghhF 7 tsqs tV P E gnrhnih sts1xtr P he E hysChstnhnsht n Q nC i thsCs# qR ht x E gVt E xs ntrns dxys 6 Hrxr2ghHrGrtnrxih 2sYnC#sFts X HNStr k P f D her E x C1 IsHrsyVhth n E gVt E xgnrhnih x H iY YnC HnVRss 6 hrigh hxF Crth n#H Ynie2xssqxhs D rxr2ghHr gGiY hxt"F#nhtnnFgC}s 22YnhsnCGgh r 7 gh hxy A F sr n 7 A nHd rxTgghx qYCrCHnhxn E nC &hFCYt x E rVnfr nrhnih 'hC nsrfs 6 ht x E gVtnr&xrng yngCbs b D t x Vrr xrt C it x dvHVe xCCr A 9 Brx C nhx 78 gVtvqn&s "rnh qBrx @ nhx 8 rVnCsn2qx 6 A 9 7 Cgx V'FxYtxr2xssts1xghHrC 2sqht nx 5 ( 4 2 0 # ( # % # " !d$0$T31)!dV'!!v&$0t! y hCxd rVn X F S Lcor,S LS k NStr S : LS Lcor,S F S F (X ) = lim F (Xi ), op S HNktr iI S S LS S S X = (Xi )iI S S ef DM f (k) S tr HNS S xgCyxY ssns#d A ' 7 rsnhe'nTrrr D th n E rytntrd P A 7 H xrd Rgqht #Vhxihr#xrngTnsnssghVs' H hCgCCrFYnxr E xH1r xrrhng ghstr hCCrFD g xrng H shC1CYdhHg1 sqrtsg sshhFxxhF2grCtr xg hstr"Vxgt9gnrhnih D nrCgrhnh r1g "ng s b i #erxh hniYn's ssrxhCss} 6 xhdeCrxx Ghqh 0gxrxitYrsnh} Fr rsngnsgs E bbhstr1Vxgtrxrxi D YC B q i Cisd s g 2 ghtCri "h$Hrhx r1hrYn htxrdst D rhxsg Cg ssGr F Q n "xrngVt1 }H"nfbysrFtr qu CVVrqgb"snnnqs s 4 xnHdtsysx D Cg g v E xr ' rd2YhhesgCrCd qQ rsrtr FV ygnriys ynr xhdeHgn hqh "ghstr A hHrFV 7 xgYstrsVsnrhstn" xrng2 qg xCC qQ H E A s b a 7 g Cr!VHns H " g#xrb i Q nxd1srHHs q& nrxih9 dersxrxCCdCsdbrsrxsnygHts xrtvCr"Cgx0ghrt}hriYstr"sVsngtgnrxih CxnCr x&esghrg n s ssgeY sr g2 r rsnyhny qQ D " C " d Rct1d B T K b (Lcor,S ) [W ] [U ] [V ] [X] [A1 ] [X] X g k p j +f Grx A xrtd h 7 rrsrt rd1CrsxgiVs#ynr Cgvrsngnsgs gF1rix xeHr s fgVx VniV q r1yhCrxg9dnghFt ngshxrvx srHfs'rhx g #Fr 6 r xrh h Cgxh n # br rsngnrxhHn1rq f D rssxrC rHn D yhns C"f ' 0 ) % ' % # A1@93&$" }rsgCrb rxdyhnhF$8sn snfxgGrQfnh qbq 1 7 F rsnhn 6 A h 7 hrqY VrnrtrD dChyhnr#rsxixssgh' rvrsnHnxeC hYn2xrt Cr erxhCd s 7 snrhstxrQ xgCstsx2 ssgeY x A 4 C2r nfC 5 Vnrhy q x'2hnt}rVn" Ggytnr gxyhrCd qt rsnhn9es ts"VnrV q r(xrh 9rxnrFt9s gTgVx Q yH 6 rsrtr 'ygnriysVrxgtr H1hdnF'hrixhdeCrxx FiY ygxgiVVngt 9i Gn q rd hn ghtgCCdq qQ ' 0 ) % ' % # A 213&$" gt rsnnFrxHr"rsnxhrhnshxqn'sn srsnitr 7 "rsnnFrx s rsnitr& rnrnCd rsrtr YfU Vn nC xts rxin qQ F s xgn 7 xhdeHgn thqh rd2HgnrqVs A ynr hnhYnCrxCdbiY YngsVnxr "i D QgytgHd# br gVx}h rgrsrxisr D dCrxx Yrqh HgHgnrqVsVngt gxrxi 9r&Hn rsrFrn yhns s#Ctb D sCY CrCHnhxnxH!gesgnnCd 'x'nC ' 0 ) % ' % b D ngC1 issrd1i s2rrsrFrx yhns nCxisr 21(&&$#" sCYnvg rsngnrxhvnrxxCr r#Cr Crhn qQ A CnCs r}gnrh s rq s 7 r D r r )rYn&}ghtngx s q i Q qt g xrh Cr rsnnFrxsrshFY1h Y E hxYnxr1rxC1 D snHnxeCgeHr hrhvCgYx2hngshn'xg2Vihy q rxytnr i gnngH si vs qQ rsnhn s Vrxgtr n YCrCHnhx rGrtnfhrxd 2qYxrh nCqrFgCxCxnCr qnxxy'b D xs E ! $$ $ $$ F S S X p K b (Lcor,S )/T W g U A1 k j xbyhrqsyhsbht'ysnrtqrCtRr#nrhnih' qQ /V f /X S tn1Chxx D CChxx issg qQ i gxg#h A ' r #ts 7 rirCrsnCtssCY nCs9nrFn90"stCg! sxix}nC1fs s nChsrsngshnGgqYV1xixgCD nChsyh' Ysgh T c ' 0 ) ' % Cg D yYngCqnrbiHrthChs 21% 3&$#" ' r pvg ( rqYV1xix'nC Ysgh'nrns !rxyhhh qQ hrirCrsHtsyY'nCb sxns s xyHVhxihr E hsGrn rnn r n2rssFg 'H 6 t x r Crth nrsxH 886642$ % % 9 5 7 5 3 H CxCxnCrdhC1bi rs !B r i s q gn Tghr' rxdbhsvxgrsCnngCtsisrd nC0s' 6 xixsxrxQsx rxd Q hfeHCsgisyh qc hns}ssgxxhrsCr rsChtxix ' # ) ) # " # ) ) ) ' " $ # ) ) ) 0 1) ) yrxCrr ns ntnCtnCrhF}ysgisyhxg Cg CrsnC t T 0 Ywd(fc V t g 0 wdfc V b( R T ( b 8x Y 7 0 Y0( X x T A 0 ( Y ' 7 0 ) A Y 0( X( 0 ) g 0 Qc V db( Y A 7 c a xT UY yv c g 0 w0 80 e 0 v t Y V u X S 0 g X( T ' ( H db( vg 0 " ) ) # # # ) 1) 0 # % ( sr' W% 0 pT S ) X ih1"b% ) A0 fc # ( e R UQd( b% `( " S @% !W( UHT) ( S R 0 ( QIHGFED4B8 Y( c a Y ) XV T P % %9 C C aq a g 0 "g T ( H Is2hnhxxy1nr D b nrfgnngC#xgf dtsnGissg#tssisrd qQ A r1 a 7 Fr#srnC#hHRgbhrnxe i s ehHn q }ssCx"VtrsnCbiHr Fgg xRCrsnCysgiCY Crx#r D CrxCC&srnxtrhr sghrxrtg#xr D rsCrx rh E HtsCYnxvrqhriygHtrs gyHt'CChxxx 5 7 A rxCr g2rsChtrsngshn' ' 7 A FVenCRgq A 7 H v gCnCtsyhRs g !qnb 4 ns nqH % 9% 9 5 7 5 3 A@86642$ 0 0 ) ( & esgCnChsyhHr}rQ dtssghhxVsrhH' 6 ' 0 # ) 1) 1' H r i HgCB h sHbnrhs xrh b CrGCrsngCdfxsnyq " 6 s s i nrC1grhnbrnrh# 2nsnsxnhxV D D g D B r i rxdVxix s q 9r&xg r"nrhgsxrCtH"b rsnY & ##"! % $ 2 # 0 " " ( CF&!$&!CCcT2 1t( B r VV q g2rsnyhnyFgHxr thnnqrq E B r i gbYVxixg#xrh 1 s q D D CnC1y9C}xgnxg}xht9VxgtH sghrx0! rf Vgs f xhs G YC9rsCrx}n Cr r gtxrb g1r CrsVy r r D hrbi G ht xCGgrhn"Gisgx hFhr"n G hnCghs D hn's}nh D ssn YstrgxxhFnsxnVn'xsnCr V9bFr YstrfsVsnrr'rs9xhT@"r}RHnxeC1nq 8 b D sgr"es hstrysnrt"rdsg#ndnHd n nxnHd xx r Crr g xrbyhx H x Chnys b D esgnnCd r S f f f ' A0 # f S 1 0 f :XS 2 isrd Cr ' # 3# # ) ) # 0 1) ) # " " $1# X X S S S S = S1 x c0 (X/S) S2 c0 (X/S) = c0 (X/S1 ) c0 (X/S2 ) f f :X S X X X/S X S 0 S f 1 cequi (X/S, 0) 0 X/S X/S 3 S m m eegqGS f xeeex'exq1egRqqsQV'xeqxkhqqexQxu Yx!Txu eyqixu qb'xkxxexsC u s o g i s w ~ y s y wu s ~ y y y y e s yuu y y y y i wu y m y u o y s y s y o u wu m m l tu hg xu gbqxgneqoxTxnRqsqqQxYxGqeeqF'tu yifkeQxebYR R fF exeqx1eeq!u qgeQY RP b'egQxeqqneeqxRqeQ"Qeqxeq1Qxu Yx e w u o s s o g w m m m l sy o y o e w ~ s y w u s y wu u w u y ou s i wu wu s y su j wu y s y i xnqxxxehW sexeeQq#deqqx!#qexhkqxfeqoYC eeexqQw A e g Yxsssn"2rsnhnxhnysf 'xx g2rsngHnys' e s s rd b! "q D B i gn 7 rx D r i rxd9isrd iqYn #Cr ysnFghVs"qrr E h sgxgbCxHrs9 t xfrsxbsqfH D x D xrtrns 6 5 "q D i snC D grhn s # r9t nCChns1 g grb bF sCr ' t 3# T p g R T ( ! g 0 Qc V db( 1) Y b a u Y 0( X( 0 a S # " 7 # ) ) ) 0 1) ) % ) 0 ) % # @( " % 0 &# # # ' A0 ( c a XY V( Qd( b Y S UQbdY V g Y 80 g a ) h 0 ! t R d( sq g T 0 a c UY Y wa b UY g a X ) 0 0 % ( !g dY 0 g U( vb T #V w0 Y Tg g ( Y T X ) 0 % ) # % V Y( c a T 80 g fd( b Y vg X e ( T ( S % " H #htsisd gxg hxgCyxgnhnnghGrxnFtrtx hFYn rsxrxCdRCsG "sgrqb D "q D i s s sV % % 5 7 5 3 r 9ssf D HrsnigCYYn 6 "9 86642$ rrsnix'nC D ixr YC hft hinCts qQ s g rsCht Cg nCts1r gCnCtsyhbs n xrt D hrixr G grsCht1 #Cr nCtsrHrCrsHtsyY9b D rytnqs n xrt 9 ssCYnxsnr Cg i 'nFn&ghb D rrg#s}&Cg srxyr g1yHtr Q qR sCD Ynxt i vyHC igCn E ssCYnxs snhC tsyhxCC rgs igCnChsyhnC' hCnix'nrq 7 "q r A snh gFV si D CgnCn &hChf1 s i VrCsC qg i ' 6 HC}x1F Cr V1xixnC'H 0 gVt p U c0 (Y /T ) T n X f () U U X U p n X/S Y q T S V = f 1 (U ) g xgCyxrsnhxrtnCrCrht ngx'H Z i Hg H 'rsnix'nC x}xixr' 4 gyhrF E ssCYnxsrhxiCstr rxysrFqsxCr} F hrigHrsChtyh'xCC xg g}yhrFtrssChxnsvgngCs gyhrFtrssChxns Q qt riCgCnCtsyh}xCCG E s r t xCnthnshC t x g Cr'h xqrsCrxH "89 % % $$ $ $$ F (zi ){i=1,...,n} S S S T R 0 ( P I 9 ED4B8 % % C C " ' W% g S 1" X Y Y = [Z]X/S S ([Z]X/S ) = [Z S T ]Y /T . T S [Z]X/S = Z Z Z X X Y q T f f () = [f ].[Y ] X f f g V lg(OZ,zi ).zi . /X p /S /X p /S S Z U Y n Y /T S X/S () = f () q f f () X S Y X S 6 S n m yu yu w QxexkiVi C0y eq! sw Ytqeeq0nxgQ y s o s w mu e y w i e s y y su wu s w y s u w m ~ qYxu xqxqxeq!}YfQx q0eqex qefxixfeqxgYt" xexex0ebqxYqxCyxeqxkhYy qsgqYhW ! i y u wu y wu i w u yu s y wu wu s y wu s ~ y i i S 0 E Q ns nTrrr bCgYts qV ssqr't x eth& trYePr E xs he n s P h qQ gnrFn F D xrtvrsnHgnsCrFnhxn s yr}r Cg P rsnYeTs9 ssgCC i 6 i C 1 fj (Zj ) = Z sg sgrb D h xVrxxstg rs}rsngshnnC'b xhngsxr g s1 C qQ A11'3&$#" g s #sC gyhrFtrtixYeP n"xHbs ' 0 ) % % A ' 7 ygCstr'xg P he E nC' g 0 ) ' xrt xnCrxx" nCghhF 'A1%13%&$#" r vnCgrxy1 A 7 H xs'nh 0s r1 i s ns nvg H )i Zj Zj [Zj ] Xj (Zj )ji (Zj )jI/i ji Z lim Zj jI/i ns nTrrr i H P he E nC}hC qQ n xrt"sCHnhxxr' "Crsg gCs H sf D ixnrg Cn CxnCr P ixnr txg"xYer#n nrhnih V#nr rsnhxr I/i Zj Zi i / Xi p S 1X xgHx D xg CrsngCyh' VhC r th nHvYy" H b n}ts i Cn e'x nFn g#b Cg 9gnssxCC# Q ih qX HrsnY!rfhHV}xH#yYniCr D yhnh 9Hrh9 tgnh yC9 rHrhyx E nrH"V#YCf h qt hriHgCnCtsyh}xCC Q # rt xCn!rxnysvhnsH xheP n} rix}xH} ' t 3# T ' c a xT UY wyv c Y 0 g Y ) # # ) ) 0 ' % 0 &# # V 80 g T ( a ( 0 vg X e ( T Y T 0 # " ) ( vg ddY T X R X ( c x ( Q( W Y a a ( x X V Y Y #x UY s0 g ( v S g w0 g 0 ( a W V g x # # # ) # 0 0 0 0 # 0 Y Y g ( !b S ( Ha T ( V 0 Tg c UY g ( ( P 0 g ( "9 871v % % 5 $7 7 ) 0 % 0 0 ) # % ! grhn2gnh yC'Hrqhh i 1ng Q h qV C1b2Cg}ts xr xCrsg E E Cvr xrdq#ns nvxnCgn0gngCx h rrx stgQf qc r E yhnh }Cg i nxYHnf s D hC Q hth n E #Tg xrrhng s ts #xtr Yh x E gnh V S Y " h 0 h0) g S P rsx#Tghe E n hsnC ' ( @% d% ( H) g " ( T t # A1% x c # Ti S T A = lim Ai op iI X = lim : lim c0 (Xi /Ti ) c0 (XT /T ) op i op iI S S iI XT T T = SpecS (A) i / Ti . gi fi Z / Xi T T X i = X S T i XT = X S T (Ti )iI OS Ai OS S (Ai )iI op XT O XT c a xT bY wy! c ( Ha d( sq 0 a ) # XT f1 , ..., fn Xi i0 I I fji () O XT T f1 , ..., fn O Xi f1 , ..., fn = 0 / XT Z Z iI Zj Zi fj ([Zj ]) = [Z] i c0 (Xi /Ti ) (i ) = 0 fji : Xj Xi Zj s ji (Zj )jI/i T lim Zj jI/i fi () = 0 Tj (Ti )iI O Xi Zi j i Xj /Tj ([Zj ]) = [Z] j T ' ) YCrrd1sgCF s "Hn xhnnyssysrVv"xrd xhe'nC" gCgxix 6 P X f ) # 0 D s r r agsC"xrdxYex i P rxr' tnhChx#snnr'f D r #tsxsnCrsrC Frxqs Cg srCyh tnxrvgbqC qu Q sYnnrFns V' qt Q # r t nCCgrhnys&Yy2' g txr gh#f D exgYsvs ' t 3# T # ) 0 " 0 " % % ) " 0 ) 0 1) " 1" 0 g ( T ( a ( !wa ( u R ( ( H g ( T ( 1) v t d( S sq c pT e g V 0 g T 0 a g h g Y Ha t Y ! Uvg g 0 T ( ( ( Ha h S UsA( ( vg ddY 0 g ( vb ( Ha Y( 0 ) V T X Y Tg 0 u TT h ( d( V ( ) 0 # # " 0 0 0 % 0 % # ) 0 % ) # " " $1# ' c a xT UY yv c g g ( S ) # 0 Y( c a Y T X Y Tg Y UQd( b Y t U( !g dY 0 ge U( S vb Re 0 v t y ( Ha T ( V 0 u u ( 1"A871v V Y( c a T 80 g UQd( b Y vg X ( T ( P % G 9% 9 5 $ 7 7 # 0 0 ) % " 0 qV sFrsxrx Q E HrsChnV1xrhgeVxnCgn sCxhxs2hnsnrYsq Cd} D tthCYYn ntsrr ns nFrx#vV H#x YCYn!qbsgGGR ! s V E q D h cgbrb bsCsr HRs qQ 889 i % 9 9% ns nYCs D exghCs 6 Cr r2shC V YCC nCxV qgtxrh" nh rsnHdtrsnhFYn hfeH d Z f f (Z) f ([Z]) = d.[f (Z)] c0 (X/S) f : c0 (Y /S) c0 (X/S) f #tsthYnC' nsrhrnb D hriFgCnCtsyhHgnCfs s gh xCnFrxnys#hns r xrt E gF h x F "89 86642$ % 9% 5 7 5 3 CrYh x E f (Z) Z f (Z) Fye nC&H Cr xCC Q ns nthCCs fC qV nCxgqFCfxnCrhF i1xCC D 0B i CnC9hrihn A s r g v 7 xhxFrx!rrxxxrsngCsgi s s i rxdstgfrdth1H#rq D i9Frx9sb D g}h xCn#hHhx ' 3# # rxxCxnYCCb 6 hnsqs ' ) @% 0 ) 0 1) 0 1) ) ) # ' 0 1) # " $@% ) % # 0 % 0 % 0 1) 0 1) ) # ) % 0 ) ' # % " 0 ) # " $% ' % t ( 0) 80 g WdY V V( T ( ! ( H ( T c a xT UY wy! c g # ) # ( H Y 0 XUs0 g ( Y V (S Y iI ( 0 ) c a xT c bY wy! ( T XUY ( 1Y ( c fd( X ab Y S UY t ( T X X y Y b( vb( T Y a T V XV 0'Y t t Q( W Y S bY WT e ( 0 ) ( c a X g V ( c a 80 g Q( W Y ( S !b( UHT 0 g ( S XV T Z f (Z) X Z % " ) @% ) D hHnb r2hC t T X aq sf T ( ! ( H ( T UY V x X 0 d( a u b( #x b( V( bdY g ( S ( P Y( c a Q( W Y ( Wy ( 1) xh 0 ( 89 ED4B8 P % % C C 0 rsxrxCCdCrvhCqqbsg ns nyr ' # Y x g ( 0 Vqr Cg bbhsrsngshnRg1rgygir' !r1C 4 A0 X a X " pT 8d( H ji lim c0 (Xi /Ti ) op tsrb s i s Q C qc i '0) ' % ex 21% 3&$#" r s ysrFxC" rx ns ntrns r} $$ $ $$ F Zj = Z f (Z) S S S Z/S fji () = 0 S Y S D f :Y X Z S f () # S S S g ( ) () = f (). f (Z) X Y S X p q t f (Z) X S i c0 (X/S) /X /Y / S. 0 Y Y f :X Y q:S S Y S Z f (Z) ' 0 # # ) @% 0 S 0 0 0 1) nC}s D ns nTghsgh#sx 6 rirCrsnCtssCY gnngC& gCnChsyhnCr"xgixnrghsxhr qQ X ' 0 H rxgtissrd fhC niHtr sxYHnhx2Cg yhrFtrrsCnngC 1snhsnC D CChxxxg Hg D i 1gtnn # Q rb qa ht xCn!rxnys2hnsxg Cr i Cxr r" 6 ' t 3# T ' 0 ( ( H g ( T ( ) g Y b8A( V ( u Y 0 V( 0 ) h # " % % ) " 0 0 1) ) 0 @ " 1" # ) % 0 ) 0 ) 0 @% X T !V 8x 0 # ( dY T ( 0 ( a R 0 I Y T B( a x ) 0 ) 0 0 ) # 0 0 # A0 e 0 V! T ) ' # % % " ih T ( A ' 7 V 80 g Y " h a U( % d% ( sq A 7 W# sx"T ( a0 # W% g ( @% dY T ( 0 0 ) x T a 0 ( t# R (fd( b% Y a0 &# UY ( S R R 0 ( P I 98`ED4B8 V 80 g c a c % %9 C C ' # hCChfq YCrCrFnhxn9xHyr q'rsnnF9 0g i ystthqsisrd qQ Cxhe'nCFrrCg #V2hCCs P (XY )Z p(Y )Z () xgCyxrsnhxr qrd D rsnhe'nrrCg P b D ht n E fenr Cr S X Y V'nrb D ht x gVtn1xg V'nrf2Cr xigsisrd1 fr' 6 S H n rssFg " x ndhHgCHnhxxr E nCg rxrv CC 6 ht n E gVtntbeF Cr % 886642$ 9% 9 5 7 5 3 hHgCHnhxxr E xH 1Hnxsf D BrYy q i rvrsnC srxh s E 5 7 $ 7 C 7 1 t nrsx}F #4# B v # 5 3 7 A81 8 %$ 5 57 $ 7 "q D agf e7 sC"xrdstssnd s i A D 'C qQ D rsnCvCYYnytrxg vsnCFrx' # xnxeYnv hritFrxs r qQ rnrFn F ' t 3# T T ( W 8x ( !g b( 0 0 x T T V # " # ) # ) % 0 0 ) 0 5 " # # ) 0 ) # " " $1# 0 (dY T ( 0 ( a T ( s( a u YvV a 0 g Xdb( e R 0 v t ( a R d( sq (0 0 a u (P V s0 g 0 g ( !W Y ( T ( a u Y Tg ' 0 &# # a c UY c WT e g V ( a T #V w0 ( Y g ) 0 ) # % V Y( c a T s0 g Q( W Y !g X e ( T ( S % " V s0 g X X R ( P A9 871v % 9% 5 $ 7 7 c0 (Y /X) p c0 (Y /S) ' ) 0 % ) X S Y Y V f |V : V Y X cS (X, Y ) = c0 (X S Y /X) . X Y pXY : XY X X (XX )Y Y p(X)Y f (). Y <S y aCC yy f C /Y Y p /X X g S = .f ( ). V S S S S XY = X X Y D D X Y Y (X) XY Z (Z) = (X) (Z). X Y Z Z (XY )Z (X)Y (Z) p(Y )Z ().p(X)Y () XX Y Y XX Y / /Y X. S 0 (X)Y (Z) p(X)Y () / XY /X /Z cS (X, Y ) XY Z XY Z/X XY Z Y cS (Y, Z) H xYnCs Tsgx 6 CxrsnYxy brrt D nqCYn' r CC2h Crth n}g 2s nsg"C 'ssVRhCgCCrFYnxr ' # 3# ) # ' % " 1" 0 # ' # ) 1) 0 hn qnFr}9i i "nstrsngshnf brixxhs qQ t T R V s0 g S c a xT bY wy! c v F 7 g T A 0 ) Y W g X x ( a v t 0 A( V e 0 X T S f :XY f = fZ () f ( a e 0 T ( V w0 Y 0 ) ) # % ) # " 1" Y wa 80 R ( !g dY 0 g U( vb Y Tg g ( T X c a xT UY wyv c g s0 g V g v 7 T A X a X x ( Ha ! t 0 A( V e 0 X Y 0 ) Y T # ) 0 % " 1" # ' A0 # 0 # ' # ) 1) 0 ) Y 0 # ) 0 g " ( T e 0 ) v$#1"# t ( Ha0 d( sq ' UQd( b% Y a0 &# UY ( S R R 0 ( 989 ED4B8 0 a Y( c a c P % % C C u " # Vr2nCd g# s fhCgCCrFYnxr E nC!ri YsCsr# xrt E r xnFtr ti inss&#ts Cisd Q qu CgCCrFYnxr E nC} } frd nnstgx1 nCFs} 6 xhe#sHrsyVs P " # Q rsnihssrg'Cnfisrs qu xyrsF sv D ns n r P rsnhe'nfghsgh ' r#t xCn Yy" g gr E D YngxrCn t x E gVtn hbeF xrt E F % 9% C " 89 w" #ts nC YCbsb" 6 2hCC r C' qnCr} Hg Cg D t x E rVn#H Cr D 8r86642$ % 9% 9 5 7 5 3 xg#xtshhx' rxdbisrd2 yHCrhn qV D ysrF Q eC} RYrix s D s nCts0gHr}xgt C qQ igHrsChtyhnC gyHt&x D gsxrC $$ $ $$ F X Y Z cS (Y, Z) (X)Y Z p(X)Z : c0 (XY Z/X) c0 (XZ/X) = p(X)Z (X)Y Z (X) (Z) dim(Y ) + dim(Z) CYYn rxd n ndCgHrFYnxr E p(Y )Z (XY )Z cS (X, Y ) Y /S S cS (X, Y ) cS (Y, Z) cS (Z, T ) ( ) = ( ) . cS (X, Y ) S f :Y Z R R c a T UY yRx v c ) # " " 1 V g s0 g R cS (Y, Z) fZ : XZ X X Y Z f : X Y S f f X HomLS (X, Y ) cS (X, Y ) f = (1X S f ) () S f S / YZ /Y S S f ().p(X)Y S S (X)Y (Z) f : X Y R XY ( ) = pXZ Z p(Y )Z XZT = pXT pXY ZT p(XZ)T XZT XZT = pXT p(Z)T XZT = pXT p(Z)T = pXY ZT p(Z)T XT " 1" # ) " 1" # " " 1 = pXY ZT p(Z)T XT (XZ)T (XZ)T (XY Z)T (XY Z)T ().pXY ZT p(X)Y Z XZT ().p(X)Z (XY Z)T (XZ)T (X)Y Z(T ) p(Z)T ().p(X)Y Z (). p(Y )Z ().p(Y )Z (X)Z(T ) XY Z pXZ (XY )Z(T ) (XY )Z (XY )Z(T ) (X)Y Z(T ) ().p(X)Y ().p(X)Y ().p(X)Y p(Y )Z (X)Y (Z) (X)Y (ZT ) (XY )Z (X)Y (ZT ) p(Y )Z XY ZT XT g : Y Z [g ] [f ] = [gf ]. (XY )Z () ().p(X)Y () ().p(X)Y p(Y )Z () S XY Z (X)Y (Z) X (XY )Z (X)Y (Z) () Z [f ]XY /X ().p(X)Y () cS (X, Y ) () (X)Y (Z) [f ]XY /X S XY () X (a) (d) (b) (c) g T A v ' 7 g v A 7 T # V a CgY D s Q H Ihsgrrq 6 "q B i qV eCx rsnhnxhnys" e h"r FsC"xrd Q 7 hsnsn rsnYxy" nx r A e D rx D rnsxgH# hhsrfH x hYgC HnrC hhC s r i sH s g esxC" rsnhnxhny& D iD xr yHtr srCn n hHhxs Cr g xhnnysv gnngH t x grxyrqsqfC qV ns nbs Q g fYstf qQ xrtrn2rs C qu gnhnngh Q nsrqsnrCynts bYsts gnhnngh gF}xgbhxgCyxq xCr#xrbVC 6 xrtrn g2s n rsnhxr2s HC2s snrCynvnd qQ CrsnxhtsYy xrtrnrghsCrrf 1V'hCCs! nh rrCgb nr Cg D p Xf XY _ i Xf VVV VVVV V VV Xf / Z _ JJ JJ a /% _ / XY Z i JJ J% / XY f Y p Q Cfh qa hCsHr esgCyY g Cg x2 rdVxrb xixtr2tgx2 &sxgts "q D i s A 7 CnC xhnnys'g eVxixr CnC rqs xixr# qQ A 7 rCsC CxxYxs gessxiCt Hg D r CCrrHsRgesgnnHd sq qQ A 7 P rd" sr sC"xrdxYex' sq qQ A 7 XY ZT / XY Z / XZ XZT /X XT p8 X eKKK ppp / XZT XY ZT / XZ. XY Z f : Y Y Z i = (1X S f ) p (1X S f ) = i p p XY XY pXZ Z ([Xf ].[Z]) = pXZ Z (1X S f ) () = (1X S f ) ns ntrns2g r C qQ ns nt gr XY 1 X S f H hssgCCC nh H fstssnrsCnngCqV g't nCC0rxnysvhns#s hxgHxrsnhxr qrd sr rsnyHnsq qQ 7 A H pCxiFxeCeP@sisrd Xf XY Z Z _ VVVV * b XY XY XY f = pXZ Z pY Z Z [f ].pXY Z () XY = pXZ Z ([Xf ].[Z]) . i i D [Xf ]XY Z .[Z]XY Z = [Xf XY Z Z]XY Z = i ([Xf XY Z Z]Xf ) = i ([Xf XY ]Xf ) = i p (). 2shrVsxhnnys qQ H V#issrd1 xtr i txrr" 6 W b W = Xf XY Z Z p Xf T i TT) ukkk / XY Z xiY$HxrxrQ nC i 6 Xf a Z W XY 1 g yhhh g Cg s hC&sb s s Y D #h CYYn hsst x s xrtrn b Cg sgr th n E rytnHCe D D D 8"G 886642$ % 9 % 9 5 7 5 3 hCCrvCgFrx2rsnYxyCrFYnxr rgHrsChtyh nC th nv C qQ 6 7 rCrsnCtssCYfxCCxginxgVghsxhrV X X Y Y cS (X, Y ) cS (X , Y ) X X X XX . /X S H rxg qHr 9 t nCCgrhnysYyxg tnxr} t T 0 ( h ( H g ( T ( 01) g Y ( 0 (dY T ( 0 ( a 80 g2ih T ( xW 8x Y dY T ( 0 V h T ( V80 g YU( hd ( Ha R VS 80 g h 0 ! t d( sq g T 0 a ( a Qcd( b Y a cUY ( w R R R ( P HG9 ED4B8 % % C C % $ 7 5 v 1 67 ' % 0 &# # D Cg ht n E rytnVgrdCr'rsnE r#xrrhng qQ D P gheCHrFYnxr} yxghb t n rytns A qsr G 7 rxqx#n3ns nt E CCnHgxhe P exyv trxCdd gQ2 V2nr 6 hCgCCrFYnxr E nC xg#ns ntCrqht x E hxigCh rytnbFVebnCxgGher9n xrng P ybnrb 6 t xqgrhntF 886642$ % 9% 9 5 7 5 3 ythhgnh' bCrCHn xrt E t x E rVn#VgrdtHg}xixrq xCC9r CVxt #tsCYYnt rxdisrd1 E nxrbr2n YCrCrFnhxn E X Lcor,S Y] : LS Lcor,S S S S S S [X] X S S X S 1X Y Lcor,S CsRgesgnnHd Q CrsrF} Cg2ersxrxCdCgyH nd hisd qC Q nCrhF1CsC'fndx qV ns ntrns2gq (XY )Z D q D B i rvrsnC s s n $$ $ $$ F f = p(X)Z (XY )Z p(Y )Z ().[f Z]XY Z (XY )Z p xgCyxrsnhxr fhC 6 nt 6 [X] [Y ] = [X S p XY p(Y )Z ().[f Z]XY Z = pY Z Z () = p (f Z 1Z ) (). X X Y Y cS (X, Y ) p = (p )1 XY Z f Z S cS (X , Y ) " @% % 0 # 0 " % % ) " 0 0 1) ) " % % # ) @% 0 0 ) 0 " # 0 % 0 ) pX YX Y XY () XY X Y /XX ' cS (XX , Y Y ) n ' X Y XY tr = pXY X S pXY Z YZ ().pXYYX X / / XZ f S 1Z / YZ X X XX Y (). pXY X XY XY pXZ Z = p Y () XY # 3# # rsngshn' 2rxf D #hRgysrFeC qrd!rVn' Rgqhtrxg nC D hD qQ CVx2C r b q #tsgxytnr sH D g nr#x snVsCv " G 89 % % g D B r i s q T c s nC1r}gVx s YH sntsts rsnnFrxts qQ pvg ( c a xT UY wyv c Y 0 g Y ' ' ) # # ) ) ' c a xT c UY wyv ( a d( sq 0 a ( ! S g w0 g Qs0 g Y U( Ha Wa ( Ha T #V 0 % G 89 871v 0 V Y xh ( Y % 5 $7 7 Y ) # 0 # # ) 0 0 # ) 0 # 0 ) # H nrssrxrCisd hngsYtst r rsnyHnCh qQ sgngCsC E Q VVrnr#ns ntnh q xrt rCssV hHCs P #ns ntxhe'nC D ixr R sr xCCtChnV'x sbf D th n E gVtnVrGd D nrhg " gnh yCbrsxqs tnn 6 Yt x E #TRr S ss'9t9her E x9 D rsr hnrq ff G Yt x P E xrxi1 gVt #tr2he E n F S P % 7 $ $ 5 5 b 1v 64663 % nhx hrsr Cg xghsnCx rsnhnxhnys" g hssts qQ hHhx}H Cr nt xrt" g rx xt xrtrns Cr gvrsnhnxhnys g ngC'f sr #tsgysrFvxs RgGgVx &r#ss"tgx# isisr E g rx E D D Cg r!rCrg 'F n xrt gCr ht n E rVnd esgCyh2 trix nhChxCr D rsnnnnr r CVxrsnYxyGg#eVxixr#xbhHhx A P Cr D sr Hgt sr1 n 7 snd2rsnYex" tsHfH CxvrsnyHtr geVsngsVnxr}drryx'tgx rxdqisrdeVsngsVnx F "q D i A s ' # 0 5% # # ) 0 % 0 ) 7 Hn rsnhnxhnysbr eVsngxsHCrsVy' eVsngx t T 0 Y Tg T ( ! vV sx 0 g ( !b ( Ha Y X T X!V s"T! w0d( ( Ha ( T ( a uw g V 0 cBY 0 1)#V t 0 ( Tx Y T 0 a ! X t ( sq 0 A( V h S 0 ) ( c a fd( b Y a bY ! t c T ' # 3# 0 @% X T T Y0 !V sx v wd( a u y Qc h Y g V 0 c Y T ( # # 0 0 1) % ) 0 " ) # # hrYn' tnbrx ' 6 rCsC P Crqr C!rtesgnnCd Cg Cr gCnd#rsnhexfe nC b D sr ryH1eC" rgVnC" s} rnhx" } CCr rsnhnxhnys rrx 9xC 9YTrCsg1r i x # sCrVn D YCrCrFnhxnnCHgrsnyHtrFr1eVxinn RrdR ' t 3# T S : X Y, : Y Z, : X Y , : Y Z ( ) tr ( ) = ( tr ) ( tr ). S S S ' Y V Y TT 0 ) 0 ( %G % C C ( 0 g 80 x ( Uv ( f( H e dY x x X "HG9 ED4B8 % # # % 0 1) ) # XZX Z [X] tr [Y ] = [X S Y ] S 0 Y V Y TT ( H e ( 0 g 80 x ( Uv ( ) t 0 ) % # # % 0 1) # 0 % X T T Y0 V vV 8x v ( s0 g # # 0 ) V 80 g % % hT g a % % 5 $ 7 7 Uv e ( b ( sq G 89 871v # 0 S 0 @% # # 0 0 " ) # # ) ' # ) 1) 0 ( ) = (). hCgHrFYnxr E xCCtnqYHnhx sgxiCbA x7 xrtrn D ersxrxCd xsnCr D srCs YCrCrFnhxn E nC xfhFYn FrsxrxCCd9 ngsFg' D gnngC Y ' A # 0 2% ' # ) 1) 0 0 &# # S XX Y LS S S Y /S LS Lcor,S iI S (X , Y ) Z cS (Y, Z) S (X , Z) , ( , ) . c c X X X Y f S iI op Y lim p : lim cS (Xi , Y ) S (X , Y ) c i op p : cS (Xi , Y ) S (X , Y ) c i Y S iI X Lcor,S S (X , Y ) c (Xi )iI S S f Y XY ZX Y Z f S 1Y S x7 f tr 1Y = f S 1Y S f :XX Y S (X , Y ) = c0 (X S Y /X ) c pi : X X i S X 1 f Y Y XY S # # E serCyhCg A '' 7 esgCyh D . (3) nCdrr 'C' CC'b D #tsthYnC' s Yt xgrhnRg ns nttF % % 9 5 7 5 3 G 886642$ rsnnFrx "q D B a 7 Cxxhnnys rCssressngC s i CsCfg&esgnnCdt rxd1isrd}A 7 esgHYtnh :T S S Y XY = XZ pXZ Z (p(Y )Z (XY )Z H # rsngxC#sisrd vg'b2hC ' t 3# T g 0 R # 0 X T X x RY! T # 0 A # # $" ' nC b Yw0 g Y g c W% Y U#V 0 ( 10) # HG9 ED4B8 V % % C C a Y( r rsnC CrCrFnhxn E xCC nhr2r CrsnhexbCVVr' 2C1r# P XX Y Y XX (XX )Y Y p(X)Y H hxgCyx D D D gnhnngh sisdv nCrvb Yt x E rVn Vg#r V'ngC rHb s Cg ht n xrtrnrghsCrrq rx dnyhsff!ht n E gVt9e xg Cr hth nqgxRr xrtF ga ( ' ( 0 W% dY g % WwAv !65 7 Yh n E E xrq# xrt#nqhFYn s!rxxiCA x7 n xrtrns i Qsxhx rxg"xix"tr} qQ X Y XT T Y T S :T S S X T Y X X cS (Xi , Y ) fi rxg'rsni"t sgrb D YV'rsnixrqrCsC P srxvsnhC&V qX Q Cr D fg"tsxYex# P rsnhe'nTrrr P rts1rsnhe'n' F # xrt2rsnHgn sngCtrxg S Ct gVtqg (Xi )iI S E ! # xrt E rstgQ2CgYt x P % % her E xChn2 xgtb iyHC I G 89 $$ $ $$ F X cS (Y, Z) X f:X X XY (Y Z ) (XY ) = qXZ Z q(Y )Z S X / XY /X XY = qXZ Z XY Z (p(Y )Z XY = qXZ Z XY Z (p(Y )Z : Lcor,S X/S cS (X, Y ) cS (Xi , Y ) XXT S T T = ( )T . XX Y YT (XY )Z f i : Xi X i (Xi )iI pi : X Xi (fi )iI : X Y Y Y pi p i (XX )Y Y q(X)Y (XY )Z (XY )Z (Y Z ).q(X)Y f / S (X , Y ) . c / S (X , Y ) c Lcor,T XT /T T . LS / XYT / XT ).(p(X)Y ).XY Z (p(X)Y ).(p(X)Y (X)Y (Z) (X)Y (Z) (XY )Z (XY ) (X)Y (Z) XYT ) XT ) T = XY () XYT XY ) cS (X, Y ) (2) / XY /X (1) 0 ( Cnxhtshnsghsgh P xgixxg gvVg Cr}CxYex!rrr"xg}inxgRgxyH n xnh qY Z qXY / Y T Z X T Y T Z X T Y o EE PPP EEE n y yy PPP EEE nn E yy n EE PP' E yyy nnna yy y b wn y p q / Y T Z XY Z X T Y P Z X T E o Y X S Y T Z S PPPc EE y dnnnn EE PPP yy P' wnnnn XY E" |yy Y Z / Y S Z, X S Y o X S Y S Z XY Z XY Z pXY pY Z % H Hrsnigsisrd sCxnysVqngeb rsnnnxr2Crhn br D CVVrsrsnnnnC1 qQ CYnVntsFrq # x hrVhth n1gHrsChnV2 &sxyt rxe2f D rVn ' t 3# T T f : X Y XY [f ]T = [f ]S cT (X, Y ) cT (Y, Z) XZ ( ) = (Y Z ()) (XY ()). R ' R V s0 g Z ( !g 0 # ) 0 g " ( T e 0 ) !$#1"" # t ( sf' Q( W% Y a0 &# UY ( S 80 g R 0 ( P 8" ED4B8 c V %9 %9 C C T Y u a q Y( c a # xCC#snC gGytsr sq 0snCrsf 6 g2rsnxhts rCrg" nd1rCg "nrHVvhChvntt2hnsrsCrxghsCrr" Vxgh 6 ht n E rytn H D ht xgrhnGg xrt gVtnF 0 T Y ' 0 # ) @% ) y0 U( hsrrCsCresgxrnCd rxdbisdYCrCrFxxrxCC Csgr"erxxiCx xrtVggh'sqx hhC n xrtrnsTrrr# Vgrd}g Q ndisrdsrF qC Q 0 nh ns n E YY Q h qV " Y( c a T g b 0 0 b g ( !wa ( u R Q( W Y !g X e ( T t g g % ) # % " # Y v X t t UY yv c Y c a xT T 0 c a xT UY yv c g Y t 7 A # 0 A@% # ) # # ) ) # ) E cY T #0 0 Ha3# UD0 # vA@% X t ( e 0 W% dY g S " b% ) bY g % ( a0 ( S 0 # ga ( g Y " 0 ( P A ' 7 % % C C 9 ED4B8 ' " g V # 0 # c % ) A0 Qc h Y Y ) ) T ( T #0 0 a vA% X t ( sq A e7 #tsCis d }hHgCHnhxxr'd"g tnrCbghsnnFrx"nrCt C # 6 0 : Lcor,S Lcor,T ' ( H g c X c ) 0 0 # % Y c pT e g V e 0 v t ( sQ Q( W Y g u a q Y( c a ) ) ) # " " $1# ' % : T T cS (X, Y ) cS (X , Y ) D D ( tr )T = XX XX = qXY = XX YY YY YY ( ) XX pXY 0 XX pXY LT LS XX (T ) . qX Y Y Y () . YY YY T S / Lcor,T . / Lcor,S () . XX . ().pXX Y Y () X Y S XX pX Y Y Y () f :XY D T /S cT (X, Y ) # G ' t 3# T X Y X Y :T S T XY Z fT f S T T XT XY () (XT )T XY Z " 1" " 1" # c a xT bY wy! c ) # g T A v ' 7 g T v A 7 # ( T X y XY : X T Y X S Y cS (X, Y ) ns nvg CshC1Yn' 2beF2g# hHCs' Cg sydnix xV'nh D yr}rdg Cn Cr#rx 9 rssrQ9"rsrFrx sVC dGifg CrY x# a ) ( 0 6 7 7 7 5 $ xnb ' h e $#" # WA0 b% 3 8UY ) 0hsnCsb x% 81 `"46A` th nqgxF0sqHrsYnh nd " # " 0V!!$ XC C ( 4 # xX ) ) 0 0 # 3# ) 0 Yn 7 rsrFeC Hgr r gyHChn' Cg#nrq srbCr A hth nTgR# xrtd 7 Hg gb dRghs"xhxrD Crhn qQ A t x E gVt"frdfi xg#xb D rsnnnxrfxH r ' t 3# T # h x E ' c a xT UY wy! c Y 0 g Y # # ) # ) # # ) # ) % 0 # & ) # 0 % ' % % ) " ' # 0 A@% 0 # 0 250 ) # 0 " # 0 2% ' " # ) # 0 0 1) 0 &# # 0 0X V g h S Vbs0 g S UY wyv c ( c a xT R Y X S ( Ha (fd( b Y x U( T Q( W Y pT S ( e g aH UY UT ( ( v ' 7 c a Y ( c a c h g T A 0 V ' t ( Y ! X t ( sq A 7 T v 0X t ( Ha g T 0 a Y( c a fd( b Y c xT c ( Pf %89 871v % 5 $7 7 T !g X e ( T t bY awy! c2( WA ( 1) a bY g ( S xh 0 Y U( T ( W T x : T S ( ) = . T X R T S ' ) 0 # q snysVb D Yt x rsCrx hfeH # xrt gVtg#CyY"d D hrihn C' snnndqrnCdFghsnxr qs g#xnxeYn D #tssYhx rfyHfeC rxd9g }ng "889 % % x : Lcor,T XT cT (X, Y ) Lcor,S (X T S) XY (). rxCCd"CC'b D #tsthYnC' s ht ngxg xrt gVtn#F % G % 9 5 7 5 3 "s886642$ D P sr r"xrd rsnYext Cg C Hg rHs9resgxrnCd snC XY XY XY = pXZ Z pY Z Z Y Z ().pXY Z XY () H phCsCisd r2YsgxrqesgHY qQ $$ $ $$ F S LT XY XZ qXZ Z Y ( X) S Y XY XY XY XY XY qY Z Z ().qXY Z () = pXZ Z XY Z qY Z Z ().qXY Z () S :T S ( X T Y ) X S Y X T ( Y ) y i Yi (x) (yi ) XY = pXZ Z d q ().c p () XY = pXZ Z d q ().d c p () XY = pXZ Z d q ().b a p () XY = pXZ Z d b b q ().a p () pi (yi ) = x S T Y LS p i : Yi X D rsnhn X % eysnV ssChxns tnnq# 6 xrtsg 9 xheP n#}snCr # S xqh}s D srxCrsg G th n E srxCrsg rVnt'H ' t 3# T 0 g Y ) g& ' t g ( a Y ( " g 0 ) V 0 T Y Y T a ( c a c % C C # 20 b( % ) A0 ( T 0 # t g ( a ( 8x ( sq ' Q( W% Y Ha3# UY g ( S 0 ( P % "G ED4B8 Vhxynhxx rih Yn Vnr& 6 t x E gVtn H xhdeHgn Yrqh g r'nrhnihFCCsdq Vbxgh 6 gh Vn s rnCdq i C xhdeHgn Hgh hx1 iGxhdeHgn Hgh nrhnihCHrFYnxr Vnr 6 ifHrxr2ghHr % % 5 7 5 3 giY hx rsn g i xhdeHgn ih Yn "G 86642$ sr rsnC $ C 5 5 7 5 3 g g#fggh2 9sghh % w" w 86642$ A rfrrsrFrx Vn qrdfhqh Yn 7 nhx 7 Yrqh Yn1g xrrhng A 6 % G% 5 7 5 3 Vvnrsv IHIG 86642$ P he E nC' g}ts' b T c h qQ i g Cx}r& 'r xixssnCh F ' s pvg ( hribhqh Vn gtnrhnih frdrsniixnCrqs th n E gVtnhxyH#yhCCsfxnHdbFC gvstrQ' D hrixr #nxgnnr YntrHgr}s HrxrghHrtxYrqY sVh ndqnCd1 i tnFrxr iCf h V Vx (X) 'g rxr2gsFg br g ysrF r"FC# #Hb D sxr" qr'CrsngxrgCr 2sisr x X X h F (Xx ) = t x E x' b rrsngnsgs E H 6 snhnCt2Crsg#sgnynxh D etvrC#nrxih} qQ h xhnysrFRr #ns nt' 9inxg gVV H sC Vns rtnrxih F ' 6 n xrtrnsvrs xrt2hCCs' g Cn s r rsnhxr ri ge ysrFt Cgt x E rx 9 g}s x E s# V pT X S a s0 b 8UY " g'ysrF Cgh x rytnvtF ( a (0 a # # & # ) % 3 ) ) xgHx D riFg1xgCyxY sDsVgdsr2CrGgh sVC iY hxi rb rx Br q i ndHsgYn D hrixr R i Vns s hhCss stgQ qQ n xrtrns gt ns n"hCCsf 'Hg ns nfrx 9 gG D nttFrrR i Cn p W q U k j YnrCyn2gYnng'xg s !Br q vr ( vg ddY s i Y T X V b( a Y X e 0 Y V V hCCsxhrir ssrQ 'x D i Cn ) ) 0 ) F Spec h F F (Xx ) x j p1 (X U )red (X U )red x h OX,x X X X Y X h OX,x X Lcor,S S (V, y) S S F (X) F (U ) j S x X V X (x) (y) x X LS S f :V Y x PS k p NStr h Xx /V p /X q / F (W ) F tr PS / F (V ) S V Vx (X)op LS lim h NS Y LS [X] S (X, x) F (V ) h Xx = LS [X] F : LS Lcor,S y (p, j) lim Y X tr PS V. x F F V h Vx (X) Lcor,S LS LS 1rix#xysns}C qQ S snrhxH"s str ixr & (Z) ZF C . rsnY#t D i D hri sVC s q )sisr ht nrssnC rr ghCh hrixCC D nh gFy sisr ssnxxyr}s str q fvHg xYHnhx gVCTn&" h qQ F Y S X C (Z) C (Z) nfYnsfs 2Cg snCr f gCnChsyh'xHxg ght xCn& hns hCYng D nCsCs V hxx D n F d CyY !rqnYnr' i nrq bqC qQ ns ntrns2gfA s rsnhnG 7 i rFC rxdxrx h OY,y Cn Hrt ) X Y g d g Y a (0 W 8UY g ( S ) % & ) ) xyHg stgQrsniixnCr2 rsnnnn h n srq 6 t T c a xT T ( v ( vg ( a Y b sbY t U! e ( b ( a 0 UY yv c Y Y a (0 hT g c a xT UY yv c 0 g Qc e X ` pT X g 0 ( sq 0( g g a T(v ( fI IG 871v V ( c a s0 g fd( b Y a bY g ( S c P % % 5 $7 7 ' # 3# ' # % & ) ) ) # ( # 0 % 0 ) ) # # ) # # ) 0 0 " 0 % 0 &# # rrYy q i 1 rs rxTgrsnixrxhGCrsVytg9rsnyHnsisdq qQ s hriYnytgssgngC xhdeHgn Yrqh gstr'rb D strs n rxCdqxr sys hth nGg#xrng #VYngxhxrxi'xCr shxHr2s E rvsg 2hxixr V'nrs 6 rsny E th n0rsssts hfU fhC r nhngsxr ri gCVxsd E V1xgh 6 hth n E rytn $ 5 7 $ $ % v $6wv f 6$ #"% r ns n E H rqyhhhr i t #ts D ssgCs sg 9 s CCtb9 g Cn Vgd gqysS V' i Cn9nys Vtnr b D r Cg i }riFgtesgHY} thhC h nr Hg Vgd1g txg x= j=1 n snh gFV Vqrb D sg 9 nr Cr rixheP n Cr nCnCy gysrF rsngCC"r ghCsbr xx 6 g H1V"H D rsnheTsqs P rHs' gHssd gQb fX c0 (V S X/V ) n = i=1 i .zi Y pX qX 0 c0 (Y S X/Y ) X c0 (V S X/V ) X X c0 (W S X/W ) . f Hyhn rfxhCn' i #n1rq b D sr #tsRr#ehHnCs s P CxYex!rrrt }nCr Cg $$ $ $$ F W = V X V i n+1 Sn (U/X) = UX S n+1 ... 0 S (Y, U ) S (Y, X) 0. c c C = ... n S Y, UX c 6 th n E xrtTghsgh g rsnnFrx 2Cr ysrF }i g#Cx0ghrssnC gVt hxsrFtF 1rs xY YCVxnys (Y, y) Y y 0 I LS {fX (z1 ), ..., fX (zn )} f (zi ) = w = f (zj ) i = j x = (zi , zj ) W S X i j w = f (zi ) = wI (w).w fX () = c0 (Y S X/Y ) Z s : Z Z X U C (U/X) i=1 n n+1 n+1 (Z) . S Y, UX c Cn = S Y, Z X UX c X LS [X] i . Z xp1 (zi ) X cS Yyh , X S (Y, X) c C C = LS [.] n UX zi (Y, y) fX p : U X S n U X S (U/X) U/X C (U/X) S d LS C (U/X) LS [X] S Y Y j . p () = qX () X V S X p q 1 yqX (zj ) Y fX p:U X p LS [X] ZF wI y. S h Y = Spec OY,y p, q : W V w I C Z pZ : Z X U Z (Z) ' # % i (w) = i j YCvh HgYH i !gs vCF1gChVverCyh'riFg i rixxrq f D srnCh A YH nhnxr xiCyY0ghVsissrd' tnysYninCgnFvgxn2s qQ H 0 F (U ) O F (U ) A Xhn Tr9ytxix ts'xgYHCgrsnC9xH bxgHx rxgF g1eysngxtrb 7 rxg9rsngxtrCisd 1hsgrq C 6 g Cn i sVh "xs#x C qQ ns nth D r tthsVC 2 ythh r n Ynt#F i yhns "ex D snhCsts ' 0 Htr F cS (Y, X) a FNis (X) 0 F (X) = lim Ker F (U ) F (U X U ) . H U X HgCHnhxxr Cr rs r i Cn r 'gxxCxn& r i TrY Cr hC hC nFn} A xhdeCrxx iY hx rsni#xrdrxxqgngC # i rb 6 rsniHY tsnC}CgHrFYnxr1 E n E v1nr Q A YnCnHnxe bet qX Xns ntrv"s D s 7 Cx rsnyHtr xhFYnGrrVnr rRdCgn Fih YnFgqxnCnxe'nGysgisyhG F HomPS (LS [.] , F ) r!gh hxGgrsni#xrdCgnFrxxiCt D sghVxsY F F (X) ' # 3# ' 0 1) g a X Ha W UY 0 % rsCsHTghsCrrt'rq b D dCgn gh hx t T ( a u #V 0 Y 0 g c v t 0 py pT X g Y ' 7 a W a u Y T ( t Y0 pTy a u ( !g ( a YU( sx t e g T T Y gT g 0 A 0 h Yt g Y Y T ( t w0 WA Ha u t g ( a Y g Y Y gT A 7 ( x X d X g Y Y ( T ( a d( sq (X 0 ( 0 a c a xT g UY yv c W 0 0 b g 0 ( a a W( % V sgrf D srghV n 1 V 80 S g Ha10) u W( 0 g&# bY g V )% Y 0 # t g ( a U( 8h e $## c # % b% & sbY ) Y Tx " #a a ) (0 ( Y gT Y Tx P % % C C h ( # 0d& ' Y T ( t w0 WA0 Ha10) u t g ( a ( 8Qg ( S 0 ( "G ED4B8 0 #ts q i #ts9gryx 9gY #sisr rxCCd RgrsnihssrVbisrdqnhx' qQ ' ) # " % ) # % d xrtRg2rsnYsr V#yn"r CV ht xgxrng V hxihrxrrhngrsn tsxrny str hfU hnyhg fg bQCrxr ifsq sn = s X 1U n+1 : Z X n+1 X UX cS (Y, .) n+2 Z X UX . C (Z X U/Z) Z H 0F 0 : F H 0F a ( a 0 A 0 F LS 0 (Htr F, ) 0 " # % ) 0 ) 0 Htr F 0 Htr F = H 0 F ' 0 F Htr F % ) 0 0 1) # # ) 0 # 0 " 0 ) ) 0 0 0 1) U cS (V, U ) ) @% ) # % ) ) 0 0 Htr F cS (Y, X) , a F (X), F ().a = X (a)Y . U 0 Htr F (|V ).a = Y (F (U ).aU ) F (V ). H 0 F (X) aU F (U ) p U = |V tr HomPS (LS [X] , F ) HomPS (LS [X] , F ) . / HomP / HomP LS [U ] , H 0 F S O / HomPS (LS [U ] , F ) . S 0 Htr F LS [X] , H 0 F V LS F F Y U X a LS [U ] LS [X] A 6 7 nrhnihghHrhsC gx1# g bgY Yrrt rsnxxh' qQ dCrxx biY hxixr# 2Vrx xhdeCrxx hrqY hx}g nrhnih2 s# snt#eC V hxCxnCrts"xhdeCrxx RYrqh g9tsxCHtg D nrrCrsnxx s G A xrng Gr 7 ' 3# # xstrVqs g bYsts ' % ' 0 ) # ' 0 ) " " 0 1) Y U( vg ( a Y ( Ha h S ( H e Y 0 0 V g Y c Y U! e ( b b ( 8d( a T 0 g hT g V 0 ) 0 ' # 0 % % ) 0 # ) " r1xg}tsxCHtsnhnCi CvCrA xngxh' Rgq#n hx9g1hxgy1}Yer2Vg 7 xniCrrn1r D tChxxgnnr P xr s 'nrxihtrssFg#rs'xrxi#rssFg"YC gx1'fi VrY tr PS T T g (0 ! t Y ! pT s( (t g Y vV g c ( H ( s g Tx (Sg 0g Y # # 0 2 ) % 0 # " " 1 ) rxCCdfrq sn !g1HnV qQ t T ( c a Q( W Y Ha UY ! t c g T e c 0(YY t ih c g t g UY ih g ddW( 0 T 0 ! X t X t ( e v t ( sq T a c h Te g T T UY U!pA S vg Y !V g ( ( ( x c c hT g a % % 5 $ 7 7 U! ( b ( 8q "G 871v NStr NS # ( 1) " " 1" " 1" # 0 2% " 1" 0 1) 0 1) # 0 % 0 # ) g@&# bY g ( a0 R g d( " X # & ( 8x ( a0 t # 0 # ) 0 g 0 # ( a0 a10) ""G A8 Y c c Y ) T 0 % % 77 0 ) % nrssrxr#issrd Cr rsngtxrdCrxx g x hFYn 1esgngC9 0Ysts#tsshhxf 'sythnie'ef qQ D CCxy rsnCxnCrnhr' # 6 hCgHrFYnxrGrqCx ressngCtr rsngCyhrs (rxdsnrY i}CY} 6 D sntCgfA gVxv g neCv st H A 6 g9A Cg 7 rsngCyh#xsrhr rqdCrxxbhCnCn2 Nis : H 0 F (X) = lim X U X sgr1b g hri yhns sgqits "sVg t Q / F (U ) rxg'rsni"t bhHCs D gngCq X F (X) HomPS (LS [X] , F ) HomPS LS [X] , H 0 F = H(LS [X]). ixnr bsnCrC U Chyhnsisd qYsts ##ts Cg rqri Vn H D tsq $$ $ $$ F 0 0 0 H(LS [X]) H(LS [U ]) H(LS [U X U ]). / H(LS [X]) O / KerU U X / H(LS [U ]) O H = HomPS ., H 0 F / H(LS [U X U ]) O / F (U X U ), U H X D a H 0 F (X) U V 0 Htr F () : F (X) F (Y ), a Nis (a)Y .. X KerU H(LS [X]) = HomPS LS [X] , H 0 F cS (Y, X) X Y D 0 Htr F (|V ).a = Y (F (U ).aU ). nhx 7 D sghsrsrngx2h qQ nhx 7 i"1yV hC inxg sTgngCq F tr 0 0 atr : PS NStr , F Htr Htr F ( a R T ( ! ( v T 0 # # ' T 0 v X t X t ( e v t ( a T # 0 A@% " 0 # 0 0 F Htr F NStr H 0F aNis / NS . PS atr / N tr S tr NStr PS aU F (U ) U cS (V, U) Nis LS [.] 0 ) ) 0 1) 0 # % # 20 0 Y X c ( c c pT e g V e 0 v t g u 0 V g t ( v X t g 81)#V Y( 0 ( T 0 ) ) # " " $1# ) # 0 " # 0 A@% NStr ) 0 0 ' ' ) S tr PS X atr NStr NStr 0 # 3# # Nis LS [X] # 0 " # % D E qiY Vg"d hhC hqh ghHrqr"Hn"rnCxgCnC Crxhd T c Crxx HnGChnb "hbeFbCxT$8 xhx"r 6 ' pvg ( #tsqCrsVx' !gqHrsnCr 6 % G 9% 5 7 5 3 sVdengn CxChn& VngC Iw"G 86642$ sbbrvh Cgxhqe' Cg xrxi Rg#enFrx sCHnhxn# ndisd h nrhnihRrrnxtV2fg#rsit qQ dengnC"bdCgn fhqh GrCxrnChn D sxgxnYh nnrf D 2rsnyHns dCgn hqh H ' $" ) # " Cg ) gV 0 c Y # 0 xg X V g 0 X TvA0% X t aWpT e ( a0 gwa0 ab% bYQ80 '% vV # T8xBT!# Y0( 0 0@% e g ( 0 a e ) Qg Ha10) QU!# e ( 0 b% # 0 T u hT g ( Y X c g V # 0 # c % ) TA0 Qc h Y g t # ( T X 0% X TA0 '(d ) 0X g Y 10) vV g ) hUv# ( 0 b% ( sq 8AIG ED4B8 T g a % 9 9% C C 1xxCxn0grCrt F i HNis (., FNis ) ' 0 0 1) # 0 @% 0 " % ) # % ( 2g r rsnyHnC s hHtrq bsgY v $ 6DC A % 7 5 7 $ 7 Y T ( t 0 py a u t g ( a U( sx g t ( T X X y Y b 0 0 b g Y a Y gT Y T Y T T g g g t g ( a ( 8x ( a 0 R Y gT Y Tx P % % 77 T ( e ( 0 0 ) 1"g v# t 0d( sq ' Y T ( t w0 WA0 a10) u t g ( a U( sg ( S 0 ( 9"G A8 " T a rsnyHnC rxd isrd i 6 Cnrsrtr YfU rsnyH' sgg#s E rq n9yhrqsyhtsgygbsehHnf D n rx D Tr fsisr s i ss s iY ' D P xrng' sxheb dCgn !gh #yr rdi invxhCYnnr#f D xhyhnCr g rsnsrnhxtrsnheTs P rsyy vVYngssr"xg s}hS V Cn1V 5 D gnngC rsnheTs Crhr D xrngrssFgthsCh rn1"hH D P xrxi qQ nrh ' ) ) ) # 0 0 ) ( # 0 1) 0 1hsnC rdyV enFrxt D #tsCh 2s t T c a xT UY yv c Y 0 g e V R Y WT a a xT c R h g h R Y fd( b Y Uc Y wayv Y ( aUT ( ( ! t R T ( ( v( S g0 b(UT ( v V ( y80 0 g 0 T g ( v 0 ( a Y Te ( 0 t w0 A" IG u87t g ( aY 1UT v(7 ( ( c a c U h Y % % 5 $7 T V X T % 0 &# # # # % # 0 ) # ) # 0 0 0 ) xhdeCrxx iY Cg rxCniH D n xrt rrCgCYfCrsHnVbrnsr dCrxx YrqY Hghqh rd'Hrxr#V 5 r rsnxxhv s hqh g ns ntYCC ns n gr qQ t n E rVn rF rsnnnn}es' bghsYrrt rnCd !rxCxnCr' qQ dCgn bhqh gtnrhnih 2s YnisrsFts"xCC" D vstn" 6 g Cnrd ns nt#nhrqhri2rrxb rytntyYnynhxxiY rssFgxd s nh sghsnxr D qQ F F = X/F LS /F ts Hg t x E lim ih Vns }hC Lcor,S NStr ZS (X) i>0 X S X : ZS (X) Otr LS [X] i X : Exti tr (LS [X] , F ) Exti S (ZS (X) , Otr F ) = H i (X; Otr F ). N N F NStr F = S G LS [F ] = F tr G = S X/F LS /F X F Lcor,S I Lcor,S lim F X/F LS /F NStr NStr ZS (X) F lim LS [X] LS [X] , X/F,Y /G S tr S lim LS [.] (LS [X] tr LS [Y ]). S G= Y /F LS /G H i (X; Otr I) F lim F S X iN LS [Y ] . NStr i=0 NStr # ' t 3# T X ZS (X) , S Otr I iN i X ' # 3# ) # # rnCdg grrsxi"tr rsnthrsrnFnf D ssrhH Fg9snh gFV xrHsn1 tr OS nHd dnnd b yeBr snHg P hrqY 2rsnCxyr sgrf Cr}Cxi Vn 2srrnrx1xhFYnnHd1s &hC nHd rx "xfhsCgxCxnCrtgx' D sxgts : NStr NS S : NS NStr S : LS Lcor,S Ggqi sVh #n1t n E gRrqri Vn Ch ( g gT a rnCd9rsnrGxxeCr"gRnCr dY W pA Y S g ( sq C 7 C $6 E 3 %$ C #E $ A 7 % WwAv !65 7 P A enFrxyeBrvhxYH #snrYqCrxr D nG 7 xhdeCrxx Yrqh Yn2sghYnCxrssrsnHCs rtxhdeHgn ih F HomNStr (F, G) (X) = HomNStr F tr LS [X] , G . S G ' # 3# ) ' Y !V g c 0 1) V( T 0 ! X t S ( a bY # 0 2% ) 0 C' 6 xhdeCrxx 9Yrqh F Cr t T 0 V ' ae T g g c a % 5 $7 7 hT g Uv e ( W g V 0 '( 8q % 9"G 871v # " $% Y ) # 0 % " ) # Crxr2gsFgRr ns nsh2rbA sD rsnY 7 iqFC} gHs trq b D th nhxsrFyrbd gnngH X,X LS (F (X) X,X A Yrqh g n xrtC2gHh'b D r snq nhx 7 g#rqA nhx 7 XLS XLS dCgn bhqh Rr xrtsvr1q f $$ $ $$ F X,X LS F tr S Z LS [X]) (G(X ) Z LS [X ]) F tr G S F (X) Z G(X ) Z cS (Y, X X ) (F tr G)(Y ) S LS ( ) tr ( ). S HomNStr (., .) F (X) Z G(X ) Z cS Yyh , X X (F tr G)(Yyh ) S ( ) tr ( ). S :T S F (X) Z LS [X] F F (X) Z cS (Y, X) F (Y ) , LS [X] F (Y, y) NStr LS [Y ] LS [X] : Lcor,S Lcor,T T h x E dCrxx Vgh f iC9b D ri0xhdeCrxx ih s A 7 H prnCdfesisd H0sb xiCxn D S S T ' % 0 7@% 0 F (F ) = lim LT [(X)] # X/F xhdeCrxx ih b nqyeBr sxs g hngsYtsqhsstt#tsh qQ P nCdfxhdeHgn Rgh Ynynhxhnhngsxr VYh xr rxCd gnCxVrfdCgn hqh nHd i rQ V hCnCe@s q8 A e7 rsniggnCr rx ng P Q A ri0xhdeCrxx ih !s ' 7 hriH9b G (G) = G T D 0 LT LS T S / Lcor,T . / Lcor,S S tr S Yyh 0 ) # 0 r D C qQ sybnh xhdeCrxx A F rryxG 7 i dCrxx qhrqY rxrrhng" sxtsxCHgvxixtr rxd issrdnhx" D dCrxx hqh gxrxi sCqCn1s n1riFg sxCxtxtr A gbqqbe1 hrVbCxYex 7 YCrCrFnhxn'nCnqhFYn P Cr rxxiC"s xrtrn 1i hCgHrFYnxrnC'drr nrH HgHn xnFtr'g xCC qQ xrtrn r"YCH n xrtrnsrrCg qQ S F = lim LS [X] X/F s hriqdCgn Rih Yn' YninnRgh ' 3# # CxH sis hri'dCgn bgh F F : T S ( ) . S D c xT ( cY % % C C " g g # t # UY ) ayv# c # Y ) b% ) 0 # 0 b% g Y 0 Y ) 2( T ( a0 R a0 &# UQY ) t 9"G ED4B8 xYex gVtns P esgHY gxyhrCd rxqrsrxCCd etn2hh'b1nnh E rg#s 4 T c hxgCxgHC ssghx'rxCCds D sgx 9 h 6 ' Ys pvg ( rsrtq nHd1rg#sYns!rCrsnh2Ynse' qsV qQ r sYC # 0 A@% gVt1 nCdTgCsrt' xxCxnCe rnCr #sr 6 r ) 0 T " 0 n xrt" i"nF ' ( g c G% ( H) V g 0 # ) x ( % = ( ) : dCgnq bhqh rd CgHC' sC# Chn CrgrtstC qf Q P ysreB2 sx Hg nCdg C sy qQ g r sYC rnCdVrsrt nbiHrxxeCr "Cg9xH 6 ' s ns nTghsght1q f D rsnr grD tsgr nChnV2 qQ gCsrts i nFn sgCs 0 trxgrsni"t VYnixnrxnCdnh i 9HrsyV iq sgHhrsrxhx"trnt Cg}snhri sVh xhFYn 0 0 : NS N T , 0 : NT N S . Y S X Y T (X S T ) cS ( Y, X) cT (Y, X S T ) ( e 0 wb dY g ga ( % P rxCCdsreTrRrrC' xbhsCgxg#hn1tgx qQ HomNStr (F, (G)) D P rxCCdsreTrRr nh gFy 6 G ! ! = NTtr NStr NT NS S Y lim cT (Y, X S T ) . T : Lcor,S Lcor,T : Lcor,T Lcor,S /N / N tr S X/F tr T , : NStr NTtr HomNTtr ((F ), G) . 0 NT o NO S o OS OT T NTtr . NStr O tr (S , OS ) : NTtr NStr :T S : NStr NTtr F X Y t T Y v X t T 0 m ~ yu e qeQ} nQxee~ wu y u wu ~ s u wu e e y s u s Q Qer qxRxixyeqex qYq'x Q xeeeqqGy eqetI s ~ s ) # # ) % ) " ' # # % # c UY Y Y T a xT yv c Y b S 0 0 bSg g ( va ( u R T ( v Y T ( t w0 WA Ha u t g ( a ( 8x h 0 g ! t d( sq g g gT T 0 a ( v g b( ( Y ( a Wa ( a 0 ( vg ( u dY #x X 9"G 871v ( x % % 5 $7 7 V T Y V Y xh T ) % ) 0 # 0 ) i gVt}s S nhx}xHgrssxrC#s99r S F = atr ( F ) rq sgHC 0 gVts tnn ghb D srxg V D VD Vi D s r1 i iC&s Cb D yr'r s'i q Q t x i C xs'x D rsni h x gVtn D sCY hri T Xi /Ti iI X X /T X = X i Ti T G = G i T j i E xihYnCnCtx2hHxh"HD vr sts'xrtq f i dCgn qiY nrssrxrqrxhdeCrxx bgh hngsxr CnC rq f D hribdCgn Gih t D rQ P yeBr snxCr nHd nHdrr nrH" rx"b D ''hnrH D gnngH" xhdeHgn 0hrqY hxdhxCxnCfFsgHYgh dCrxx GhrqY rdqriFghxCxnCesgnnHd i qnrf D xs S : Lcor,S Lcor,T tr tr : PT P S F T = SpecS (A) :T S (Ti )iI n xrtTggh V'nr' 6 gnh yCbrsCrxnFn 6 ht x E t x qQ S CtFr1xrxi "s P rtsfrsnhex Rs E Hg nnf D rsr xYCC Yt x Y c a xT a # c S P # gVtng he E n F ' bY ) wy!# c 0 &# UY # T hth n E g xrt D rrCg nvYChx} n xrt" qQ hqh sghYnCx rns ntrns Y nhs D xss rsnCCs S nYrsrysRxnCd1nhrhG G Cg sVgngCGs n xrtq qQ t T c a xT UY yv c Y c a xT UY wyv c ( Ha R 0 g 0 80 g Y ( a Wa ( ! S g ( a T #80 I 9"ED4B8 V Y xh (V % %G C C 0g Y ) ) # 0 # ) 0 0 # ) P n xrtr'bxCCeBrvCr 1b ns nts n hC 6 A #tsqCrsVx' rxd"sisd xrtvrsnCeB P Yn 7 hri'dCrxx bgh FA nhx 7 "xrdrsnhex' 2n i 6 P F G S 'sC Cr 7 q ! dCrxxr bYrqh vrnCdrr 'CgCnC' D P ixr q A xtsrsnhex2CrrsnHCsnhrh hx"tr Q C qV nqysreBq sxqs D gVtns h D rsCnngC2 P $$ $ F = LS [X] G = LT [Y ] (X S T ) T Y X S Y 0 # # 0 r#ehHnCgrt' s $$ F ( F ) tr (.) S : (( F ) tr G) F tr ( G). S S ( F ) tr G ( F ) tr ( G). S S G G. (Ti )iI Ti = SpecS (Ai ) F G S X j = X i Ti T j Xi /Ti 6 rnCd tsys Xj F (X ) X = lim Xj F jI/iop jI/i lim S F (Xj ). T tr tr : PS P T S A = lim Ai op iI 0 0 1) # # yr'nh G ' # 3# ' ) # # ) ht x E gVtnGrxgCyxtCr ' ' HC gh2bti rsnyHnCriFg xrdF#eHePhh hxrrq 6 t T g a X wa b UY # 3# 0 0 T ( ! Q( W Y a Y( c a # % 0 &# # c bY t ( vg ddY a Y X e 0 Y V g 80 g V T X # ) ) 0 ) Y Y ( T ( a d( sq ( 0 a 0 ) 0 E }xgtsxCHhxnCqi qQ} Hgtthissrd1 g1Chy D D Hr1s s rxdiY sVC fg2rsngnnhnrgt sH hC ' Y( c a fd( b Y % a 0 &# # c bY t ( vg dY a Y X e 0 Y V T ( V w0 HGIG ED4B8 Y % % C C T X g # ) ) 0 ) ) # " % ) 0 # Y T ( t w0 py a u t g ( a Y g Y Y gT 0 0 1) ) R T(v # R T !g X !wx 0 Tg Y T ( t w0 py a u t g ( a Y h 0 ! t d( sq Y gT g T 0 a 0 0 1) 0 # ) 10) Y # W# sx e 0 ) b( % ( 8x Y ) ( a0 # Wa ( a0 0 ) ( vg ( u dY #x X 9"G 871v V T Y xh ( #x % % 5 $7 7 ' ' x T T rsnnFrx sisdv sgxrvf D CCYHr xthFnhx gnnHd s xnFrx shhx& 'g xrt& D r rsnYH nC h qQ ri gVtFtn Cr tnn rb D VVi D s i nCV h " C 6 i Cn Cr xs}nh G9g qQ xrt E D xihYnC}nCTrht n xvhCnYFgh g H xs"nh }C 2Cr rsngxynhx"xCC rts ns ntts qQ n xrt E xgvf i2yHC U/F jI/iop hCgCCrFYnxr E q fq'CsCvgb D ssgCs C r rsnhnCn rxd nC x#YHnhx }gnnHd sFns n S (X , U ) = S c c jI/i Yst iti r rsnyHn ns E rtrngghtq b D rsr s YCVxnys rsnig# vsHi E tng th n rytn # xrtTsg9hriCnbxts1 }nh bddCgn hqh r D cT (X , U S T ) S U S (X , U ) . c F (X ) = lim cT (X , U S T ) U/F xCC tnsrh t T ' # 3# lim lim cS (Xj , U ) = lim lim cS (Xj , U ) = lim F (Xj ). op op T lim Xj , U jI/i U LS [U ] F U/F f : X X f iI % Xi /Ti Xi /Ti f i : Xi X i f = f i Ti T f j = f i Ti T j Xi Ti = i Ti T F (X ) iI (fj )jI/i jI/iop lim Ti F F (Xj ). Wi gi Ui Wi gi Ui g W U i vi vi v / Vi i f i / ui X i . / Vi fi / Xi . ui /V f / u X. iI S F jI/iop lim cS (Xj , U ) , T jI/i S Ti Ti Xi Ti A ' 7 A 7 i H CgCHnhxxrt n ndynr YCrCrFnhxnbenCr R hth n E gVtbH Hg 8 86642$ % 9% 5 7 5 3 t xgrhnsnCr 186642$ % 5 7 5 3 2 # 0 " " ( ( " 2 % # ( d CF&!$&!CCcT2 1tr c F&y&'4 1c 0! H i0 = H i1 = # isd2 "#ts' qV xts'rsnCCs Q CxxCyg0nYnr 1Crb rsnyHn rxdTisderCyh1Crhn9 Gnh = lim op jI/i jI/i Wj DXj kI/j op Wj DXj i qhsCsq qQ gCHqC D rsnYeP C nCd2rsHC' qi #snshxbVqx##tssYhx' 9hrihn Htr F (X ) = lim 0 op jI/i op W DX h qQ ndq RgytCCx Q xtrsxri sVh Vg}snhnC1tsnrhnih qx h x E g} A Xi A t n E i Cn xgCyx2Y ssns#}xs'x g Cn A Yh x E rnrhnihFC" 'F 'b D t xqgx V 7 Vgr 6 i C th n E gVt'tCr rytn}H Cg iGxhdeCrxx HiY hxH ' t 3# T # T # 0 2% # ) # # ) " % ) # % Y Y Tx T ( !g ( a U( 8v# t T ( Ha10) ( ' e g ( " 0 ) b( @% X y0 w0 # % vA@% X t ( a0 c c V 0 T Y T #0 0 0 V h S ( 0 # d( `( 0 # ) 10) Y b# 8x 0 ) b( % ( sx Y ) U( HaWa ( Ha0 0 ) ( !g ( u dY # x x X 8IHGIG ED4B8 V T Y 0# xh ( %9 % C C ' ' # x T T H #ts1trn' CVsx rsnyHnCCYYn' !r D ythrdsx#sisrd' 2hhsnrsCnngCs'f D vrsnhng 2 0 Htr S ' # # 0 0 1) Y ! X t t bY wy! c Y b 0 0 W g ( !wa ( u d( sq c a xT 0 a T 0 g g g Y gT T ( ! v T ( v Y T ( t 0 py a u T E A s 7 VVi D s r1 a 7 rx 9 s i A A ns n' F 7 D D 7 A A s s r1 a 7 nxhts2Frvg i A ns n' 7 D ' 0 ) # A A1@%'&%3$" 7 xrtvg D y2YCH D A ns n' ' 7 7 i A A s s a 7 xrt gs n xrtt g D gYnng"xgCyx's A 7 H tnn2rb D rxn#tCHC# 6 D RgtsxYex' s h qQ P D }b Vrfr Hg $$ $ $$ F ji F j = i Ti Tj Zj = (Xj Uj )red Tj = (Vj Tj Zj )red j ji W j U j Xj Vj Tj V j uj fj Htr 0 S X X = X i Ti T DA U X V / V /A U 0 Htr . fj tr tr PS P T S X j = X i Ti T j A W W =U V , cS (X, Y ) iI 0 Htr F (X ) = lim op Ti X jI/i lim lim Y D Xj D X , W j W j Xj X Ker F (Wj Xj X ) F (Wj Xj Wj Xj X ) lim lim Ker F (Wj Xj Xk ) F (Wj Xj Xk ) j S Ker F (W ) F (W Xj W ) . S H cS A1 X, Y rHdxiFxrH2ys s t x E r i xiFxrHVV G nC E rs g CnbsgChns'H S X X S Yh xGrxgCyxtrrsnhxrrsnhxrFx Q gx xrt" qC YCgrHntsCHnhxxr rvgYnngh g grrs'xix"tr sgChnsg xrt gt xCn2hnsts Cr % % 5 7 5 3 h x" i C sr VgrC hns9sgh 6 "" 86642$ #TD hsCs9A sV D s C v 7 hn S i $ssgi f Cg hribrsnD YeP CG e ssCYnxs ifCtHgGFrx S T c sqfr ixH g#xihHxCtrVVbs ' pvg ( rxRr xihHxCtr2"xsnCrh 6 qqr0s1f D ri#s gt nCCvFr2gsYrxyr S V #&# e g rsngHnrCtr qn 6 gh xCnhns YChx}Rrn 9 gxiFxrH1bF A ' 7 t xCnvFr2g r Csgyr rxTr#nFrxs rxiFxrH A 7 rx E srxhrsgvgF Crth nrsxH % 5 7 5 3 % 86642$ 0 & 1) ' ' rsnix#Vrxgtr qrdbYCrCHn E nxrnCrhnxr"Csgisyh xtrnqissg bA CrsngHnrCtr Vrt 7 rVnrsrvn xhns 9rnFxt qQ #$ 8 !w 8 % 5 7 3 C 7 rsngshn' qrdhHgCHnhxxr E xCC!rfYxr'Csgisyh' btn xrt Cr t x E gVtxYer nrxih& P yxgh& 6 xix#Vrnrtr #V rn2rssFgysnry V'ngCb D Cr th n E rVnbeqr 86642$ % % 5 7 5 3 hHgCHnhxxrnC!r qsvxnFtr 'sxiCtvCgxrsHrsyVs xix' qQ n nd#Vngt xsx D nys D nhrh1d Hg i Cn YCrCrFnhxnbr CCY enh "s V Ynr D xrrxgtr qx 6 Hgqht n E gVtnH Cg "G 86642$ % % 5 7 5 3 H rsnC sisdt 1rr'C t 6 rQeysnnCgn D rhfi xixsxntVn Cr H xq}nCVVr!sFhCgCCrFYnxrfehbeFVrxgtr VrChns qQ gbA yH' Yn 7 ntt2hns sA Yn 7 nh srF #nsCHnhxxr i0 0i<n h i1 1 Lcor,S 0 , ..., n cS (X, Y ) X A1 X S cS (X, Y ) Y S S X A1 X X , cS (X, Y ) h 0 = n = i i+1 Y h S S (X, Y ) h (X, Z) (X, Z) S (X Z)/S 0 # ) 1) 0 X Z =X X X X/S X/S S S S X (X, Z) X (f, g) : (Y, T ) (X, Z) X/S S T Z X/S /Y f /X X X/S X X/S X X = X X X/S X X/S X/S S X S X/S S S X X/S 0 Cr r ' " Y w0 ( s g 0 e Ha ! ( Ha t 0 g Y A Qc p!g Tx T T ( gT rsChtRg resxgiVrhss1 rtr 0 Y x g ( 0 pT 8d( e s}hny gxhe'nC0gCr " tsnhsnCvy HC 6 P f n r X An k # # 0 # % hF"hrtCxnCrtssnrxxxgy hxCs#nbC isg' 6 g bxhs2nChsGg th nH D eCsgrsxFV & r rsnhnxhny D A ' 7 sbA nhx 7 g1nns2hCYn' fA nhx 7 A 7 HIV1xghC CChF"s C srg#sxtn dnys' 6 S rsnxhtsYy#CC2ghf D t x E ghrsg#Trs rnCts F H rxCxnC A X X X Z = X X X S r Z Vi k An k X Z Z Pn k X x1 = x An A n k k sqrnsVs#tnnb D nrsC2g'bqr D sgCs srFstg nhHhxC xg 6 rsysreP h hrg yeP thF i Cn V rF s vsHhx qr1g y H h& s xiFxrH VV H gb D C V hngxrCn rY!g V rF s"s#rsnihHnCtry'HC#gh}b! CxixssgYHn"xgxnYhCTsVg"Yynr rtxrr}b D hrihn sCYnxq tnxrvg 6 H rsnCh A #nr#x D s g # % # &# 0 1) ) # ' # # ) 0 % ) @% 0 # ) ) ) " 0 ) # # # & # ) % ' ) # # 50 0 1) ' % C 7 ' CxCCsrH % A hrsg 6 7 v 6 ( g % " g % #" c ' dY b7W&$A ) ( S P V rF s#T hH g1xtr' D xYex1s rCC Yy nC Q qV nChsg r1th nH2hns#F rh xCnvhH D nCh"sgygqbHr#n9f g}th nH2Fr1Hb rx1 qQ s V V s C i rxd i}nbrsCrx}xhe'nC"v fs D g#xrbq R rx E gr P hHnCv fsq inCfs h qV r xixssg#xr 9hC Q hri1rxrsnhexrxnysrs s rxns2hHhx P rsnxhts hns& CC r2f D gr grhnysCr#T ' 3# # S ' ( UT X % g t 0 b ) g T g a X '! t S Hae b bY g pT ( g !g T # # ) 0 0 % # 0 % % ) " " g x c V e T ( v ( UT X X e ( T s0 g Y g ( # &# ) CnxhFn CGg b $$ x nFn nrh nqsxnhxr 6 t T (Y0 V g Y R 0 dwd( ( T ( a ! 0 t u T !g X e ( T ( H ( s1g Y Y ( ( T ( sq Tx a ( P " 871v X g ( S % % 5 $7 7 V ( 1dY g S g t dY b ( sq ) ( ( g a " 6 ' " $ $$ F C/k T(v ( c a X V( Q( W Y S bY bdY ' % C # @% 1 " " " # % C C # ) % 0 ) ) # # ) % 0 0 # 0 ) # % Z C C H C C/k Z k C C C C C/k Pn k k Y ) isrd s CgVx}sisd9 bCsCs9b D hrYnq get r t T c T ( v 0 gb ) wx c V e g Y vV g g g a X wa b UY c A 0 0Qc Vf( H g ( T t Y (R c a xT UY wyv c Ha UY g 7 0 (0 ( 0 x a t V R pT X S a e 880 g ( W 0 g ' 7 A ( c Q( ( c a W Y 80 g Ha UY g A 7 Y Y ( T ( a d( sq ( 0 a (P ( 0 t Y 0 x t dY ( 1) g ( S 0 dwd( ( T ( a v 0 (Y0 V u (P g a X g V( Ha W UY HT wx bdY g ( S ' # 3# V ( c a 80 g Qd( b Y s0 Ea ( g % 0 &# # c UY g Y ) 0 0 % ) # " % $ {x1 , ..., xn } (X, Z) (U, U Z) k X k x1 , ..., xn f :U S C X S C xi C X X C C An A r k k k S 0 X # ) % 0 # " # # 0 # 0 # # ) 0 ) 0 # ) 1) 0 # An k r S C An k k C Vi (C C) (X, Z) 1 k xi Z xi X An k C k k k Z 6 agshr' i 0 0 % 0 # 3# # 0 # 3# 0 ) 0 0 ) # 0 rq 6 xYex1s D rsnYex1s P P ChnV P xYex rrCgt vCg g"th nH Fr"nCh1s h qQ r rx1 0rnCy snCr1b D xrt}xYePx"nys Cxy" Q A Yn 7 D ' t x E xCC nC r rsnsxnhxt H}A ' s n# 6 nhx 7 srFRgCxi't H Q hsts qt ynY gH ssd gQtdnixtth !gbCxCr1hn 1 fr D shCsHq s g#g#shxs gH sd rQ' g nCn hH nCh#httthtshhx D h xysrFgCxi C'r# rFC' i CnCxVxHH ( g% "g ( 0 fdY b7pT 8d( A s Anr k n n,i n2 Anr k n n k /k An k 2 V Crx D nrhnvYsrFgh hn #shhx vCg ssxrHnCHn nxr gtesnriyCrhs s gtsYH1Yrx q Crx qQ ' " $ Y gT g Y T ( 0 py Y 0 0 ) # ) 1) 0 Y pT 8d( e 0 x g (0 # " ) ( b( T e ( V t # U( % g Tt bY T ( Wa Ha10) u t # Y X xh 0 ) 0 # ) % Y T ( 0 0 ) ( sq "9 v 6 0 ( a % % C 7 9 H h D s 9 F D B i rhn Cisd tH srCC# 6 GrnCrxx' Cr g rsnhnxhnys tnFn n&Cr' r xHr s # r nCbxnFgCr2s rsHt E Rg#exgiVngYs D r fxhsvrsChtGg D exgiVn1xYex P 1g s xYxs tr2nCn Ch YC rsnCr Crhn qQ nChsGg shns rnCnnCtYCvxCrC1 qQ x g A Q nd1Cxrvghbv qt g 0 0 ) # " g 0 # ) 0 pT yHCxC H pCxetbeshnb Q nChs1rixt t nrYhHs ' 0 #Y (Y 0(V ) ) R a e X df( e vg 0( T # " Q qQ t T T 0 a ! t R d( sq ' # 3# c ' " g 0 # t # Y 0 0 ) # x ( Ha0 1"'g a0 &# UY Y ) 7 A R V ( #" # ( 0 0 R 0 0 x bY $% t 50 dY ( 1) 1) g Y ) ) # A ' 7 ( 10) 1Y ) 0 ) A 7 V( 0 ) ( X 0 x % % C C 0 wa0 b% UY Y 0 0 x ( a0 h S b8A( V g a X ) # # t 50 dY S UY d( b# ( Ha0 ( S 0 ( P " " ED4B8 H CxnCrtxfisg tthtsisrd 1ngxb2hsFqCrsngxr nh qQ xrt x hHnV H r n xrt gt n E nhChx#r ghrF" hxgh bhrihn D n P r}HrsnYex h9r}ynhH x#sCHnhxnhCg g#xhnnys 9sb xYex9 0gfnyh P V}xgh 6 FV () L E YyHtr r g nx xrtrhs E E hFn r&hC P xhe'nC}gYs xninn 6 ythhgQsbs CFrCYnbx Cg rbysrF"'F P g'yH# V&YH nxtggHxg'CxYex#Y 'rQ s D YH Anr k ()nr p p |Z() X() L p p n Anr k L n D # r 0 0 n x X () ()[t1 , ..., tr ] ()[X1 , ..., Xn ] n ti j=1 Xj i,j . n An k An k X() p1 (p (x)) L p : An A r () () X Anr k Anr k An k X Z r p : 2 Pn () r Ar () L An k A1 () (i,j ) 1ir Pn () r P() . L Pn k An k E e0 0 g 0 ) ) 0 p k k p p X k Pr XL k r p : X Pk p n p : X Ar k xi Ar /Ar k k X/k L = X L L Pr p:X k i X p X x 1jn X k k x=0 r i n,i f p X 2 # % ns ntrCss' CC 6 A q 7 rC Yyqr xrt2F A ' 7 Z Z Pic (X, Z) Z Pic (X, Z ) (L, s) (L, s|Z ). (f, g) : (Y, W ) (X, Z) ' e n rxd n xrtxnxeYn' 'Hb D r'h xCnYys A 7 gt nCC2hnst Cg't ntF 86642$ % 9% 5 7 5 3 ns ntFrrCgtnh qQ hst E gHnHn #V2hCCsqsxnCxn2n qQ rsnixrsVsnx CxiCtrqhrqY ssnnysr xrttVt D hri rrsngnsgysxns Cg gh CxhrV gt x hs1r n V2nr 6 sgHYy#F % G 9% 5 7 5 3 "w 86642$ % 7 B v A US D S S #T t x D isCChHCtr x }g xYbgC x hngsxrrnsVs }s S Cgt#T xHreC nyrxrCr s gsreP s i H g S Q V rF sCtFrg}xs'x H qa r sreP Vn xhn rCqrxg rdxsqx # P xYexbs xCCfs nb riFg rqFC} 1H ' t 3# T ( s0 g Y ' 0 0 # ) ) A' 7 A 7 # ( c a X V( " Q( W% Y S bY WdY $#@% g ( S 80 g ( UT X % ( H) @% ( # 8g ( S V P % 9 9% C C 0 Tx 0 ( 8A"@ED4B8 H hV rF s#vrxtr S tnn r2f rh CYn Vg i #h&issrd CsC& Q D i'nCq rr n rsnhex1 r P rsnsxnhxt 1i n n 6 rbxyH1 rhnFtrsnCrbxH rxdsisdxCC g xnxeYn bixr rsHtrsnixg gVt1s xrt vh qQ n r"xnxeYnq Vfxghf"Cg qb'sgCs tthtshhx' s YyHtsxCr2s nC rytn s tnxr r#b2xgxxhYf sHh ritxH1s i H s S g y H h#T'hH'g0ex TC qQ nC1xg riFg r D Crgnhnnghqs grrs1riFg nrCyn 9 A H xiFxrH rxCxnC F xhHH} hri1nCfs p H $$ $ $$ F Ar k rxg#issrd Yrx Z U S p n U = p1 (S) S U U (X X) p X X (U U ) Z U p:U S F/S V 80 g wa b UY 0 v Y H V g a X T V pT X S a e 80 S 80 g d( W 0 g Y Y ( ( T ( a d( sq ( ( 0 x 0 a a ( P ( Y 0 ) g a X Ha W UY t t 0 x g ( % ) ) 3 ) # # & # ) 0 ) 0 U U = p1 (S) f Ar k S X / X L S S /X xx x p xx |xx p / Pr k n f :U S 1 f :U S S p(x) L F s = p(x) s S ' # G0 ) # x F ( H ( ( 0 g s0 g V ) % 0 0 ) t # 0 ' 0 1) 0 0 % Fs D X FS U S D US D S X FS Fs i (X, Z) Pic (X, Z) s : OZ L|Z (f, g) : Pic (X, Z) Pic (Y, W ) (L, s) (f (L), g (s)) . X F Pic (X, Z) Pic(X). Z s f r Z Fs (U , OU (i)) S S (L, s) L Z S OX p Ar k s S L U /S S X p 1 p(x) D k p1 (S) (S) S DS US p D U Z # Q V E CnC a i g rsngnrsVsnxnqCCxy g rsngnsgysxn qf s Q g V rF st# hH rhC S ss&vF S iYnx nysntst ghnsYn}xCxnC"b D xrtgs xrtYCH E rix# Q YCrCHnhxn gxix ynr n rnrQ YCs D ht xgx"rd n xr br D x nrh x#h9CrsV E Cgnriys"Vngt 'sH sgH '0) ' % qQ A 21% 3&$#" g txg#xnsrh 7 D nC gnrhbxH #ChnVnGisrf s i r Is2hHqrnsVs snhnxhnysgrsnngFr H Cr VvhCCs Q Cg#2rCsC v qa V E qsnng rCCsn"#nhxinn1srfHrsCsHnr Hg i ng r A rsnY nhx 7 xYnh5 fFA H nhx 7 ' ' t T c a xT UY wyv c Y 0 g Y # 3# ) # # ) ) ) # % # 0 ) ' # ) 0 " # A0 # # 0 # 0 # 0 A@% ) ' % 0 &# # c a xT V X UY wy! c b( !V c a xT UY yv c ( Ha d( sq 0 a 0 ( asq 0 g ( 8W c a ( a a e X a Y v g t T hx T T ( c a fd( b Y 80 g a bY g ( c ( s0 Qg d( X ( 8x ( Ha ( ! S g 0 g 0 ( a T ( V 0 " 9D871v V g c c Y Y % % 5 $7 7 T Y (S 0 " " # & ) 0 # # ) 0 0 # 0 ) # rxCxnCVqisdrsnCD Cr bH qV g Q rsnyHnC2V h qQ th nHYy x2hxixrt }s s t nCC Yy9r}xrt nF r"srsngshnnC D srFqeC1 fr H yhCCs xrtqiH }F xCxnCVbCrsVVrsrFCrhn qQ ' t 3# T # X c0 (X Z/S) c0 (X/S) f :X X X/S j X/S 0 # % ) ) # " $1" # 0 ' # # 0 b ) wx c g g # ) 0 % ) @% 0 Y X c U( e c V g Y # 3# ) ) 0 0 # % ) ) ) ) # " " $1# 0 ' ) # 0 % 0 a V XV ( H g Xcc Y c gpT e g V e 0 uv t ( Ha ( 8q h S b( vs0 Y HT x bdY g V( ( Y x g T #V w0 ( t cUY awyTv c 0 g YU( Tvb ( a V X V c a xT h S b( vs0 UY wyv c ( a ( S V s0 g V 80 g X cUY yTv c g g ( S V ( c a T s0 g fd( b Y vg X e ( T g ( S ( P A 7 ax Y 0 g ( v S g ( a T ( V w0 " A" ED4B8 0 Y % 9% C C ' % ) ) # " $% ' # ) 0 0 # # n xrt rrCg YCtq C t 6 n&rsnxnhxx#F xixssgVxnHghsghq gh b D gHC' fs D ixr ysC s}ns ns ntrnsn r gh "sxhrVs g2nqHrFYnxr1 D r V rsCht 'r hxsnCr D qQ r#s#xixvF xiFxrH2yxr}1nFnvCg S n# E Cy gVt2Hr# 6 h x#1rsx#F S % 9% ! X/S X XZ S S X = X S S s() X/S c0 (X/S) Pic X, X , (L(), s()). Z X/S :S S X = X S S 0 (f , f ) : (X , X ) (X, X ) # ) # 0 % 0 c0 (X /S ) c0 (X/S) ' X X/S /S / Pic X , X . X / Pic X, X L() (f ,f ) X X/S X # % c WT e g V e 0 ! t ( a ( 8f T ( ! 0 aq g u ( x X V dY #x bY 80 g t fd( b Y S UY WdY g ( S ( c a X V( # # % # " $@% 0 t ( P ' 7 A ) # % % ) ) # 0 " 0 0 ) # X f 1 (Z) A1 Y i1 /Y Z (X, Z) S i0 : Y A 1 Y i0 S i1 S (Y, X) Pic Y S X, Y S X S Y S X/Y i1 : Y A 1 Y c0 A1 S X/A1 c0 (Y S X/Y ) Y Y s Y = S i0 XZ/S L X i1 V / Pic X, X V / Pic X, X . (L, s) X X r(Y S X ) : X /S X/S Z Pic X, X s L (L, X V, s) 1 Y Z i0 n sCY Cr S nC#T E nrCy S n#T vxeC hssg Y xnFrx#nC 6 ' # # ) 0 % ) 3 ) " ) # 0 # % ) 0 % ) # % 0 0 1) " ) 3 ) 0 ) ' t 0 ' # # ) # 0 0 Y V Y TT ( T t 0 g Y g H y ( Ha Ha u ( 0 g s0 x ( U! 80 0 s0 h Yt g Y b a u U( 0 g s0 x ( U! ( ) ( Ha R T ( ! ( v V Y V Y TT 0 a T 0 Y g ( a a u R g H y Y T a A t g ( a ( S T ( 0 ( sq ' 7 # ) 1) 0 # A0 # # # 0 2 ) 0 0 # % % # # % ) ) # " $1" # 0 % ) % # # % 0 1) 0 ) 0 " ) ) # " " $1# ' % # # % 0 1) % X c Y V Y TT hx sW c x X ( H g c U( 0 g s0 x ( U! a c WT e g V e 0 ! t ( a ( g c b a u ( 0 g 80 x ( Uv ( ) g Y Y ( ( T ( Y a V Y TT 0 g u 0 g X( T d( vH b( vg e 0 v t ( s ( 0 g s0 x ( U! ( ) g V Y TT 0 V ( c a s0 g Q( W Y s0 ( u aq g (S 0 ' " ) $ 3 ) ( P ' 0 # ) 10) Y # W# sx e 0 ) W( % ( sx ( Ha0 t # Y ) ( a0 # Wa ( Ha0 T #V ) w0 # 9 ED4B8 V T Y xh ( Y % % C C x T Q #tsYnxrt' #VYstssrsnyHnC' bC qt Vngt g H y Y T 0 ) " n r rsnhnr#Chns nysgisyhtxg Cg CrsnC t T Y a A t g ( a ( S T ( 0 ( sq 7 ( H ( T UY Y X c a xT c a A bY wy! ( sq ' 7 ( H ( T UY Y X ) # ' ) % 0 ) 0 &# # # ) # 0 % 0 " ) # ) 1) 0 # ) 0 % ) % 0 # % # &# 0 1) 0 0 # ' # ) ) # " % ) # % 0 # 0 ' % 0 # 3# ) 0 0 % c a xT bY wy! c ( a R fd( b Y s0 ( c a ( 0 g X( T ' ( vH b( vg e 0 80 x ( Uv t g ( a '( 0 g 0 ( a 0 V Y TT Y 0 b ) x c V e g Y !V g c g g g ( x Ha dY #x X 0 0(V V X ( UT X 80 ( h S ( #s0 g R V ( c a T 80 g fd( b Y vg X e ( T s0 g 0 g ( S ( HT wx bdY g g V( # ) 0 0 # 0 ) # A0 ) # # % ) # @% " % " c g T g a UY h 0 v A 7 Y0 80 e 0 v t ( sq V u a t S (P T ( 0 ( Ha ( S T(! Y T Qc d( W b 0 0 b ( a ( c 0 x g g c g Ha UY g Y g a X wa b UY ( " 9 871v P % % 5 $7 7 # % ) # 1" # " ' ' # 0 Y ' w0 # ) 0 g Y ) !2% g T #0 % IE87 # w61 v 7 B6 6 5!6 $ 8 77 C 7 $ 5 57 $ $ 57 rs ! D s r i s q n g#rsrtr rsssnC T c gxitr RrnrrsCnngC#12xnFrxHrsyhnC qQ ' qs pvg ( Cg sgrb D gVx' gtssrF' CnC D h qQ E rxtggCCggh nh rnsVs ChCv x hxixr2v H i r rsngnrsVsnx sgxrb D r ns n2 r r CsC r C iHr hFxxhxhVx & hsgrhsrsnhFYn F Cr & n g Cg CnCxVbe rsryHC gs ChHhrhV s gns Vngt g irix 6 r's1xix'nC}Qsq Cg hrinCs g}nrFn D srCs ixH Ysts s D Frx S Cr#T E nrHs D nC GgFC" sg nCh} ixr 1hss gnrFn qQ s YngsVnxr# F s i #ts D BsC fissrd V9snng CYninn9 ! $$ $ $$ F L : A1 X i = 0, 1 X 1 si L AV H D(L , A1 , t s0 + (1 t) s1 ) t A1 XV X H i0 = 0 H i1 = 1 s 1 X/S X/S S X S L(1X ) " ) 0 X 1 X V S 2 H ) 0 ) ' ) % 0 ) 0 S S X/S S k5 X Z TTT i TTT) k k / X. Y L()|Y S Z ( 80 0 ( vg T # 0 2A # ) U/S 0 S L(1X )|XS Z :Y X S (X, U ) S (X, X) S (Y, U ) S (Y, X) S S D(L, X V, s) i i S Y X/S V X/S cS (S, X) s0 s1 s U =X Z (X, Z) L()|Y S Z S 1X cS (X, X) X X/S 0 1) % # # % # (X V ) (X, Z) 1 0 1) " 0 # # ) 0 ) # # ) 1) 0 # % t asq A 7 Y 0 s0 V c g H a UY 0 # 3# ' # 3# V i:U X i 0 L Y ) 0 ' # 0 Ysts s 9gh E thsC t x gs} hri1nC9s Q h x qt ri1nC9b D n Frx Yy CgCt" vhs xnFrx xCr S xtCHnhxxr s hri rY sxhrVs g H ri rsnihHnCtrVrr Yst Q tnn r#b'C qX d1s b D rd1Cs hn t 1Yn (X, Z) S V % grrs1xix"tr n r xiFxrHtVr19 i C t x E rtCg D rsHtqxix r n xrt gVt2 D s g V CrF h&hHCt r D S th n E gVt1#Trfe}x D 3xr Cn S ' ' # 3# Y 0 Y T Qc d( b X S ( a ( vg ( c 0 x Y T # ) ) # # & ) # 0 nCs xHyHC t T V s0 g ( T ( a u i j # 0 2# # # A0 ) 0 0 # % ) ) ) # " " $1# 0 0 0 % # # % 0 1) 0 ) 0 ' # G0 ) # 0 X 0 x (Y 0( S UY ( W #V g # X c h8b c x X ( HR g c Y x a R 0 t ( ( 0 aR q R Q( W Y Y Y a ( T b( Y a g d( sf( ) ( c a c V % 0 &# # " c pT e g V e 0 v t ( Ha wa b UY g u g a X 0 A 0 g s0 x US( U!eT ( 1) g ' 7 V Y T V pT X a 8( W 0 g A 7 (0 0 x a V ( c a t 0 x g 80 g R t fd( b Y P %9 % 5 $7 7 g ( S ( "G 871v # # & # ) H s i VTrrYV q ag qs rsnyHnVYnsnCxhxs 0 ) w0 # ) Y T fc ) d( b# t # 0 # ) % dY b3# P Y ( c 0 x 0 ( " g % s xnFrx Cisd qQ ' Cr sbCrstts2hH nhFYn 9gnnHdtnr s Crsnxhts Frfrrd'Crsngn E gVxnC"f D sg1rdCgC rxixssgVxnt#snV D hrihn S Hg'#T D grsngHnrCtr Vr'b r s#rsnYft ntthH #h qQ i Cn shC2'ri S ssCt 9g Cr Yh xCn hH % % C benCr r#b D xnFrx' bg rsngCg&nrY g " G E w" gVx' rxdfrhs#h' RgyHr qQ grsnixrsVsnx"vYCs h qQ rsngx rxgCisd Vrt D C qQ xrt rxVvnh D s' gbxr' b s r yts r2xqCHnhxxr HgCHnhxxr E nC" D srxh H YCengn D Ys t#tsx"Cr D CgCCrFYnxr E hxCYnb nC hC s sr tg ns qQ g xixssgVxn Hr H 2 1 s L()|Y S Z (L(), s() s) Pic Y S X, Y S X Y S Z S i = 12 S Pic Y S X, Y S X Y S Z (L(), s()) i = : L() L() Z = (X U )red s()|Y S Z 1 |Y S Z X L()|Y S X X x i:U X (X, Z) X X k U U WWWWW W+ s() X V ) # # # S L(1X ) S (X, X) X/S Spec (OX,x ) (V, V Z) k 1 % k Z = (X U )red S Lcor,k Pic (X k Z) = 0 L(1A1 ) k i: U X x X :V U V V U vV j v {v / X. i U O Y S X |Y S X U X x X v X X =V X S X S X 1X Z Spec (OX,x ) S Z Pic(Spec (OX,x ) S Z) = X /V / X, Lcor,k A1 k / L()|Y S X . g ggggg Y X s s()| k X/S S L()|Y S Z U k k f :V S P1 k U X G! ' Y 0 Y T Qc d( b b 0 0 b ( a ( vg ( c 0 x g g T # ) ) # 7" % ) # % 0 V s0 g ( T ( a u i c a x X ( H g Xcc Y c pT e g V e 0 v t g u ( 0 V Y g s0 ex U( U! ( TT R 0 T ( v ( 0 Y Y ( T ( Ha ( 8f ( 0 aq # # A0 ) 0 0 # % ) ) ) # " " $1# 0 ) 0 ' % 0 # 3# 0 x (Y 0( ( W d#V g # V ( c a 80 g fd( b Y Ha k j c UY g D svrsnY#xtr i s Q hrsrg 'stg' qu xYexTggh P 1k p R V " B) g ( S 0 ( P "8G A8 ( %G % 77 CCrnfisrCr nttvFr" RgD vrsnhn# rnQ1f sgCs rd D Vngt n2rsngx X # % i grrs qi C CrCHnhxn E nCvrx C 6 s rtnsnysntrdrsngx rxg1iHr' sVg# S ynr 1Cg CrCrFnhxn E xH gFts1r9b D gsxrC' sgnnCdq r Yh xCn2hHet2r' friqHnts' x D th n E Vgd A1 k k k Y cL (VL , YL ) = lim ck (V k W, Y ) , 1 Y 0 : V k W U W Ak xrtrnFrrCgtvrx'f D D xnFrxtsys nttTghsCrr srsngx rd Lcor,L j:V X p VL ppp jL xppp U L i L / XL . |VL Q rxg#sisrd1 qV g Cn s g V rF s hH r CCqbsg ghf D rsrF2 rxd1htr nxhts Fr# g Cn s r CrCHnhxn E nC Cr V rF s HCb D ysrF 2n&Cr nxhts Frt vx"nrh2CYYn# 2sys VYnrsn g}ysrF#x#srrrCCrFYnxr ysrF qQ t nD E yrdsnrsstn Hg C& 6 Hr f nCs h 6 ngsxrxr CgCCrFYnxrnC E h'b D rxgsYhx "xfnCdb sys Hsbf'y xCCRg rxCCd'rsnsxnhxq D gVtG R xrtygghb bH :Sk k k k x iL L = k(t) x () U i H xi"tr grrssisd hrE r# CgHrFYnxr E nCb!ex CgCCrFYnxr nC' nCg nhssgt #tth n hrirsVsnxqs i Cn g D y H h hH rxV"'C qQ hrirsVsnxs rsCnngC $$ ! ! $ $$ F Lcor,k x : UL X VL 0 V L(1X ) x X S k j k p p:W X X :W U % # # % t # ) # % k hx 8b U # 0 2# j 0 1) # Y =V :V U U V k W V X W A1 1 k p k W o oo j wo /X i U nn vnnn X 0 nn i ( Ha wa b UY g a X 0 A 1) g ' 7 x W 0 g A 7 S ( c a X t Qd( b Y UY 0 0 0 % p : W Spec (k) V k W jk p / X. L x V j / X. XL Spec (OX,x ) S Z L(1X ) iL : U L X L L XL x V x X XL = X k Spec (L) S |VL V S Z W A1 k V X j XL k W s yuu s wu u y y s w xqqx!exxu yesu eq! S (C, C) ' 0 # ) 1) 0 Y V Y TT ( 0 g 80 x ( Uv ( 1) Y Y ( T ( a d( sq 0 ( 0 a Y b 8x e 0 W( ( sx ( Ha t Y U( Ha Wa ( Ha 0 ( !g ( u dY x x X % HG" ED4B8 V T Y xh ( % C C x T T % # # % 0 1) 0 ) 0 # # ) % 0 # ) 0 # 0 ) # #ts' 2sYrixHgvhxixnrCrsngshntxgxnYhC' qQ [W ] (,) ooo o D #ts bg'Hrsnig# 2hfeHVrnrtr sg # xrtfe Vhrsr#Vrxgtr sg D hhC tnhH Cg#iHr str " HnxeCRs'f D #tstissrd' v ' t 3# T # ' 0 Y ) a b au t ( cXY V QUWY g s0 g ( T X X A 2b( vb( T T Y V XV 0 % 0 % Vs0 g t HT ( Ha Ha u ) 0 0 1) ( !g dY ( H g c ( a X c T X Yw0 ( g g b ) x c V e Y Y ( T ( a V s0 g Xx ( ) 0 0 # % 0 # ) % 0 ) 0 # % # 3# 0 ) 0 ( c a fd( b% Y 80 g Ha3# UY t # ' 0 # ) % Y e t % ( vg dY b( S a Y ) X e 0 ) 0 Y ) V T ( V ) 0 # ( 0# c Y 0 ( T X V g ( c a T ( g 0 g ( 0 ( P "G v 6 % % C 7 ' Q( W% Y !g " X ( T 80 BrhV q i s g #h1g xixssghr ts# xr $VnrV q r xixnCnhnyt e ) h x# s}yH!gny" Rg'Cr2rsnyHn#Cisd qQ ' 0 # ) Y % ( 8WA0 # c # eCtr E Cy # gxyH nhr n #ts shhx sr stn& 6 ' t 3# T x H Cg S X S Z = C U T = V W (C, Z) C/S h W U j l / C. /V f V /S # 20 ) 0 % ) # # V f 1 (C ) Pic (C S Z) = 0 0 [W ] [U ] [V ] [C] 0 ( x c " Lcor,k j = 1C l = 1V f =h l =0 h = 1U j h = 1W l rfytsgbr V hT g U! e ( b ( H #V g ( a 0 ( # 0 % ) 1) 0 0 ) ' # % ( a d( sf 0 aq 0 o wooo hl / V V f f /C C [C] pp ppp wppp [U ] [V ]. [U ] [V ] W o O U o Lcor,S 1C (L(1C ), s(1C )) sD stgbns txCCrFYnxr} n xrt x Y rsnyHn#srbxH} 6 S e wu y e yu e i y wu s gbhxqxTu eekTqxQq0q}tI V C (j,f ) Pic C C, CC (V, T ) 0 w0 g ( T e 0 v t ( Ha h Yt g Y W a u Y u a ) # ) 0 " ) # " " $1# 0 ( ) 0 % ) " 0 1) @% gT pA 0 0 ) 0 # % Y ) " 1" ' 0 ( cXY ih g ) QUWY Y V h Yt g f80 g 0 ! s i s tn9CHnhxn s tn9CHnhxn CrCHnhxn"xCC D hnh 6 rx 9 s Cr xrtr r D hrixrtbi xitrfngxCbxrt G i shrxbb D x#sCrFnhxnsx rsHnV x2hxixr xr } nh N w|U V = 1 N (w)|U T N (w)|U C = 1 1 = s(1U ).t g 1U j OU C , N (w)|U C UT sr n hs rxg xrt} n RCHnhxn CrCrFnhxn} s g 1hf D D Tr i rrsngxC' "sis n nCxVr xr Q rsnHd1grhntsxhrVs# H qX g y H h CtFr#rf S Q h qt V#YCH xrtnC F tCr s g V rF st#T hH rhC S F 7 xixqs qQ HgCHnhxxr#xhH b A D Ys 2#tsCg rsnHnxeC 6 st nqCCrFYnxr OC V CV Pic C V , CV CT M M|CS Z CT l (OC V , 1 + uv 1 ). (OC V , 1) D H isrd sCrt gns ntsCHnhxn C 6 hri r xhnsxhrVD s1rsCrxv" i #xg 2 g rsngnsgysxn2 ns ngs R r g1rsnixrsVsnxighsgh #F rnys xCHnhxn iY fsnnVb !nCrf 6 HnxeC1nHrthnR #tsGg1ChyhnCrvsgrsb 7 rsngsh A C S V (L(1V ), s(1V ) + ). Pic V V , V V sCr tn"sCrFnhxn D CC gtrsngnsgysxnfs xY} gsxrC# (f S g) (L(1C )|XS Z ) = L(1V )|V S T . = (f S g) (t) rxf D xrtr n xrt' nCghh 6 Yn 7 Yn 7 Yn 7 g9th nH'Yy}Rr"hn!A g!A Yn 7 gCrg "nCrFnhxnbA iY 1 D rsnHnxeC# 6 s stgbns x9CCrFYnxr gvyts r D xrt D hs rsnnFrx rr s CrsngCgthYnC' 2sqr s ##ts rxdbisrdtsqA e7 rsngsh V C S Z C/k V /k g = f X Z : T Z 1V (L(1V ), s(1V )) L(1C ) L(1V ) (L(1C ), s(1C ) + t). Pic V V , V V C S C yhhhCisd nCrFYnxr CC 6 xixssgVxn" H'F D FrsVsnx s gh sxhrVs tYsts Y rHV 6 ! $$ $ Pic (C S Z) = 0 t Pic C C, CC CZ S (V, V ) L(1C ) C V S V A D $$ F : 0 V T w U S 0 h Pic U C, U C U T (f S g)1 (X (X S Z)) = V (V S T ) L(1U ) (L(1C )|U C )1 , s(1U ).(s(1C ) + t)1 . OW S V , 1 + s(1V ). 1 . M = (1C S f ) L(1C ) 1 (C ) D = (1C S f) u CS V D g v = (1C S g) t 1 + uv 1 h v W S V C rrHs' D xHV D hC D Ys t#h2sgr#nC D sgCs gY isfA 7 xixqC qQ Z f C 0 L(1V ) VT 1 + uv 1 f 0 = f 1 () U 0 /U V S T YrsrssgCC Vrxgtr x2 YCrCrFnhxn hrqY hx Vngt hCrs&#h sisrd qQ A hrqY hx Yn 7 Yrqh Vrnrtr Ysgh H stn'stA hrqY hx Yn 7 hqh H xyrsxgiV Vrnrtr xg hri'dCrxx GYrqh Yn Yn 7 hqh A Yn 7 g xrxi A V nr f D gx h 6 n xrt E g' P rsnhe'nGrrr# V YCC xrtt D t n E gVtFrbd9yrsxgiVVrxgtr Hntgxbs ih Yn " # " 0V !$ XC CT ! ( 4 # f t xtH 8% 86642$ % 9 5 7 5 3 % 5 7 7 C 8# wAv ws5EIE87 rxg'rsngxtsisrd }q bCr Q gxiFxrHVr"f C qu r#nbs Cnth nH2Frr r Hrxyr0snCr D i1xCCbs xixssg#xrq D ghrsg ns n'xCC Cgvrsnxhts A rYnnQHgh D F D r D T 1 i 7 s hHrb nh xrh sg#1 e VnrTsVC #T E nrCy S ns nt# qQ rxiFxrH2ys P rxxhe'nC gVtn#}xVx D xnFrxxCr D h qQ H"Gg nxYHn tnnvCg2rsnyHntsYhx' rqsnh gFV 2sxgCyxY s E b xe sr CrnrtbtyHC xr riFg s1xrhFg HrsnihHnCtr"CxnCr gf#x #stgCsnrsrb D Cv Q YCsC Cr V /k f V C C k f n Csrhstissrd qhsCs esgCyh' q 6 rx9b D sgx 9 s Hg xrtrn"gR i sgr rsChnVqxH9 1n1hxinn nq VYnrbrq 6 g 1C f = f 1 V N (f S 1C ) L(1C ), (f S 1C ) (s(1C ) + t) (1V S f ) L(1V ), N (w1 + w2 )|V C f VZ H trCyh1nr rqhstissrd i A xysgisyhF 7 rsngshn D hnhxrrq brsngxCn xbA nhx 7 rsnCxyvgtA nhx 7 xixssgVxn' D r Yn 7 S V CrF h#g9 A i ih ssnnrVs} rbA Yn 7 rsngnsgysxn bsnCr' 6 HrthnfF 7 xix1 ts 4 A L(1W ) (L(1V )|W V )1 , s(1W ).(s(1V ) + )1 . L(1V ) V V s(1V ) Pic V C, V C VT VZ 0 s(1V ) w1 w2 ) sr x9CCrFYnxr CrCHnhxn nC rQ ndisrd iA x7 rsngsh V T V S 0 V 0 /V N (w2 )|V Z = (f S 1C ) (t) X S F S LS (f S 1C ) L(1C ), (f S 1C ) (s(1C )) (1V S f ) L(1V ), N (w1 )|V C , V (C, Z) V C V HNStr f / C. /V V S j f V V C F (X) F (A1 ) X V tr HPS j 1W l S S V (V, Z) C/k ! shC21b D rsnhnCn 2 }#6 63 8!65 Awv 1 5 7 $ % $ 5 7 Cg C D g tChtb D xi#xrdCgnGrxxiC2 C qQ rsxgiVs&Vrxgtr s ih D xrdH qnf"g n h0 F (n) a r hxrxQH2ghtsxCH1riFg 2 sg xrt2rsnD Crxx1 qQ rhnysrrd xhntVYnynhxx rsnC 6 x h0 (F )(A1 ) = lim h0 F (A1 ). X X n nN H # ' A0 ) ) # 0 2# # ) # ' T 0 ! X t X t ( e v t X T Y ( ! 0 g vv 0 8b T Tg hx Y gT T ( ! Y T ( t w0 pA Ha u " 1" 0 # C Cg xh sxYHnhx Cr D th n E gVt F t T 0 V B (t Y S ( Ha g T 0 ! X t ( Ha R T ( ! Y T ( t 0 py a u t g ( a Y ( a d( sq Y gT 0 a c a Y ( ! S g b81)#V V( 0 ( V ( c a T 5 $7 7 T t g ( a Y g s0 g fd( b Y vg X e ( fg ( S ( P % % 871v # 0 0 1) % " dCgn bhqh Rr#nrhnih shrgqts' 1nh h0 (F ) = lim h0 F, nN (n) ' ' ) ) # % % ' A0 ) ) ( # A0 # # ) 0 " ) # ) 1) 0 # % nhx 7 Vnrb D t x E gVttr t T c a xT g UY yv c b R 0 0 W ( a a e X a b( ! g t ( S 0 b 7 g T V T g A ( dY h 0 Y e 0 T ( V t 0 0 R Q( yW Y X ax ( T UY T g( v(T a' 7 Y Y ( c a c x ( T A 0 g ! 0 sW c a Y A 7 Tg hx V 0 g X( T Y ' ( H db( vg w0 80 e 0 v t u ( sr Q( W Y vg X e ( g T ( v Y T ( t w0 pA a u t g ( a ( 8x g ( S ( c a T Y gT Y T ( QIHG ED4B8 P % % C C aq T ' % " # 0 0 1) 0 ! $$ $ $$ F F F S X A1 X s 0 ' ) # " % ) # % 0 # 0 # 0 2% % # # ) % 0 0 1) ' # 0 &# # " 1" # F S X s0 : X A1 s1 : X A 1 X X = s1 Lcor,S Lcor,S p X : A1 X X X : A1 X X X t n E A P Yrsx1 !rbrsngCx Yn 7 xhe'nCFrrCg' fA ' # 3# 0 ) # " " $1# A1 X A1 X #ns nGgCyYtHh' 6 g'hfiF rsnyHtr E V#nrf D rhnysgngCvr F A h 7 shC'YdhHvgnxYHn 's xhs E ygnrqVs Vrxgtr r'fgh s D ghr V nrssrxrnC ih Yn# YninnxhdeCrxx iY & V nr& 6 dCgn 'iY i#hC ehHnCxCeTr HgrsxgiVsVrnrtr P ns n r0Cnhrr Q Yst#hhYnC" qC r CtCg D dCgn FiY YnC}H isdRrthxCxnCr P yeBr#tsxtr rnCdfCdnd D nrssrxr&" GiY Vn s A n rxnxeYn' i C nhx 7 !iY YnC}t gCqsfGgh Yn 7 gh hx#Vrxgtr D rsCnngCv % A % n ndtVrxgtr }hC i Q Crsngshn ndsnrh9isrd i"#h} qQ Cr A1 X X F n 1 F (A1 ) 0 F (X) X (1) h0 Lcor,S h0 (F ) S F h0 F ... h0 F ... h0 (F ) HNStr ' 2% # 0 # & ) # 0 # A0 0 ) # 0 " ) 0 0 1) 0 X A1 /X X h0 (F )(A1 ) X gyts2gF g i nbrq f D #ts'shhx n1sx R h0 : NStr S s0 s1 s (axn ) = s (axn ) a 0 1 as (xn ) bas (xn ) = bas (xn ) 1 0 1 (1) / h0 h(n) F 0 s s s0 F 0 (F ) h S s (x) 0 (n) h0 h0 F F LS tr tr HPS PS xn s1 = (n) h0 (n) (n) s (x) 1 b h0 F (A1 ) X F / h(n+1) F. 0 (n) pX s 0 = p X s 1 = 1 X n h0 F (1) h0 () h0 (F ) (X) k as (xn ) = 0 S s = s 0 1 nN S gtrsnhn ' # 3# 0 # # 0 2% 0 H 6 D hri iY YnCVgrd D CrsnyhrVrTrxh rn'Cr D C qt Q th n E x1 }C 6 CxxCq sxCr et rQ 1n1s qQ nCsCs rxdstr1r D nhxr r E/k Mlis (E/k) = A E | Spec (A) Lk , Frac(A) = E (E) = AMlis (E/k)op 'f D shC2rsnCds h 6 tn xrt9xCrsg E ixnr bHxCCd0gtxrrhng' 1F 6 CrsnCd"rsnCxy r Cnr# 6 g C&rsChnVFyexHsggCh2s i ng xrtrnFrrCgt'rq b D D h qQ }ts}x P Yer E x g2rsngnsgs H Cg D rCVV H sC hHfr2xrxi H stfb ghCh H qV shCs s g Q Q snrs' qt ' # ' 0 ) # % ) % # (0 g'ysrF"nh#F Cr D t n E gVtn#F Y 0 x T 8d( cUY wyv c 0 c g Y c a xT UY yv c 0 A ( T ( a d( sq 0 a a xT T Y ( c a X 0 x (Y0 V t Q( W Y S bY ( W dwd( V g 80 g c UY g T #V w0 T ( ! t g ( a Y 8b c a g ( S ( Y hx ( c a Q( W Y a % 0 &# # ) # ' # # 0 2# # 0 s i rxC vb rxg n : W U0 ( P % % A8 77 hsts sys Vrxgtr x F a=0 D grrs2 i Cn CgCCrFYnxr E Chx"t nC 3Cg r i#VnrT 3CC f D r xgD n sys Q C qV Cg hbeF ns ntrns r}YCH i Cn r t xCn&hHg Cr g t xCn hH"nC1xsx D g#sxri Vns i H Cxi sVC Gsnh gFV VGxsqx D Yfeq D hrihn xt1iHr g#vCr V shnrb}H D q b HgnCsn g t nFr' qQ xrt' rYntFr g Cn rnrhqC1 FVnrT hrhCr 7 sxri A sVC 'xsx D YC ns ntrrr# V r rg#s ts i Cn yts gex 2Cxr q# 6 ' ' ) # # # ) ) # # ) @% ) 0 0 # % # % 0 &# # # # 3# ' # " # $ 20 # # % % % & ) ) 0 1) 0 ) % 3# # # ) # " $1" # 0 # # hC t T rix' 6 i H c a T UY xyv c 0 c g Y c a xT UY wy! c 0 y Y T ( c a X 0 x (Y 0 Q( W Y S bYE( W (#V h 0 g R (fd( ab Y e a UY h 0 Q! t d( sq c c g T 0 a ( T ( Ha R t h b W sUY x a (0 ( a v t a u Vb( g YbY g c b ( Ha ( a ' 7 T a a A R T t g ( a Y Y vg ( a A 7 ( u !S t ( a t 80 ( S ( P ( P " % 871v ( % 5 $7 7 htrxg'tgx bYHHg!BrhV q i s ( c 0 g r(n rCg s3xnFrx sisrd qQ ' 0 # ) Y T Qc ) ( x Y Y T T(v t g ( a ( 8x h e V Y aH u b( g bY g T(v T ( v ( !g ( a U( 8x e 0 Y Y T Y T T ( v t g ( a ( 8x g ' # 0 G F FZar Lk H 0F Lcor,k Lk F Lk 0 # 0 1) 0 ) % 3# # # $" # # 0 A 0 ) ) 0 0 U X U W0 b W U0 a=0 b F (X) a F (X) G(X) X a|W F (W ) G(W ) W X U W a|W X U = 0 G(W X U ) X W 0 b|W = 0 a G(X) b|U0 = 0 k X G(X) G(U ) F W0 U W X F (X) F (U ) ' ) # # # ) ) # # ) % ) 0 0 0 X k h OX,x = OX,x OX,x Vx (X) X x OX,x F (E) = X X x AMlis (E/k) Ek U lim F a|U = 0 Xx = lim S Lk F (Spec (A)). U Vx (X) p lim Spec (A) k U W b|W = 0 X W qq xqqq / U0 X h F (Xx ) = F (Xx ) x k X U F (W ) Spec (OX,x ) X k F U0 k p W 0 ! ' # e 0 h Yt g Y ( UT X s0 g a ( ) ) 0 % 0 &# # c UY g T(! Y T ( V w0 V ( Lcor,k ) # ' " Y V R h x T T t g ( a ( 8x g s0 g y T ( W sx ) g ( S ( P 9% 871v % 5 $7 7 0 Ds i H rYy q agxixssghr}Q D ss rsnnFrx#sisrd" rQi ndCY 6 nC}nh i g t D xCn#hnstVgbrd9g 9hsstehHnC9 ng gxrnrQ snq rsChnVqxHyrrdRi enFrxq }xTA Vr i D s r1 a 7 ysgisyhqqA x7 ehHn D xvrsnCd #T S r1th nHvFr#Vrrd1xnb D dnxbs#enFrxq R ' pvg ( T c D rsnCxy'nCTsg A x7 H n qd#enFrxCisd 1Cnxy' 6 nghrsgvgF Y a a HgvshCH ' ( UT X % t # h e $#" # c # # % W% ( % W DIE6 #"% $ 77 C7 7$$ gGhnhr xChxtrn 2Vsr&VfgVx# CCr it 6 Cr 6 7 ns ntrf C rCCnhr} 9s U X rsnyHnC h qf Q xrtghsght1q f D snCrsVy 4 CYn t x E sCYnx gVtn&vH ygxgiV#Vrxgtr s xhd E Crxx hqh hxTgxrrhng s gVhnhr hC D sghVxsY ' ) # # ' ) # # # ) ) # ) 0 " " 1 # Y c a xT x ( T UY wy! c 0 c g Y c UY a xT wy! c Y Y ( !g ( a Y Y !g W( g bY g ( a db( A ( S Y 0( u T V ' ) # # ) ) ) 0 ) % &# 0 0 " g pT X g 0 g ( S 0 V 80 g # 0 g X( T Y ' ( vH b( vg 0 T ( v ( vg ( a ( 8x ( S Y Y T 0 " ) # 0 ) # # ysVs hsst i qQ qQ t T UY yv c Y c a xT R 0 0 w0d( ( Y g v ' 7 T A x YU( T cUY yTv c 0 c g Y ax T(v c bY awxy! c ( sq A 7 T a V80 e 0 v t ( sq u a 0 g c v t 0 py T Y gT ( % ED4B8 P % C C ) # # # ) ) # ) 1) 0 # % S n xrt' xtthVC' #sr#nysnss2hhC ) # % ) 0 ) ) ' 0 0 1) ! c Y T g Y g (S # # 0 R x Y( T cUY yv c 0 c g Y d( a a xT c a xT V X V UY yv c W( !80 V 80 g R T ( v U( vg ( a Y 8b c Y hx ) # # ) # # 0 2# # c a UY x cUY yv c Y x U( T UY wyv c 0 c Y c a xT a xT ( Ha 0 V B) h 0 @! t ( g T t Y T ( t Sw0 WA R Ha u ( !g ( a Y t UY yv c Y gT Y c a xT ( f" % 871v P % 5 $7 7 a ( ' ) ) ' ) # # # YrqY Vngt rd}xnHdqFCgstgQrsniixnCrt nd}iHr YC#xnCd#hHC i VnrV q xv rsnyHn sisdv qQ rnCdqhHCbCnrssFgD xYrqh E yhns rxd xrtt D rssxrC Cr ht n xt Gsgqgtst 1 n xrtrnsbghsCrrq D f D rsrF sxhr 2hxgh D Vgd hrihn ! $$ $ $$ F Spec (A) F F (E) E/k A Mlis (E/k) F G Ek x h F (E) = F (Xx ) = F (Xx ) h (E) Xx Xx E : F (E) G(E) ) ' F G Zar : FZar GZar ' # ) # 1" # " R ' # ) 0 0 H 4 d I$" " # 0 X Lk 1 NZar (X) lim NZar (U ) = N (E) xrtTghsgh hsts gtt xCnH Fr# 1riCxts# nh N (E) = lim NZar (U ) U X N E/k N (E) 2 Ek E : F (E) G(E) k Lcor,k E : F (E) G(E) D Pic (C k Z) = 0 ' # 3# ) # # # ) # C N (X) = 0 F 0 # k C L/k Pic (C k L) = 0 A k C C/k L/k C/k X k : F G Z E Ak L C : F G N (E) = 0 X = Spec (E) C 0 H grrsxi"tr V HnxeCb qQ ns ntn hngsxr tH rsnYeP CtnC r rsHnVxCC# Q q) Hg r CvxCdxYHnhx 'H Cr b r'h xCn2Yy grxy gs Crgrhnys txr 6 n ndCrCrFnhxn E nC22H Cg D Yt x E gVt2H Hg sfxhdeHgn qiY &r H xxCxnrrr2r VYnynhxhn righ # F ri gh hx#xrd hCnix tnnC A x7 enFrxqhCnix g txgxnYY T c S nsrqbyh##srqgfbrsnnFrxbs 0gfY rHV # ' pvg ( X Z N(Z)/(X) Gm 1 1 C/k C/k H1 (C; FNis ) = 0 Lk Gm X Y k X Y X X k Y (X) (Z) (Z)/(X) Z X F Lcor,k xrh sr 9gHC# tCsr rrsrt YfU Cr D D r 1hniYn # snxn rsHt sh gi r rtr Vn hC ' t 3# T # # " # # # % 3 % ) ) 0 ' g (S V /C YxX Te V s0 g y T ( W 8x h x T 0 # h e c b 8UY ( Ha R a a (0 e 0 h Yt g Q( UT X a UY g ( S Y c ) ) 0 % 0 &# # T ( e ( 0 ! t g T T(v ( 8A P % 9 9% Lcor,k 0 ) " 1" # # 0 0 d( a sq Y T t g ( a U( sx 77 A8 F (C) ryx' !r#CF' qixg#hn' nd#xnFrx1 fYCsCqs #Hgbhxrhq g2 Cg hn2 snhri1 tYninnfHrxrrrsrt hf) fYstf qQ U sxxx 0 F (C) F (U ) F (V ) F (U X V ) 0 0 0 j +f h l C qhsstg hn D A Q7 enFrxYCengn t riFg#nrCyn h rsn i'hC xgHxvh rsnrhCrxq fxyHRgxH"}s Crgrhn}s "#T D S iC}'b g y"nq xrtrn9 s g#sx'br gfysrFVr D sxri Vn gxysrF hns}rnvnC Q h q 9 q f D rx 9 ns nts xY1t g H g nCn hHnCh1xV nh D snhri yhns # #T# S tnn qtvb D hC gyhhCn snhrit i H##T Cg S nrCyn h rsn # xs nh snhri Vn Vrrdg nrthnbxs ' t 3# T ) ) # " # $ 20 # % 3 ) ) 0 # Y d( 0 0 ) % ( # ( 1) % 0 Ha u t xX Te he # # # $" # t Y h e b b e 8UY ( a ! t 0 x a (0 T c b ( Ha ( a R a a T ( ( 0 g ! t d( sq T 0 a # # ( % % 0 A 0 0 ) " 1" # F n0 Hn (C; F ) = n F (C) 0 ( u dY UT ( Ha ) 0 # k Z W C Z U W C Z = Z V C OY (Y ) ) W C /V U V U X V h f j /X U W C W W/C U X W C U U W C U = U U Z = (C U )red W W C Z Z Z W C Z Z V = W Z V W C/k n>0 l Hn (C; FNis ) = 0. / OZ (Z) _ _ _ _ _,/ OX (X) _ _ Nk(Z)/k(X) / k(X) . k(Z) C/k n=0 C/k C U F m # 0 A% # ) " % ) ) 0 # 20 0 ) # # % 0 " ) % ) ' 20 0 ) vv 0 (8WA0 # c # a Y ) g Tg ) h x# T T v# t g ( a Y ( Ha0 R S T ( v0 # ( t P g ( % a U9Y ( % 8 x h 0 gA8 77 ' V " ( ) g ( ri iY hx#Vrnrtr nCg rx n" s rsnyHnCg#nfCsg#x D ns ntrns2g Q i 1CsCt t rq f t#h ndqg HrYnqCq % ' " " 1# # # 0 A% 0 " % ) 0 xrt g CY'f D esgnnCdHrbissrdi txg#x vHg &rsnyHn nd qQ rsnhn E F F rsnhnG 7 dCgn fhrqY hx A rdrnCdg &C &H s& 6 ns nghsgh H s Hg g shC xCCd V2nr rsnYeTsqs i }inC#n#sr1rq b2CgYrsnheTs P P g9ht nCChHtet} iCCn nh xgixnrHrxyrh$Hxr U 0 Htr F (X) s isF grrsxi"tr 2sqC qQ n xrtrtqsD n xrt D g t xCnhH etVrrd vrsnnFrxsVC ssChxns} tnn2q# 6 ns nrt i qQs1rix ' 3# # xq b D rsnhn E }s ' A0 ) ) # 0 2# # ) ' 0 g ! 0 8b c a Y Tg hx Y T V ( ) T ( ! t g ( a ( sx g ( S 80 g V B( S # t x gVt2tH t T Y T 0 a t g ( a U( 8x ( a d( sq ( P 9% 871v % 5 $7 7 " 0 A Cr srshFY 7 BrhV q xgg rxd}xhF$8Yrqh 2ynr sxCr}xnhx s i g Yn& g'ryxr x yH nhr s rsnnFrx sisD rd qQ t x E gVt Vrrd#i qsgY dCgn }gh hx# V#gC 6 dCgn TiY hx'0gxxCxnCghsCrrfr rsrFrx sVC 9rd D nhxixrgh hxrsrtr YfU D e i dCgn iY hx Y T a a a0 a yrrd tth rxd#i rY ' t g ( a ( 8x h $#" # c # # % b% ( HA ( sq q b Crygnriys ynr snvs qQ D dCrxx iY h n xrt0ghsCrr nChnVnCtH xrtrt n 'r enFrx sChHgCHnhxxr nC'r rsnnF sngCtr&nrdCgn&Y qQ grxs e inxg Ynng qQ $$ $ $$ F L/k 0 Htr F 0 X (X; Htr F ) = H 0 (X; F ) 0 F OX (X) k j : Spec (L) Spec (k) j j G m 0 k F H 1 (C; j j Gm ) = Pic (C k L) X/k s : X A1 0 X 0 tr F (A1 ) Htr F (X) 0 s :H X X 0 0 U X Htr F (X) Htr F (U ) k 0 Htr F 0 Htr F (A1 ) X F 1 0 / lim Htr F (AU ) 0 F (U ) / lim Htr U X U X X D sE : Spec (E) A1 E E Lcor,k FNis tr atr : Pk Nktr tr : HPk HNktr Y 0 V t( Y b a u T 0 v X t 0 X 0 ( a g a Y XV hT T ( 0 g U( vs0 Uvg v t A1 /E E T 0 v X t g T 0 v X t ( a R vg X vx 0 T Tg # 0 A@% 0 @% aHtr tr HNktr HPk ' 0 s : Htr F (A1 ) Htr F (Spec (E)). 0 E E 0 Htr = Htr 0 k 0 0 : lim Htr F (A1 ) lim Htr F (U ) U F U X X : Spec (E) Spec (k) tr tr : Pk P E E 0 U X M Gm = A0 (.; K )1 u o i i m o s m w v ~ u s s y u xeyqYq Yqb YtxeE 0feqeqqebb!!I Lcor,k ns ntrnsq 0 ht nGg grrsxi"tr xs s # xrtTgqCr19s sgChnsg n xrt A s rsHtV'nCRrqs nhx 7 rytn bA 0 # ) wdQc ) V # % Y a nhx 7 a0 &# UY s Y0( g # c qxFss} 6 rt xCn#hnsfs Hgt x E rVns i C s1 i}sgCvYy t xtH " 86642$ % % 5 7 5 3 Cri vCrsnC T g V( " 0 Tg"rsgCr CVxnys}b D isFxH ' Y H) x bdY $#% ( H) g " ( % Wv "B8 7 gh D ynr iY Vngt s g Chf D CCYHr# i Hrqqr9s#b D CxnghsCrr vCnC 6 g rsnhn C" V2hbisdxYexVv P n xrt" V hCCss g xxxx9ghsCrr&q rCrsxgiVs Vrxgtr snC CyY }snrq qqr'f D rsnCs 6 A1 X X G m X Gm X/X F1 (X) F (Gm X) F Z 0 F (A1 X) F (Gm X) F1 (X) 0 S S Z X (Y, T ) (X, Z) s1 j D t x E gVt0sgd xhdeCrxx Gih YnVrnrtr xngsVnxr" 6 ibxhdeCrxx fih YnVngt tF ' c b 8UY ( a d( sq 0 a a a (0 T(! # # % % 3 ) ) 0 ' V ( 0 P A 9% C 7 h x# a x t g ( a Y sWA0 # c # 2g ( S s0 g V " B) % ( Tt ( fg ( S 0 ( f% ynrhV q 7 AI v 6 H xrh sisrd rixtnqxY !g r' qQ V&1 Th F&!# &$t( 001 % 2 2 ( " #4 xhdeHgn Yrqh YnVrnrtr g xrt 'xhsCg' '#ts rxdGisrd9nhx ' 3# # Q C qc hriiY !rsxgiVs}Vrxgtr ' V " B) T ( v# t g ( a ( 8x 8WA0 # c # g ( S 0 ( P 9% A8 ( g Y T h x# % 77 ' a H rHnFVnrT qrdbdCgn fhrqY hx g2nr sBrhV q i rCYn rsgCrv g rx2b#i srCCng s r stgQ9r#ssgnynnY 7 rqxythh' nr A rxihr qQ i Yst'iHrnrssrxr" v qQ C#}'C 9nh fnrh "1 rsnnFrxGrd'YCVxnysRCrsngxgCf RHrGCG ) # # ) ) ' 0 A@% ) # 0 2% # ) " % ' 0 1) ) " 0 1) ( g Y a u hT g W a DU! e ( b 0 g ( S g b ( 8d( a T g Y V0 % ) # 0 % ) " % ) 0 # ) Y T 0 ! X t 0 X Y c Uvpy hTgT hT g a % Uv e ( b ( sq I 9% # 0 % a S 0 ( sq c T vg Y vV g A8 77 ) 0 1) HNktr k F c a xT UY wy! c Y 0 g Y k iVf D hhx!g n t T c a xT g UY wyv c b 0 0 b ( a d( sq 0 a g ) # " % ) # % 0 k T ( v 0 g ! 0 8b Tg hx # 0 ) ) # 0 2# c a Y # ) Y T t g ( a U( sx h e k # " $# Lk % % 5 7 5 3 Crtth ngx#F "G " 86642$ % 5 7 7 BA6 B8 S # S g H hri F S X F1 (X) = coKer F (A1 X) F (Gm X) . F FNis HNktr Nktr HNktr FZar FNis FNis H (.; F ) F h0 = h0 h0 L[X] (1) F F1 XLk S D Z n F (X, Z) / T Y g f Z / X. (X, Z) n X X F1 j S (f, g) k xgCyx xix"tr#2xixsxnrxrC tn1ngsVnxr" 6 xixsxnrxrCvCg sgHYyvyHC hn #enCgs' qd1C}xntr' qbnYnCsisd qQ " % % S (X, Z) (X, Z) s s Z (U, ZU ) i grsngnnxtggHgsRghyysrhF!ss' 6 i r1xixsxntggC'Hg s g V rF shH'gxsTqqr nh r srF2yr}rd D rVn} sgC2hns vh D sghVxsY yhCtxnsCxg nys D rsCnngC i nCtsx' D rihn v rsChtr gVt1 Q h q@ xixsxntggC rsr"yHC xsGgCrd xrtrx 9 gnhnngh#s i rxixsxnrxrC risgHYy Hg't xtH " 86642$ % % 5 7 5 3 VngCqqbsgsf s sgChns hCvxtgTs 6 xhn E 1F Crt n E F hriqrChnsRr Cg tn xrtsgx 9 gYnng2nhFnhx T c gnnHds rsnHnxeC shhx ti h ghhrYnv qQ ' pvg ( rsHCr q b D ssgxntD V t gt xCn&Yy Cr rsnxhts hH&r1 rsCnngC ns nsgx 9 r&rsnhn 's D Vdny nxhts hns hC # 2Fr g rix#x#Cr1rq 6 ' t 3# T ) # ) ) % ) 0 % % ) # ) @% 0 # 0 ' ' " 0 @ ) 0 0 % # ) # " $% ) % # " $% ) 0 # % # Y T ( V 0 ) Y c a xT c UY yv ( H Y 0 g U( vb ( !80 R R 0 ( e 8x ( a R T ( !R ( v ( Y Tg XV Y T T ( g Y 0 h t d( V X sq 0 Y a g a X wa W UY T ( v HT wx bdY g Y g V( a u 0 a (fcd( ab Y S XUY bY g Y g 0 g 0 80 g V( Y V ( c a X 0 x t Qd( b Y S UY d( b 0 g Y d( sq Y 0 ( 8w b 0 0 b ( a Tx g g (Y d( a 81)& (0 ( V 80 g R % # ) # % Y 0 0 V Y xh 0 0 x % % C C ' ( v# S g w0 # ) g # 0 80 g Y ) ( a0 # Wa ( a5W# V g ( "I ED4B8 Cr gCC2hnst D snrsst 9 rh xCnhnst}dnyhsb C #Hg rt nCCYyqb ri rCrCrg' H risg 9 nr Cg i H hri1rC Yyvbe Hg snCr th nv#F # xrtqrsnsRCxnCrnnYxVghrhrsr tsisdq qQ "" % % xgvgvrsnCdr# th nH&hns# Cg ht xt vdnyssrCnCGs"b D h E r T c t x rVn2ny rsnxhtsvhns rsr& 6 ' pvg ( Xns n2g Yn 7 ht xgtnrhnih 2s A ) ( 7 g Y T g rsnhxrfxgHx'shhx HA ( H) Y % ' nhx I0 ) U( 0 vb% qxCs 6 hCnTrrHnCrFYnxr rgYnnghs 0 0 ! $$ $ $$ F X S Z S X X Z S X (f, g) = X S X [(X S Z ) (Z S X )] p q Xo /X X S X Z (, Z) S /S p q ' (u, v) : (X, Z) (Ac+n , An ) S S (X, Z) (f, g) : (X, Z) (Ac+n , An ) S S 0 1) ) ) 0 # ) @% 0 # " % ) X (n, c) S s0 : X A n X n (AX , s0 (X)) X S X : X S Z f S Z : X S Z Z X S Z X S X c S S (X, Z) o Z S X (X, Z) p gred : Tred Zred X Z S S # # ) # % 0 (X, Z) (Ac+n , An ) S S n (X, Z) (X, Z) U s X Z (, Z) s0 X S X 1S v / Ac+n . S / Ac Z (X, Z) (Y, T ) (X, Z) (X , Z) i:ZX q 0 / (X , Z). (n, c) (An , X) X S (Y , T ) (X , Z) (X, Z) i(Z) S Z S Z S Z X S ) # # ) " % ) ' # % ( % ) 0 0 ) # ) ) # % # # ) 0 0 % # 50 ) # ' # 0 2# # V( bY c T UY axyv c Y gb 0 0 b g Y( v80 g XV 0 a g T ( gTg x g t 0 g Y y Qc p!ww b h 0 g d( sq Y0( g 0 0 wdfc V t Y g a X Ha W UY t 0 x g ( S ( P ( Y T hb x a S V T g R Qd( b Y T ( ! ( t g ( ( a ( g8X x e (8g ( S c (g P ( "AI80 g 87(! HTvx ( c a 0 % G 9 % 5 $ 1 7 7 T v T # " $% Ha 0 # 3# c UY g ( S % 0 " 1" % " 0 xrtrCrtsnrnxhhrq rvrsngxr g'nCr"snstss 'nr b ghxr V'nrvCr xrt2hCCs' fhC'b H xA h h h Xs Z s X s Z s . h P Cnxhts E x 7 xhe E nCgq#ns ntFrrCg'q C 6 P he E n' qsnCr#nTgxiCq S sZ h Zs = ns ntrnsrrrtrq b D ysrF#yrr Y Y T h x# ( g a h xRrsx1GhrisgC1F ' ( !g ( a U( 8x sWA0 # c # a t # dY b% ( sq hHCrs qV A Yn 7 Q g&VV H sC Vn hsts' qQ s'C A Yn 7 s s r s ' # xrtrsnsHg CxnCrf D i ' 0 ) 0 wdfc V ( a Y Y0( # ) ) # % 0 ) T ( e ( 0 ( 8 aq 0 ) ) g2rsngnnxtggH0ghVs#CHvgf D hC t T g 0 t T ( ! UQd( b Y 8x t Y( c a T ' # 3# 0 A d( V ( a Y b a u h 0 a # 0 1) 0 ) 0 ) % ) ) 0 ) 0 # % # V ( c a 80 g fd( b% Y g ( S 0 ( P 889" A8 %9 % 77 ' T ( v# HT) fHa3# UY g gx 0# c H rrsrFrx Vn sqxysrF rd xr xxCxnsisd "ChnisgnYx1s }g nr" 6 # # # xrtrsgnhnngh2sgrb D #tstshhx' rxdh qQ i Hg E rChns#fefYrsrq qQ g2rsnY nh s0 0 Ac Z (X, Z) (Ac , Z) Z n+c AS (X, Z) (, Z) (Ac , Z). Z (f,g) (Y, T ) (X, Z) (Y, T ) S S rC hns iqxsgHYyRg rxg'rsngx nyx i r xixsxnrxrC YCH &ntngsxrb h qQ T c ns n}rx 9 gnhnnghrsn9Hr ' s pvg ( (, T ) S s c a xT UY wy! c Y 0 g Y ( a R t ( Z (X, Z) # 0 0 x h 0 ! t d( sq g T 0 a ) # # Z X s S (f,g) (X, Z) o (Y, T ) o h Xs (, Z) (, T ) A c )h Z s U (1S g,g) / (Ac , Z) Z / (Ac , T ) T h Zs X D y H hhHvgqhhxi rxd # # (U, Z U ) (, Z U ) (Ac , Z U ). ZU s U (X, Z) c Ac ZU D Ac Z LS D i!iY YnCVgr s S h h h F (Xs Zs )/F (Xs ) F X h h Xs Z s = S h h h F (Xs Zs )/F (Xs ) h h h F (Xs ) F (Xs Zs ), Z V Vs (X) h V Vs (X) (X, Z) lim h lim (V V X Z). V X Z. (X, Z) s S h F1 (Zs ). # # ) 0 ) 0 " % # &$" 1 (X, Z) X s 0 # ) s S X W xCCdrsCrx2 HC&r#b'H stHtnh i''"n#nC h qQ gFCrshF2 H Cg D gt nCC Yy hHhx r g rsngHnrCt y" sg 6 rsngnr#nxVssCx D rtxrCqs g ntnh nd1txrvgh b D g#xr}s Cr}ysh}s s'Ch} tnngf D s r xnsYCYnf 1V srh ntthHgR CgbnC!s i Cn V V f V /S () S ) 0 ) # g9ex 9hstsVxgT g xrh rt} qQ ns n2g V'CCs V I f V A1 S f I #ns nt#Cg t x E n xrt g 1rixs'f D #ts CCr a sYervgH Q P g #T S tnxrgh"f D rV rF st s CYnqqqrvrb g Cn g Q t nCC hHgex C qa rsnhn 'xr V YCC D xrt# D gsnC s s r V CrF h Vns tH s hhC rxrrhngTGgHC 1 h q@ # xrt E qinxg D ns n r1 Q Cr Ct's g Cn S # xrttrx 9 r xrng 1Hr# 6 A V Hg" C D s a 7 t nyshqrsx i Tssnb i ngng R rsnyHnCfhntitxg#x ! 7 A ns nt CxiCtrxgYnCnxeHrCe1#ns ntrnsbe1 D YH yH#Yy P t x gsnC gh& nh nrh2 x hChxxg2fC qQ ns ntrnsTghsCrrtCbh D s rsnnFrx bh rn D xxq qQ h F A1 )h Zs /F A1 )h = Z s Z s h h V Vs (Z) W Vs (A1 ) V nn2ghf D hrihn xrtghsCrr} xrt rrCg rsnyHn}nCsgxrb D nr!tsxCCYnhnC gnh yCCrgrhnq i ng gfts F Cg h Z = Spec OZ,s Z h F1 A1 s F A1 )h Zs /F A1 )h . Z s Z s Zh () h Zs D ' c a xT UY wyv c Y 0 g Y # # ) # # ) % ) # " % ) % # % 0 ) ' ( c a Q( W Y Ha % 0 # 3# ( ! S g W( S d U#V UY wyv c b 0 0 b (e a R S 0 V T Y( c a xT g g c UY g 80 g Qd( b Y d( ( ( vg X ( Qg ( V ( c a 0 T T % 0 " 1" " 0 0 x UT ( ( v h T ( P AI ED4B8 % 9% C C V Vs (Z) Q ns nrrCgtfH qV rV rF E s Vn } qQ g2V CrF h yhns F sgChns Rg' tnhHhx xgbH qQ enys xrtrnggh nrssrxrrxdYCH i # g rsngnnxtggHghVs2 D hC ' t 3# T $$ $ $$ F Ws ) s g 1 V A1 S (S) S lim h s s F (A1 )h Z)/ F (A1 ) Z s Z A1 Z F (A1 V )/F (V ) V S s h Vs F (A1 S)/F (A1 ) F (V VS )/F (V ) S S V Z k A1 S (X, Z) h Xs A 1 ) h Z s (A1 , Z) Z s Z V ( F )1 (A1 ) Z I V P1 S V f A1 S () lim Ws P 1 s P1 s k V j S = Z lim F (W WZ )/F (W ). h A1 W Vs ( Z ) V g : VS S V V s /V f / P1 S h F A1 )h Zs /F A1 )h . Z s Z s F1 A1 s . Zh f :Z S lim S A1 S (V, S) s V A1 S A1 S S F (W WZ )/F (W ). g Z A1 V V W = f(V V ) W V h Zs f V VS = S V V S V Z I h nTrCyh gh Vns Ftn 6 % 9% 5 7 5 3 Hf D Cxixr1riFg' @I " 86642$ D ysrFVgdbr ns nvg ns nt#rsnHePBr1 th qt Q Cxrfx}F G ns E rtrCrt#1s ns nt rsnCeB D nrssrxrsys P CnxhtsghsCrr i9sgCtYy gVtn"F Cg nCts0g H n (X ; F ) = H n (X ; F ). S 1 j : X Z X hstsC D q D B i s rx yHn sVC g Chnh i nr# sgxrf D s rsnyHnRg2rsng sE rh rg1r&h qQ n xrt!rrrt i1 ygnriys#Vrxgtr hsCs s xnFrxi Frh R 2rsnhns i dCgn !gh 1 ns nbghsCrr 2F ht x E hCq 6 #T rytntrher E x r S P hsnC g 6 ssx'H Cg D hri S h#T"ht x E sg 9 gher D E xvH P srxhxrt rq b xrt rytnts h 6 ' 7 A pvg ( ' s T c s Vgr A e7 rq sHrsyV'f D rrsnsxnhx V1ngCsf D th n E gVtnVgqr hriRih Vrxgtr F Cr'h xqrsCrxH S f U/X H (Y X U ; FY ) F S n s hth nvgsnC rsRr2CsreP 'h x gsnC rsith n#xHnCs rh 1r D g }hsst1 qQ x2hngsxr9gh yhns r yr}rd}i rY iY Vn rssFg 7 VgvCr D A nHdhCCsHg sn' YrsnhsrxrFr rnCd hqh ghHr9gxrxi nvrsnyhng}}nxeYn"f h 6 r f f qN Rf : Db (Z.YNis ) Db (Z.XNis ). q D P rxCCdbysreBGrCrt D q D B fsisrd D q b D ht n0g ns nVgbs s i h 6 ntYninnfyHn 1F D Cg D rsrFrx sVh 9ht x E Y ( g a sg 9 g xn& vH sf t n yrr ' ( !g ( a Y t # dY b% ( sq grrsh2g2ynhxYhh hsts s rC f :Y X 0 F (A1 ) F (A1 S) F (V ) F (V VS ) 0 S S j +f CyY qQ H nrCyn n"g hn sgvgh'f D sgCs Ynsyhxqr S HhH1Ct hHCq D Q h qV S CCvFr1#T' s YCsCsfs C qQ ssnrnxhh D Frx9 P xYex n rrsChnVr"F g Cn P1 s D(h) (Ws {0}) = l h P1 S V S W D(l) = W {0S } V (h) = P1 D(h) (V V ) VS S 1 D(h) f l (X, Z) i:ZX D X ZNis U XNis H n (U ; FX ) = H n (X; F ) f :Y X (Xi )iI Z X X = (Xi )iI : X S F X C S X : X X i C X (f , f ) : YNis XNis FX j j FX Nis Z.X h s X Z C(Xs ) = 0 X XNis X FX = ( F )X (h ,l ) FX V VS g A1 S S F(X,Z) XNis f FX = F Y X FY j /V f / A1 S Y R q f FY C i i C F XNis XNis S f ' Y c a xT UY wy! c Y ( !g T ) # # ) 0 " " 1 ) 0 % 0 ) ) # ' % 0 &# # # ) # # 0 % 0 # 3# # ' % 0 # 3# ' ' ) 0 @% 0 # % ) # 0 ( dXbdY ( a 0 bY wy! ( a g R d( sq 0( ( Y c a xT c 0 a Y( c a fd( b Y a UY t bY wy! c 8d0 g UsA( Y a c c a xT V 0( Y( 0 ) V X x ( u R fd( b Y Ha UY h 0 g T! fd( ab Y Ha UY g ( S ( P ( c a c ( c c aWpT e gWvgx 0 b( X y w0 UY awy! c( a ( S V T Y Y c xT (P xg T T ( v 0 g Y y Qc p!wx g T ( gTg (X, Z) S p q (X, Z) (, Z) (A1 , Z) Z T S S S S (A1 ,Z ) ' # V 80 g T ( v# HT) x bdY # @% g ( S % 9 % A8 77 g V( " 0 ( P AI HIxhn gYnyrxnrFs hstst#hbCVhntbe1 qQ g1CrsxgiVs#ynr tnC s #tseCets D hs #tsr P # ) 0 ) 0 ' # 3# ' ) # # ) ) % C s iYrqh Yng ' ) # # ) ) ) # " % ) # % # ) # " $% # ) % ) # % 0 &# # # ) # ) ) % CxiCsgi&y rsnnnn vY nsq 6 gysrF#F t T c a xT UY yv c Y 0 g Y c a xT g UY wyv c b 0 0 b ( a d( sq 0 a g 0 wdfc V a bY t UY wy! c ( H Y ( 0 g ( S Y0( c c a xT T ( v Y HT wx bR( Y g V V 80 g fd( b Y vg X e ( T T ( v t g ( a Y sW c 2g ( S ( c a T g hx a " # # A0 # # 0 7 Chyhnr' A # ) 1) 0 # # ) % ( PfI 9" ED4B8 % % C C CrsgFC # ) 0 Q CVg#Vbisrdnhx' bC qV ' ) # # ) " % ) # % ( (S E V s0 g R (fd( ab% Y Tvg " X e ( TfT ( !# t g ( a Y sWA0 # c g ( S 0 ( P AI ED4B8 c % 9% C C g h x# # a gxrnCd1HyhniHr bYrrt ' # ) # " $% c a xT UY wy! c Y b 0 0 b g ( !wa ( u R g g g h 0 ! t d( sq g T 0 a 0 wdfc V a bY g Y0( c T ( v HT wx bY g V( sZ # rt hCCs g#nh ns xrt gYnng Vgr A XNis Yrqh GgChyhnqrTrrr1q f D xCxnCriFg' y D C qQ $$ $ $$ F F 1 h h h h F(X,Z) (Zs ) = F (Xs Zs )/F (Xs ) ' D rsnC t T 0 Y W 8x e 0 W( ( sx 0 g 0 ( Ha e 0 X V T Y Y x T 6 ' # 3# 0 # 0 ) # ) (Y, T ) (X, Z) S ) s h Xs F h h F(X,Z) (Zs ) = (i FX )(Xs ) Z (f, g) D gh hx' tn#YngsVnxrRgh bs hxn Vn sg#nbhribhqh Q CsC qg rsnCxnyr V D h E qI sngCtr s'# xrt E 1xg sqixnr" qQ # xrt s Cgth n E rx 9 r s i Cn st xhe xrngt "H P D h x E rx 9 yrr t x E gVttF Cg riGiY ynr F #tsthYnC' ndbissrd iCqnhx qV Q g Cn r'srF sF ns ntrns ghCCs C qQ r#y H h yhns q D nsbs r9hrqY 1 0 ' # 3# # % 0 &# # X ' Z f g(t) = s F A1 Z V h Zs IV U G : V /Z 0 FX j FU i F(X,Z) 0. : V UZ F(X,Z) g F(Y,T ) (f, g) : (Y, T ) (X, Z) F1 |Z G ZNis Y h Yth Xs F(X,Z) g F(Y,T ) (U,)IV F(A1 ,Z) Z lim S S Z F (U UZ )/F (U ). Z t S (U, ) 1 ZNis D (A1 ,Z) Z : F1,Z F(A1 ,Z) . Z 0 g Y ( v S g b( S d U#V V T Y( ) # xixtr' nC2 qQt hxCrrsnnnn th tx#sr1rq 6 t T c a xT UY yv c Y c a xT UY wy! c ( sq " 9" ED4B8 a % % C C (A1 ,Z) Z ) # ifYrqY Gg n xrtthCCs xrtCVVr g#vnh qQ F(X ,Z ) F( ,Z ) F(A1 (p ) (q ) : F1 |Z F(A1 ,Z) Z Z ,Z ) F1 |Z Z Y S : (X, Z) (An+1 , An ) S S S Y F1 IV (X, Z) s (f, g) : U X Y = Y S T 0 0 7 qxh Cg Cr rsnYex9rrCg2 H P 0 ) ' 0 ) 0 ) # % R V ( c a c T ( e ( 0 h 0 g 80 g Q( W Y Ha UY R h 0 g V 0 fh 0 g R Q( W Y Ha UY h ( c ae c T ( e ( 0 h 0 g s0 g R x T ( ( 0 ( H Y x g % ' 0 ) ) 1) 0 # 0 2v t t g T (S ( P # F $ A!6 9 5 7 rnsVs gVtnGgxg E tht' !g ntt2FrtVgrdCg rd gs rq rixnxyrtr D iHr1 xsxhr D r nYnChs2GA rnsVs 7 gCgxgi E gVtqrd CCYq"#xixssgh sgeYCr tVrxgtr in rHsrsnxhxx ibeHr#VnrV q g0gVx1 qQ shCYdhHtGg CnxhFn F i W"v % 7 7 n#YngsVnxr'ChyhnrxixssghVs' brQ2ss xrd br'Hyhn d g&C gi iVbti xgf D CsC& Q s D ChyhniHr tsnrshsgrCyY D rsnHnxeC&x# 6 xixsxnrxrC hCCs T c rC#Yy1 hxnCrrq gs 6 ' pvg ( #Cg #ns nt E VrnqYHnhx sCrsxrxCCdChyhn9 qQ stCg9tn xrt E rx 9 nhFYn V = V 0 S T ... H n1 (V ; FX ) H n1 (V ; j FU ) H n1 (VZ ; F1 |Z ) S H n (V ; FX ) H n (V ; j FU ) ... (V0 , V0 X Z) (An+1 , An ) S S V0 X0 R m j (V0 , V0 X Z) m > 0 T rbhqh gChyhnrbghsgh&hCCs xgn s YnHnxeC xrt qQ CChF" hnsrrCg &F Cg Cr t x E gVt F i xixsxnrxrC Cr rsnCtsg risgHYy gVtn1H t xgrhnF % # 6 $ !" A 8 "B8 $ 5 5 7 5 7 $ 7 xixsxnrxrCTgCtRr1Cxe' "CxCYnr rsHtVRrrC Q Yy rytnVg1rd n xrtrnsr CnCxnCx2hnC"Fvr} qC r s rChHhgGYV i srFG Gi i ghf D rg#n x rsng#ndCnC2y1i qixrtnr esgnnCd'nC}rssf1 D nCn rhxn s qgxrnCd srxhxrtt xrtts qQ ' r pvg ( T c xqhFnhx g#esgngC gVtn srsxrxCCd sCVVr2 n xrtrns qQ YrqY rtns ntrns r xixsxntggC x Yninnrq C # ZNis ,T : F(Xk T,Zk T ) F1 |Zk T . nC h x E S T D (X0 , Z0 ) (An+1 , An ) S S S (X, Z) = (X0 S T, Z0 S T ) j : U X i : Z X D XNis T 1 0 FX j FU i F1 |Z 0. X0 V0 r "sVgx ... H n1 (V ; F ) H n1 (V VZ ; F ) H n1 (VZ ; F1 ) H n (V ; FX ) ... D ChyhnrCrsrsngnsgsr'b t n E sg 9 yrfd D C qQ g CrY i D n H n (.; F ) R m j j k S n m<n X T S s X T S k (V, VZ ) 1 j S : (X0 , Z0 ) 0<in R 0 # 3# H i (A1 h ; F ) = 0 X s 0 # 3# T 0 v t d( a # 0 0 H (X; F ) i 6 i : A1 X X FA1 X k p,q E2 = H p (X; Rq FA1 ) H p+q (A1 ; FA1 ). X X X H nCr xCCYD0rxxYHn sb hC X 0 < i n H i (A1 ; F ) = X A x7 y y s u s y ~ o u s o s su y w~ wu xu qs xeeeqbY9Hexeq!iqeeYxu qs qqxV hnrsnHhx9 D sgCHC qt Q qQ ns nVrrr qH xrtTghsgh hsts Vgd enFrx qQ tsrd th n q ht n E hsg 9 g P her E x' qQ r ysrF#Cg t x E rytn i2 Hxxd' F r#enFrxrt Cr D r rC"Vgfrd snrhtsgrb D tth9s 6 Cg d Cr CxHr}hCsxix r't xCnvhnsg Hyhn" D h qQ Cr d Cr rd"Cr HrsnCr sVdengn r1th nHvYyGr D Hyhn#shxCsvgvCg s g V rF s"f}x rty H hs Hg ntYsr xtnn"rsnCCs G snCx i D 6 rsnCts"x q D C 6 s rbxns#YCYn} F E hdnFbs gbCbgrCn' F t n gVtn#tx n xrt rsHt1xix rVntCr g t xCn2hH rxsnh D snrnxhh th nrCrsnCts E rVnH Cg YdhHq 0gVxnb nrh qQ # nCts' rxCC1nCCrYnYbs tnn1 6 ' t 3# T g ' # # ) & ) 0 # 3# ) ( T ( a !# 0 g s0 g V " ( ) @% ( Tt ( x g T ( v# Q( W% Y a0 &# UY g ( S 0 ( P I8G " ED4B8 V c %G % C C u 0 ( c a # H @A h BrhV q i Ds 7 VV q g"tthCisDd "snhxCf D s fr ri rC rsnCxr }CCi tsVin2V}ngx" 6 Yy2Vg1rd'Cnx ' # % # " $% ' 0 ) # % 0 ) h x T y T ( b 8x ( Ha 0 # 0 # # # 0 wdfc V Y0( # ) ) ) s s # % t # # ) # " $@% 0 &# # # 0 0 0 x h 0 g V80 g R Q( W Y Ha UY R h 0 v t d( a ( c a c g T 0 ( T X A Y c HT x VWdY a UY h 0 g v t Ha dY x x X g ( T g ( c a UY x c a xT V X V UY yv c b( v80 ( a ( u R T(v e HT wx WdY g v g V( T T ( ( 0 ( H Y x g ( S ( P ) # % ) 0 0 ) # % # ) # " $@% ' 0 ) ) 1) 0 # ChdnF}hdnF'Fx1C $ 5 7 Yb#x srF &snr"s hn&sisrd qI G drA!6 Q D yHyrrd D hC hsCssnh gFV g 3iCxg nh## S $$ R q F A 1 = 0 X h s X Rq FA1 (Xs ) = H q (A1 h ; F ) X X $ $$ F n Y h x T A T ( W sx ( Ha T ( V 0 0 s (X, Z) k T ! c 0 c g Y P (X, Z) P (X, Z) k X H n (A1 h ; F ) H n (A1 h Z h ; F ) X X s (X, Z) s X H n (A1 h ; F ) = 0 X R 0 # 3# (X, Z) k X X sX a X W UY V 80 g % 0 ) c T t v Y H V Ha UY g Y ( c a XY V( Q( W Y S UWdY T! t t ( 0 x a t V pT X S a e 80 ( W 0 g Y Y ( ( T ( a R # k P (X, Z) k U U # # % # 3# # ) # 0 ) 1 i r Yi Yi1 Z U Yr ' # # " $@% # % ) 0 R h 0 v A 7 g T t g wa 0( X( T t ( dbdY e 0 Y g ( 0 0 g 0 x UT ( ( v t d( sq h T 0 a ( c a XY V( (Y 0( t Q( W Y S UQbdY #V 0 0 % ) 0 ) # # # Spec (E) P (X, T ) P (X, Z) # h OX,s A ' 7 ' # 0 Y T V U = p1 (V ) Yi = p1 (Yi ) Y0 ... Yr+1 Z U T X n=1 n2 X U0 X k Y Zsing Y P (X, Z) T : Spec (E) Spec (k) H n (A1 ; F ) = H n (A1 ; F ) E (E) n1 T H k n rs tnrssrxrsCbisd hsst' xg#x ns nrt E/k mi gYtu oe xqekq0 s y su j w P (X, Z) P (X, Z) k X sX h h (E) = lim Xs Zs (A1 s ; F ) Xh ZX Z p(Zsing ) n p(s) Y0 ... Yr Z 0ir H n (A1 ; F ) (E) k Y Y = Y0 ... Yr Yr+1 = Yr (Z U ) U X dim(Zsing ) < dim(Z) < n dim(T ) n 2 Y X (X, Z) 1 E U Yi1 s 1 Z Z T X V n1 x X r>0 V 0 < q n p : U0 Y p(s) X 0 ) nCsCsCisd 2sgHCsgxr 6 xrtgght H r#rsrFrx Vns }d}nHd1FC#sxe1s Q C qp FyenC9g HrsChnV E E n r D ri&HVevnC1g ssnrnxhhr# D x1r rs snhnCD v ts CgsrFs YH ns ntr " n xrt YCC D nrssrxr"sysr D C qc rstt E x"9 Q n xrt rrCg qQ g C xxd F k E Lk : Spec (E) Spec (k) s E h h F (Xs Zs ) h h h h h h Rq j FU (Xs ) = H(Xs Zs ) = H(Xs Zs )/H(Xs ) h (E) Zs h H1 (Zs ) H1 (E) Q n xrtrns2grq D bqhsst s rsnyHnC2h qV iiY hx Vrnrtr s &nrssrxr Cgsnh gFV rsnHCs 6 H xEs g rFC1 D Vgd}inCCr A snCr#f 7 inC2 Q Y rHV rsnHCs &Cg hC HgCxevrsnCYnynrf D rsngnxntrxrC rChns gVtnvVr 1 ns nrt1 C qQ (1) F (A1 h Z h ) X s sghVsxr s H rsngx D }xgYs s 2s D grrssisd qi fhstsbA x7 CCY' 0g s#ersxrxCd1 qt V rF s sVC nrxxCnrV Crx Q VCCYrt gtsv snCr iv 6 rtysrF vF A x7 Chyhn RgesgnnHd rxd9issrd iCqtn xrt sg 9 xYHnhx esgnnC A dq 7 enFrxg'Hyhn" sgrC 'bCr g }hsCs qQ g HCr xrtrCrt&t n xrt Cr rdCyY}rxxYHn q sys rrCg rx f D 0 F (X Z ) F (X ) F1 (X ) H 1 (X ; F ) H 1 (X ; j FX X j FX Z b H 1 (X ; j FX Z ) H 1 (X Z ; FX Z ) j = b Ker() = Ker(j ) (X , Z ) X j xiCnh YninnA Q7 Chyhn qQ nttrrr D Cr D r rsngnsgs# r rsnixnxt E gHv Cg rsnCtsgsgH gVtH yHC rsxgiVs#ynr A s 7 A V2hHCsrsngnnxtggH' tsF1h x E sg 9 r!gtrsngnnxtggH 7 tn xrt}sg 9 xYHnhx HgnnCd H n1 (V VZ ; F ) H n1 (VZ ; F1 ) H n (V ; F ) H n (V VZ ; F ) 1 A1 (X , Z ) = (X k T, Z k T ) k j Chyhnr2g}hHCsbA x7 CCYrrsnixD r' D rsnix t n E sg 9 g Cr D rstt Fr E xntrxrC"hrsr nChsFg sgC'YyD gVt'Vg 7 A H rq b nxhts2Fr sgC hns gVt Vr1 nxrsisd n1 E (X, Z) 1 j : XZ X 0 < m < n, Rm j (FXZ ) = 0 (X, Z) 1 j : X Z X V X D rsHtVfg rhnysg&hrshx'rixx iChx'nr A 7 H Cx 6 T X X H n (.; F ) n=1 (X, Z) (X, Z) T = Spec (k) U = X Z j :U X H1 (E) H(Gm (E)) = H q (Gm (E); F ) = H q (Gm,E ; F ) n>1 k E 0<q<n s Z / F1 (A1 h ) Z / F1 (Z h ) s O Zs h s / H 1 (A1 h ; F ) X R q j FU k F F (E) /0 s h h (E) = lim Zs Ts (1) T Z h Xs H = H m (.; F ) s / H 1 (A1 h Z h ; F ) X (X, Z) X h Zs s h H1 (Zs ). Z sX ). m a a y I y y s Y @4qsyt}tu Qeg qY y w E w m " D W $D 0" D t x A 9 g 3 D D m | m X p w nxequ iHTnX@D @h26v@YtS`q!B@2 BR@wG!FEReepF!x'w x ba u| m vvvz f su y m y s YqqRryxkYsxCnxexCrg qY s u y m z % ~ i A W V $D 9 9 P { e g " D 0 H 3 D keqsQH2Y h ei'Xr#X"i6BdR e qx@Yf@D @D @#iG!FD A 9 5 " $ 9 " 5 3 0 ( sI m o B26k!@QF6 21vtYye en0eg GeQexexxqqQgYhkFbYF ul g y u e y ~ u o u ~ s y m p o g l m l w v v m v v v z f s u y m y s Y s u y YiqTrykeqsQHBeQHrg q!ukeqsQH m % ~ i A W V $D 9 9 P { e g " D 0 H 3 D D D A 9 5" h2 42eiY'#rX"R61dRQb eqY2Iq!@D 2r@XqUFUrB@F64e$ t 9 " 5 3 0 ( sI m s u w y g w i w u y ~ e w P m p o g l m l w v v @xU}6 21st!nQxYx2xu eenYexYbdeqxGnYGxqeHTbxF ul m vvvz f su y m y s YqqRryxkYsxCnxexCrg qY s u y m % v ~ i A W V $D 9 9 P { e g " D 0 H 3 D keqsQHdY hqi'#r)X"RIdRrf e qq4s@Y2ID @D @#CG!FD A 9 5 " $ 9 " 5 3 0 ( sI m g w w P o s y u B m p o g l m l o s sy W m & w v v l B2Hqk!@xU64 21EtneqxnqTYFeeYk0g xx0xqeHxb9qfqhfpx uW m aa {a% z Y @4 qn n Y X" 9 F 9 c m y u y g y ~ u u ~ e w ~ sy W m p o g l m l o s sy W m & w a l vixnexkxfxqekxxqFYRqqVqkhHibY'qrfpx q uW m s y y o o y wu m o a ~ y i W mp 3 3 W 5 3 3 "sm 3 P m W m m f w yxu qe1QqiY@24 eeqsQqhg B nri @D Q4G'H Qxehi qlxeh uW m v a ~ ~ y i Y24 qQeFslQeqsQqhW m D j " W h a e g 9 3 p 9 g W 3 g W 9 9 P m o s B m w v 4UbY"k!9tqCQ#yib@ YY4s 4q@@D 2@gRdqih!qw xe uB m a a a v v z { % v a 3 s W Y@@4 n 1 e { YXX"f4X8e m p o g l m l o s w m wa a l qkhHYfqVQTA x@q1w u V A W d W 9c m e w wu i u e w 4RXTth@)neeqxdYq m v a y o u m wu w s u y m e g 0 H 3 D D D 3 h m y w m m f w v y Y4eqqxxiyxx'pxkYsxCBY qg2ID @r2#G @t0XqsTrW qxw u m z a ~ y i W m v v a z e g p V Y vx24 eqsQqhrY Y2z Y@ Gqq@q4#RA 202ID @D @#G@#B42IXxFU qCeqQHqV9`eqqeqxxRVw sdRSx 'w u 3 h H 3 D D ( D D 7 A " Dp 7 m m f y o l o s m & p y o s wu S m yu & w f & m z m yu e wu Ygq!ixxeqxx'w sy u j u o S m a p ~ l ~ y i W m 0" D H " s 3 P m s wu u m w xhkxxiqxRCh@24 gxHkCqQeFdeqsQqhVnb2Iw@Q4H RqQqxdRB n@IRu u m a W X u o m wu w m m z { v Y 4 rTvI qiyxx'TqYHBY Yz qz Yz e g c W " s p 3 " p Dp H A 9 3 h m s s o y E m f o s p y o s wu S m & w E q'Yt42H @vnr@'ibp 8r qqqeR9qyq eqYYQRffex G uS m a a W X u o m wu Y 4 k 4 rTvI qqx'w m mIII e g W " s p 3 " Dp H A 9 3 h m s s o y E m f o s p y o s wu S m & w E qqC'xQqqt4G'H 2p @'xxi4qp 8# qqYqwqeqqeqxQRfex G uS m a W X u o m wu 4 rTvI qqx'w m mII e g W " s p 3 " p Dp H A 9 3 h m s s o y E m f o s p y o s wu S m & w E qqCXQRYhqt4G'H 29nr2I'qRibp iB qYqeRqqgqYqeqQRYfek G uS m a a s y y o o s ~ y i W m 0" D 9 " 9c m s A Y@4 u qeYexgeqsQqhi2Iq!"@DnIktidyxu qiYm | xbaqi uA w m v v v z m y s Y s u y m A W V $D 3 9 9 P { e g " D YqneQHXg q!`keqsQHX@XrR4UbY"TIS 2gRQq e YY4s@Y 0 H 3 D F D A 9 " $ 9 " 7 5 3 0 ( m p o g l m l o s sy W m & o s o y A m m X w v v l W 2ID @D @#8G!ECB@!k!@Q864 21)qkhHxfbqyxqhf'eqYqexinw !2xH uA m vvz {% YY { n Y " m u s y w o i ~ u s m ~ E m w v ~ $ #! YbY"nxeeqq9qGegqx!eRqxGxQexqxiyegA xe uE m e e u s y wu o s yu e m ~ E m A o s y p y y P m P m yQs!xxkqqj qxqeg eQHxy i1qqsxqRQ fE uP w E 2 " CF&V r CgtChn xCYn rr2nC iC ghfr rsnCxrrsnCCsv yYCsC# qQ H n1 (A1 s ; F1 ) H n (A1 s ; F ) H n (A1 s Zs ; F ). h Zh Xh Xh A1 CCYrsisrd sgrf D gCVV H sC yhns "x9hFYn 9YChyhn1nh gfts} qrrxb h x E rx 9 Vg9dG 7 enFrxFrHyhn9 A Q hCrfb D h}s1rsngnnxtggH! qu grsngnnxtggH hCCs s D g V rF srsngnnxtggH'9tD b ysrF &Cr rsChtbg sgC&hns gVtn2snCrb i1s 7 r A ns E rtrCrt'rY' dChyhn0gnYHnq CrA 7 enFrx2hC S A nrh vr tshx'rn Cg2nysn i 7 r A7 rdr1nr1b D xgnV"xhH srxrqxgs1 1C G s nr#xsnC $$ $ $$ F sZ H n (X; R0 j FX ) H n (X Z; FXZ ) (A1 , A1 Z ) V V b V (X, Z) j V s s V 1 n=1

Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents: