Calculs explicites dans les groupes de Grotendieck et de Chow des variétés homogènes projectives

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(1)Calculs explicites dans les groupes de Grotendieck et de Chow des variétés homogènes projectives Franck Doray. To cite this version: Franck Doray. Calculs explicites dans les groupes de Grotendieck et de Chow des variétés homogènes projectives. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2006. Français. �tel00120949�. HAL Id: tel-00120949 https://tel.archives-ouvertes.fr/tel-00120949 Submitted on 19 Dec 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) ` ´ THESE DE DOCTORAT DE MATHEMATIQUES ´ JOSEPH FOURIER (GRENOBLE I) DE L’UNIVERSITE pr´ epar´ ee a ` l’Institut Fourier Laboratoire de math´ ematiques UMR 5582 CNRS - UJF. Calculs explicites dans les groupes de Grothendieck et de Chow des vari´ et´ es homog` enes projectives. Franck Doray. Soutenue a ` Grenoble le 9 octobre 2006 devant le jury : Emmanuel Peyre (Universit´ e de Grenoble I), Directeur Michel Brion (Universit´ e de Grenoble I) Bruno Kahn (Institut math´ ematique de Jussieu) Nikita Karpenko (Institut math´ ematique de Jussieu) Laurent Manivel (Universit´ e de Grenoble I). Au vu des rapports de Bruno Kahn et Nikita Karpenko.

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(73) & VX8& 07& P DU!" <$ #87# D < : P !"$"#H# :# Q!#%"# k f : Y → X 7 #

(74) ;#% #% !O%"#U#%1 $Z7$%:; P *!E !#%"! %1#' 7 : : # p8 # $#'$ !" < U#% :; !1 !" 7#87U#%$! "#$:;@!# & !"@ d %#'":$K#%1$#%" P G_! 7! d U!" <$ UW # $:; 7# D < : 7# 7U#%2D f : Y → X ! "# :; # P 5_!*# ! =D ;:" ' $1O !"# 7k# 7# 7 #'! P 9# :$O'"#@<d#%"$ p#;& f ∗([V ]) #% # J U#@7V#% 'J X >!:1#%n"" '7 iD 4$#' 7 !D < D : a #%" −1 & : !# H'_# '$#'p%& $\ # %2# %#%1 7T # : %J :1# $"" %7 ' 4# f (V )−1 #% :K T _W " V >W V %V " 7# 7 :; Z ⊂ f (V ) Z P 0E!#' $#%$ 'J !:1#'"" %7 4$#'@$"" %7 ' 4#% 7 # L #' : '# f −1 J (VJ )!2# D : %#'pS7# −1 7 # :*" '"#7# Z < :8" '7 !1L >!"E7 #'! P !f$(V 7 '")!> L :#%:R 6 %: 7# 6 %: $ %$# O Y, Z : :" T : 8 $ ! ! % # " N a P . 4;& Y :": ": <# $ 7# 6P ^* #'"d+i −1 Z m(Z, f (V )) 3!PN 0 7 D U27 $# &!H >!# : $!"! : 7 M3 $! ;:1 # OY, Z. !#"%$'&($'&>P. p#R#TU!" <$ # 7# &H> !# f. Of −1 (V ),Z. . #'"! #% :;!7 "%7#'p#%'PJ^* ": >#$# #%* >!9 :;%P E ? !"!#H!D ;:"' $1O!"#. X. f ∗ ([V ]) =. V. X. m(Z, f −1 (V )) [Z] ∈ Zi+d (Y ). Z. : J # #%] "$# ",!#'O#%O'J !:1#',""'7 4$#% 7# Z P # '% #'"U#'87# 7 M3 " #U: %:;! ∗ f >−1 f : Zi (X) → Zi+d (Y ) !(V "@) P i. ^* :,: \ # "J "' 7 # a ! !" : . ! "T#%.U!" <$ #'8 :;%P p "!$ D < : H# & !1 7#'65 _G X, Y, Z k f : Y → X g : Z → Y U!" <$U#% :; :$" ∗ ∗ ∗P. . (f ◦ g) = g ◦ f .

(75) . C #W!"! ># 7#%9M

(76) 6$#' :J4"2#% " Do < : #' "J E!":;7 X :"H#K"# :;$! >!"R- "#Kp%"#%:1 P*9L #% >!"p! & !E$#\k2#%1#A 74L p @:$#%%#P J U #%?%! :"*": #$#'"+ : 73 ! 7#9 :.

(77)

(78)

(79)

(80) ?02" f2" V

(81) P a ! ! ord. !#"%$'&($fP. 2G !$#'p #(:;" $1O!"#I" #' #9! D X k Y ;:"' $""%7'4$#S7#%27U#%$ 3!PQC LN:;# :; 6%: #%S7#7U#%_D ! 3 #' ∗ 7 '#!$#%" 7#% a !O 'X,Y! ":;$!##% φ ∈ k(X). k(X).

(82)

(83)

(84)

(85) . . " . X. &!Y >!# ordV (φ) = `(OX, Y /(φ)). `. #', :R!!#'" 74L Y 27#P.

(86) "$'& $ P !#%" ! !#% -U#% :;!%P;G__! >!#@7# #' :;# :; 7 # ;:$ :$ 7$%"O #U#' n! # :" 82# AJ'% ":;O #'X," Y 7# L $7%:/ :&5_ : &2:$!"@ #M5_$# K#'p$#'" #'# π # $2 # P 4&#G 5# $#:HP FPN0 P n n #'. u. φ = uπ. ordV (φ) = n. !#"%$'&($P. G2!$#'pU7 D < :& # ! D X k Y ⊂ X ;:"' $1O!"#H7# 7# 7 #'! "* 7 #' P i+ 1 U#+':1(i$: J # φ ∈ k(Y )∗ ^*K73 , L U#%1X %J div(φ) ∈ Zi (X) X div(φ) = ordV (φ)[V ] V ⊂X. 9$:I U# #' "$# " !#%L#% !D ;:"' ' $1O!"#% 7# 7U#%$! V 7# P. i. Y. "$'& $2P.

(87) . CZ:9 U# : #%'& :;". #' p2:6a# !". 4 &: >#%7'# =IP CP F P ord (φ) V

(88) "$'& $pP !#%" !!Y$#' :;! 7#D : 7M3 7#'p#&f #%I :"#8U#%1 7# & ,: ψ k(Y ). JR4"#+3 7#. V. $ " D. ∗. div(φψ) = div(φ) + div(ψ). #' div(φ−1 ) = − div(φ).. ^* #%":K:!"( :". . div(φ) Y ∗P φ ∈ k(Y ). $# ! D !"! ># 7# #'!#'7"9 :"$#' Z7i# (X) #%K# R !i (X) D=;:"' $1O!"#J7# 7U#%$! #' X. !#"%$'&($fP. !"! ># 7#. <.7#. X. i+1. 2G !$#'p D < : '# #'1$#'"9 a P C # X i $#!"! >#82#%1 7#87U#%$! k& '# , i CHi (X). CHi (X) = Zi (X)/Ri (X)..

(89) . TU7'

(90) & 3 VX & 2"-&.

(91) . &2". . . VXTU7'

(92) & V>

(93) . .

(94) . K. & P . . G_! . . . U!" <$U# 7# D < :%P6*I! : !!N_\2#& #%! f"J: "#Y&2!→ 7$X >!# k f 74L #, :!# 7"# # " $#' !"! >#%U7#OM

(95) ;$#'%PNG a : & :# :; p$#%1%&% J #$#Uf!∗1"##*<'!"O #8;:1 &0 <'!"O f∗# 3!P C "%$'&($

(96) _PB. , 6 TU . 0 ;$. 0 &2" 0 V. &;02". fV. α ∈ Ri (Y ). 2BT. & " :& 70 %" ,. f

(97) 2: Y2 → XV

(98) V V. i.

(99) $2V10 T6& >

(100) > ;&.3

(101) & R k @W$

(102) ;0. f∗ (α) ∈ Ri (X). U!" <$ #X "J "#,7# D !#"%$'&($)" P G_! < :%PIC L :!#H7$"#'# 7f#%8:

(103) ;Y$#%→ :X J4"p#'+ :#U:Fp$#%18k# 7M3 #H: $ :;$! f"∗ ! i. f∗ : CHi (Y ) → CHi (X).. ? "X L :!#,$Q!#'"#;& !K: !H# "' :; ;:1. 3!PNA . . , . . "%$'&($)""P. 0 V

(104) & W " V%8& d 3

(105) & k :R 06;$ W$

(106) 0. f (α) ∈ R (Y ) ∗. . fV. TUf0 : &2Y" → 0V X. &;02". _,T. α ∈ Ri (X). . 4&!0. <%!"OU#.

(107) $2V10 T6& >W "=3

(108) & 3VXT6& R

(109) 2 _ & " :V &

(110) 70 %V " V , i. i+d. !#"%$'&($)"!&fP. G_ #' [U!" <$ #X $:; 7# 7 D f : Y → X U#%$!,"#$:;@!# 7# D < :%&/ L :!# "' "6p# 7#%(

(111) ;#%+ :;# : p$#%1 # #'"Ud#' 7# k73$" # : %:;! :# "% ";p# "9$#% !"! >#% 7# < 7#87 #'! " ! #'1$#'"$ a i. i. f ∗ : CHi (X) → CHi+d (Y ).. " f2" 2;& 3 & !P(.

(112) . D ;:"' $$#;P. *I! !#%"! %,2#. X. k CZ: '!"'! 7#%, "27 74L $1#'"#'$!,7 :,$#'!"! #'I7#. #' #. <. #% 7 $ :;$ # :;$L >#%" # S7#( $" 7L #9" "# CH∗ (X) = ⊕i CHi (X) 74L :#%:Y!":7 4& /_PNF P .

(113) . . K. . *INa #'"!SW4"#aZ": >#f7#% "% :$#'%!"'$!S#% D <'!"#P K ? " 7#$ n: $ #' $6a !" :;!'& > !"":+'!#'"L L :"$#a !7 :;#%" 7# <*$ #' P.

(114)

(115)

(116)

(117) . . . . .

(118)

(119) . . !#"%$ $)"P G2 #' #E%:; !!"# :4/$#%#&9 73 D$# !"! ># 7# ` "<#%7# ? :A ;%\ ' J # #%R$# !"! ># :4/$#% #'!#'7"8 :;"$$#%

(120) ;R4 $ #% A #%I,JK 4 0#'(A) 7# & : # [A] A A :65 "#$:;!. . . [A0 ] + [A00 ] = [A]. 7O',p4L #c5_$#8#8 #8#M5: #8%"# 0. /. A0. /. /. A. A00. /. 0.. G H :"%$$#'"%& #87 >#%7Op#87#8 :R :#874L $ !" <U#*7# P ^ W >#%@:6'$#%[A] * "S7#%;5X!"! >#% 7# ` "<#%7# ? H < :&6!"!A #' p #'"!1I$J " <#' 7 : # : #%$# & G_# $ 3h-2P3 P ? "J < : ' <% "$#% 2 #' !Xp # &( # !"!> # #%]# 0 X K (X) 0 ` !"!> #,7# "<#%7# ? :;% H $ :Z% :; !!"$# 7#% a :$% # :;65 % !<' "#'p. ". P. C X #+!"! >#. #%$#+"! ># 7# ` "<#%7$# ? :;%8R$:. : !"#. 7#'+34"'+!#'K 0"(X) $#$'& L #%D hD7"#H7#%+a :%#%:65]$; :#U#%14"#%I7# ":. 3 " P X C #+ "_7@#%!"# :;"@X34"+!# !"$# ':1 #M5:'%& !X >#%U#'"# # " "#H74L :# :]'J :aE" pE#%*:;> Z74L # K (X) 0 $Q J$ $! $7# :;"R : 7 : ; P I#W - #;& 0 :J# " "# 7# DU _7$ #;PQ* ^ :8 :;"#$ # #'1n# O <#IK$0 (X) 7#, :" L $$$! 7#' K P :0 (X) !"#% 0 K0 (X) #%→ 0 (X) G_ IK H !" <$ #*7# D < :%&f! : Y " <U U # f : Y →X. k. f ∗ : K0 (X) → K0 (Y ) [E] 7→ [f ∗ (E)].. :9 & '# IXa !'#%"'!1": ;:" :1 P 9#% K0 # kD ;:"' X #% $#&%:$!" $# !" <U# :;"# K (X) → #%8 $ !" #%K# H#% ;#%"!K $#U#%1 K0 (X). . P. 7

(121) . K. . CZ: D < '!"#H 2'"$#'"#H4L 1#%" 2$#%1 :*7 :;+%# ": ;: :($: # ># "K :$# 7# =S" _D ` #%"#%_D-< #% Ma PI3!P FP F "#$#A$#' "! >#% 7# D K <%!"$# :65H!" #'*7# < # a !" #T$#% #%1"#R#%97#%65O :"$#'I7# %# M# 52#;P #%8J 4 # a !187!9 :"$# 7#'8!$*#$ >!"$#. :% 7#.

(122) . . . K.

(123) . . . !"! >#% 7# <8PJCZ:'!" ! 7#$: D <'!"# 2'"$#'"#@:('!U#%'# H4$#% :" <*$#% 7 : PE? :" :1 7KL #W :;!"$#U:4/$#'# &Z! M %" :W%:; !!"# 7!18$#'*J4 # 8!18#% J4 # 7# # 7!1 M!" <$ # $#'N U!" <$ #'n#%1"#*7QM #%;5 4 #' # !1$#%$:#%@7L JU A B 7#87$: !": U#% Ao. . p. C. i. /. B. #% J4 #' 7# & #%@W U!" <$U##' WU!JU!" <$ # M p i 7 :C P M ^*]%" :!"( L #% :%#R :@3f:1 >& < : "# 3 & # ! 7M3 #% !"! >#% 7# Do<%!"$#* /%"#%B(QM) "#T'J #* K. Ki M = πi+1 (B(QM), 0). #%+# DoOU#T"! >#Ta 7 :U#%1 :E7#R L #% :%# J >JJ!p# #' πn7 '!#8n$# >!$1 7# %!""#' >!7 :1( L J4 #' :f7B(QM) # P 0 B(QM) M *I "#%Q

(124) ! <*#%W >!"$#' 7':@7#U :R'!" !4P C #I" % :; ;:1N p", # p # '#'#97M3 $ #%7X4$#'\ :87M3 $! < : 4# #*7#', "! >#% 7# ` "<#%7$# ? P fV &;02" 2 & , "%$ $'&

(125) h( W$

(126) 0 W1& '

(127) & ; & 2 0 @V106

(128) AT

(129) $& " π (B(QM), 0) 3

(130) &8W " :'2

(131) V:& 1 3 n V &

(132)

(133) A. . T

(134) ?02" " V

(135) P(. f&fG_#'!J0_&0. &2" V%"+&*" :'2

(136) V:& ) W V:& & '

(137) & 3

(138) & @

(139) "$& 3 V:& . <'!"O # 3 P. G_ !! p# ! < : _# <'"#% :";P ? !" #%1$#%" >! a & ! #'"X:Z :" $# DoOU#H"! ># 7# D <'!"#J:;% n K (X) n K n S$ :Z% :; !!"#,7#%\34"% !# !"$#$ " &n# 0 &$# DoOU#O!"! ># n (X) 7# Do<%!"$# :6'R : :;!!"$#'I7#'X a :%#%:K 65Y '!<%"#%1n " & K X < : "# A P C #%] "J " % ; :1#% 7!% #'D $ <:D> !" #' K0 (X) K00 (X) %L ' #%7#%1 :;5A!" #' 7# Do<% !"$# 2' "#%"#P K' :1 \a ' #%"@#c C L !U7#%!% :; !!"$#% 5 :' %&_! :HWU !" <$ # >!"@ #%1#%"> ! a n. Kn (X) → Kn0 (X).. #'# : $%:;! #' $ !" <U#9 X #%, < : " $#%"%P ? " ! #%1$#%" >! a & 0 #% D= 27#PQG ! "#& n K (X) K (X) 0 n #%@ a ! #%"%p": ;:"$:1 #' 0 #%%!1": ;:"$:1S"#% " <UK#%n Kn $:; && < : "# A_&f0_P 3 P.

(140)

(141)

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(143) . G 3 4& 0 #%n.a ! #'"L% ;:" :1L >!" $ #%> " <U#%I3 # . >!K"Un$#'TU!" <$ #' #% R " # @a f& < f: : Y"# → A_&. %7 ? . Y →X. X 0_PNA P. Y. G_! #,;:"' $#I" 7#I7U#%$! >!" %#%" : #'p$#'" & X n #','!7%"! L :# :K7# Dok<%"$# Pn K K (X) 0 M3 $!Y $: 3 ":$[ :"\ :_%27 #'! 7B >!"K7#D LN:;# :; ;P ^*. >! # >!" #'1$#'" i #'$#!"! ># #%!#'7"( :;" K $#' 0 (X) # $ p8 # 9 : '_7 i #'! 7KF (K 0 (X)) "7# ! P [F ] F > i ^* : 7! K0 (X) = F 0 K0 (X) ⊃ F 1 K0 (X) ⊃ . . . ⊃ F n K0 (X) ⊃ F n+1 K0 (X) = 0.. ^*K#'":9 :". LN:;# :;H!":78:;;%P. Gr#'F C LN:;# :; K 0(X) : 74L # : "#.3":;!J:;"#$#&/ : D K 0L (X) γ 3 ":$Wp#T #'": :" i & < : "# P Fγ K0 (X).

(144) . . *

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(147) VX. 3

(148) &;0. . "%0 3

(149) &.

(150) 2 2 & " V:& p I

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(152) "+&

(153) ?0 X W & ?07& T)%W1& 3

(154) &

(155) *3VXT6& ?0 V

(156) p 13 ?0 X W & !V102"+& 2 & 0 _(p)V%"+& 0 & 2" W1& Y. "%$ $ B P . X. X. E1p,q (X) =. a. 0 K−p−q (κ(x)) =⇒ K−p−q (X).

(157) &;02"*W1&,

(158) 0 ;0 V3 & WS

(159) VX"

(160) &2""-&02V%"-&.&;02"

(161) " 8 V "+&

(162) 2κ(x) SW1&;0 T

(163) $2V10 T6&;0 >W "%0 nW " " V

(164) , x

(165) 2 :V &=02 0 &;02"W nW " " V

(166) Kn (X) "

(167)

(168) W$

(169) 4'V & x∈X(p). . C #I#%"U#. . ! -_P /. . . 7#%#'#*#9#c5:'#9#% :%$:4# f&G_#'!KA_&_? "J >;D !"J :"#*-_P3 5 . #' E2 f&>G2#'$!,A_&. ,

(170) (>! "$($ 7

(171) hP 4

(172) V%" X

(173) 2 "

(174) " W1& ,'V106;& 0_ 0;06

(175) 6 V n > 0 K0 X n. 2. . & k R+8 V 2" W V10;07&

(176) "

(177) ?0 ', V106;& U 7→ K (U ) . > . 0 n.

(178) . ! . W & !V102"-&HW$

(179) 0 2 V106

(180) T

(181) $2V10 T6&H

(182) V . &. .

(183) . . Y. . 0 E2p,q ' H p (X, Kp+q ).. #'#U # ># ":$# >#%"U#' #'p"# :"#%'&Z74LN: !$"8# #c5 "#%! 7#' !"! >#% 7# < &>G_# !,A_&0 <%"OU# -_P 37 , . 2. %" &. "%$ $PB. . 4

(184) V V . V106

(185) T

(186) $2V10 T6&H

(187) . 2. X Y. & k +R 8 V 2" *VX 43 2 " V)%W1& &2"nW V10;07& % W& !V102"+&. H p (X, Kp0 ) ' CHp (X).. .

(188) .

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(190) . . 7 + . #'#,# ! #% #]4"OM!#] "%#%1 :$ 7#%\ a '& %L :

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(192) *

(193) CZ. : %:; !!"#87#%;:" %,$#%#'+ " #'@!#'*"IO'!" L #'+ : $ # %:; !!"$#@:;77#PQ^*9 >#%$: !!#'" 7 : #$ : !"#@:k77@!#;& : : !"# 7#%8 '!""#% >!7: %#' #' " : ":7$:1N1 L #'#R4$#H7#%8U!"D < U#% #'p"# 7#%; 5A:;" '%P . . *I;!. Xi. "%$B$)"P. G2!$#'p. #. &17#'65. $#'%J :1#'%X !#M5_Y#%7#. D ;:"' '% " #'#%n#'$#%%P. kP/^*K# X. C(X, Y ) = ⊕i CHdim Xi (Xi × Y ).. . #8!1,$#'%!""#% 7 :'#% b 3 $#% @. 7# G_!#%1. &. #. "!. X. #%". Y. P. D=;:"'%! " # @!#% # $$#%@". P*!. X1 X2 & X3 :+ " k # ! 7# k PpG2!$#'p " 16i<j6 3 π X X ij 1 × X2 × X3 i × Xj #' P ^ D >#%_8:$"973 "*DU#%1 7# α ∈ C(X1 , %X β ∈ C(X 2 ) U# J :" 2 ×:X !# 3 )$Q#%"#N :;#& ∗ #' ∗ !1E7#%65 C(X 3 )7# U1#%,1X8 PZ n H:$" 7p# 12%(α) !$7%p"23 #'"T(β) #%"8 "_7 74L $1#'"#'$! CH(X #' 7#U1 × # X2,X #%3" ) $#U!" <$ # #', "J "# p13. >!". . . . γ = p13 ∗ (p∗12 (α).p∗23 (β)).. ^* : : %!" YU#%1. γ. 7#. C(X1 × X3 ). P.

(194)

(195)

(196)

(197) . !#"%$B$'&> P ^* 7M3 $ :R%:; !!"#97#'%!""#% 7 :'#% " k %# %J # Corr UC #' J4 k# 7# Corrk !19#% [X] X #' #\;:"' " # @!# # $# " P ' ?E!" k[X] # [Y ] 7#'65 J4 #' 7# Corrk &p!U73 HomCorr ([X], [Y ]) = k P. C(X, Y ). ^* : Da !'#%" &47#9$:\ : D ProjSm → Corr k k !!"$# 7#' D=;:"'%8 " # #% #'T$$#' " #%"T :X : !"#U7#%U%!" D "#' >!7 :;k%#'U" P & S :6' $# # k k X [X] f ∈ Hom (X, Y ) ProjSm k :;%$##!": <#87# &; &2 #%YU#%17# P

(198) . "$ $ P . f. C(X, Y ). Γf. C :X :;!!"$# Z %# :;"#$# #'p8# :;!!"$# :;77# #'8#';D Corr k ` "#$#9! # P. . . [X] ⊕ [Y ] = [X. . [X] ⊗ [Y ] = [X ×k Y ]. Y]. + . .

(199) . 9 +: >#$ $p#T L Y '!" :9%:; !!"# 7#%,U@a @7# < .# # a " #'

(200) V > #%

(201) !:65 7#%N " #'#%" b!$"

(202) Z&G_# $ - PZk+: > # !p 4L SCorr " #k #'" 7 :;8# :;!!"$# #' Z #'1 p C 7# > !" ,J4 #' 7# &#7p # 2 P HomC (X, X) X C p =p !#"%$B$ZP (# :;!"$# :77 # #%Z7#% #'7;D:4/$ #%# 2 " @J4 #' &#' ! " #' #%" C & O'7# X ∈ C p ∈ Hom (X, X) p C 1

(203) : 7 : #'7$ #U !" <$U #+% :!p # ker p C. . 3 ' ". #%,JU!" <$ #;P.

(204) . ker p ⊕ ker(IdX −p) → X. ,

(205) (>! "$ $P 4

(206) V%" C 2 & " :'2

(207) V:&* 3 3V%" V%8& W & !V102"+& 2 & " R '2

(208) V:& 3 3V%" V%8& 07& 32

(209) R ) W V:& & C 2& " _ ,

(210) 2"+&_ 3 3V%" V , 3 W1& ps. α : C → Cps. "+& W 0 & 0 V F : C → D &;02" _ ,

(211) 2"+&2 3 3V%" V , 8& ;0 _ &H" :'2

(212) V:& 07&?32

(213) R ) W V:& & VXW & !V102"+& 2 2 V & ,

(214) 2"+&_ "+& W & F ◦ α F : 07C& ,'2→ &2" F 06

(215) V:& D" 2V%8W1& "0

(216) " ;& T6& " 3V%" C → 2"

(217) D V107& Y D ps. C }. Cps. }. }. /D }>. .. ps. ps.

(218) . ! . " :'2

(219) :V & &;02"8 & W ;& W & 8& W$

(220)

(221) )%V:& & C3

(222) ps& W K" :'2

(223) V:& .

(224) . . & 07& 32

(225) R ) W V:& &

(226) !& 8& W$

(227) &. . C. P C #% J4 # 7#. T

(228) ?02" " V

(229) ( P ^*S;:O%!"$"# $:H%:; !!"# C ps p #%R :$"#% #' #' E " #'#'Cps "%P (X, '# p) X 7∈ G_!#%1 7! #'6C5K#$p#'$∈ :Hom(X, "#%'P#^*,X) 73 (X, p). (Y, q). HomCps ((X, p), (Y, q)) = q HomC (X, Y )p ⊂ HomC (X, Y ).. ^* %"3 #p4L 'LN:4#% 74L #, :;!!"$#;&J #'7;D:4/$#%#PQC #$a ! #%" #% 73 % U# Z #% J4 # H7# & # α X P C α(X) = (X, Id) :$ " f ∈ HomC (X, Y ). α(f ) = f. CZ:Z :;!!"$# 7#% ;a 7# < #

(230) #'a " #%R L #%Q!#$ J >#\ #%7;D :42$$#'#U7#\$:H%:; !!"# 7#%.'!"kD. !#"%$B$_P. %# eff "#' >!Chow 7 :;%#'kI". &. k Corrk. P. *!#%" #8a '#%"(%J h Q P Z C : ; : ! " $ # < 74L # ProjSm " "k#→ 7# eff eff '"#@7# Corr → Chow Chow Corr k k k k : !"#@#%!"$# $#;JP G 3 ! #%" P 9L #%# #%"# >!"$$# "27 #%!"#47#U'#'#U :;!!"$1#;P = h(Spec(k)) . . "%$B$2P. C #RU@a 7# < 7#9$: 7"!#. " # # " # 7% >!# 7:+$: : !"# 7#%U@a @7# <.#

(231) # @a b!$" Z &>kG_#'$! &#C # U# . ,

(232) (>! "$ $ P SW 0 > V10 T6& "

(233) Y. oV.

(234) ) &2". h(P1 ). 07& 3.

(235) T .

(236) 707& 13 ?0. Choweff k. h(P1 ) = ([P1 ], p1 ) ⊕ ([P1 ], p2 ),.

(237) . &;02" W1& >

(238) & 2"+&_

(239) &;0

(240) 13 " R p W1& 3 ;0 ' & " &2" 1& & 2"+&. &;0. 3. 1 p1 (P1 × V _

(241) VX W >

(242) 1 _ [{pt} × _PV ]

(243) ∈ CH 1

(244) 1 p2 Id −p1 PR )1 W pt 1oVXV [P1 × {pt}] ∈ CH1 (P1 × P1 ). ([P1 ], p1 ) ' h(Spec(k)). . 1&. T

(245) ?02" " V

(246) P( 9 7#9!!"p# PI* 1 Id −p = P × {pt} 1 :" :*7$: !!:$#@7: 1 &QL #% :;/#$!": <# 7# L : $%:;! ∆P 1 P ×k P1 P > * ^ : Y 2 # #%T!: 1 1 1 1] Id : P1 : !→ P [P × {pt}] + [{pt} × P ] − [∆ P # . . div(φ). φ=. x1 y 1 x1 y 2 − y 1 x2.

(247)

(248)

(249)

(250) . M. . #. !1U$#'8%2!"7%#' <J !O%#'87#%8a :'#%" 7#. [x1 : xP 2 ] [y1 : y2 ] 1 P 1× P CZk# D

(251) ;$# a !" HU!" <$ #*7: 0 {pt} × {pt} Choweff k. '. #',#. 1. D M

(252) ;$#. . h(Spec(k)) → h(P1 ). P1 × {pt}. a !" Y !" <U#. h(P1 ) → h(Spec(k)). :9 &!,: 7#%,U!" <$U#%. . (Spec(k), Id) → (P1 , p1 ). #'. (P1 , p1 ) → (Spec(k), Id).. #%I7#%;5OU !" <$ #' !19$Q!#'"#' L 7#T L :"#8#',a !"$#%1IJ;D U!" <$U# ([P1 ], p1 ) ' ([Spec(k)], Id). !#"%$B$

(253) P

(254) . C #8U@a. "$ $)" P. ([P1 ], p2 ). #',$#UaZ7#80 :;#;& . CZ:9 "J $! 3P CP / #8"'". . L. P. h(P1 ) = 1 ⊕ L. . "%$B$)"%"_P. ? " ,#'1$#'". ,

(255) (>! "$ $)" &>P W K" :'2

(256) V:& Y Choweff. m. >! a &2! #%":. L⊗m = L ..⊗L. | ⊗ .{z a ! } fV. m. m>0. n

(257) ,. W. 3.

(258) T .

(259) 70 V%" V

(260) 60_V%8"+& 13 ?0. k. . T

(261) ?02" " V

(262) P(.

(263) . h(Pm ) = 1 ⊕ L ⊕ . . . ⊕ L⊗m .. E&fG_#'! & !"H$# / P. G_ #' # D=;:"'W$#X " #'@!#K#' E34"X#'!"#@" 7# ":X & !" k %#'" :;$ #%1$#%" & :!E" : #Wa !"H$# !X" : %$#%" $n#N ;a_7# p%'":n$#> :,1 " !$!U3!P CP360 E&G_#'! P(E) A_&& "J :"# .

(264) .

(265) . .

(266) ?07& 8

(267) ?0 W1&;0*

(268) "o" V

(269) ?0 VXR eff Y. , "%$B$)" fP T2W1& 0_V%8"+& 13 ?0.

(270). . 3

(271) &;0;0 0 . @ W ,

(272) R. Chowk. h(P(E)) =. . + . n X. h(X) ⊗ L⊗i .. i=0. . ? " J4#%"% :+ : !"#@7#%U@a 7# < B" &1 VX !$#$U@a 7#0 :# , :"$" 7#N : :;!"$#n7#' U @a 7 # < k # #'a 4&;G_# $ L 3 P . . . !#"%$B$)" P . & #%9:;$473$#. $#% J4 # 7# #' & ' nm ∈(A,Z n) # . Chowk. Chowk (B, m). . . CZ:_ :;!"$# 7#'X a U7#. !1 $#% :"#'. (A, m). . A. <. ". #% HJ4 #' 7#. !17#%;5K#$#'$ :$"#%'&!Y !#. HomChowk ((A, n), (B, m)) =. 8& 07& !).

(273). . k. &S '#. Choweff k. HomChoweff (A⊗L⊗n+i , B⊗L⊗m+i ). k. lim −→. i>max(−m,−n). A :9U##%#8$ #97 @!#P .

(274) .

(275)

(276) ' 9

(277)

(278) . #'#n# '$!U "'#'p#N#% :1:!# 7 #'I!"! #' :J4"2#% #'E7#' ; :" % <JUJ!O'#%$ " #'@!#'%P/C #%, "% : $#'@"Mab%"#%'#%!1 & &) &

(279) ) nPP P

(280) '

(281) ) &2"0 P.

(282)

(283)

(284)

(285)

(286) n V%" V

(287) ?0 3

(288) &;0 4 :#%'P . . U#% ' !W :"\#% 73 ! 7#. !#"%$%$)"P 4& < : "# &2? :;":!": <#83!P 3 (I kD=!" #:J D 4"p#8#%I#:;" " H$# 7# YU #%1 e ∈ G(k)k& ' Y !" <U # ; & $ HU !" <$ # iµ: :GG→×GG → Gx 7→ x −1(x, y) → xy #$ Up #\ #%U !" <U #% & # k p #'R$:X%J %# 7SU!" <$ # p " ": : i! #µ. @"#'7#%1$#%7$:!": U#%. G → Spec(k). e : Spec(k) → G.

(289)

(290)