1227040

Un isomorphisme motivique entre deux variétés
homogènes projectives sous l’action d’un groupe de type
G_2
Jean-Paul Bonnet
To cite this version:
Jean-Paul Bonnet. Un isomorphisme motivique entre deux variétés homogènes projectives sous l’action
d’un groupe de type G_2. Mathématiques [math]. Université des Sciences et Technologie de Lille Lille I, 2003. Français. �tel-00004214�
HAL Id: tel-00004214
https://tel.archives-ouvertes.fr/tel-00004214
Submitted on 19 Jan 2004
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"p$'59(#*A$6(#56+\*,+,O4OC(#OC!3C59+M1#O4(#"Y+gJK1#OC34*,34$6+Œ^`+A$Y1#O4(#" 3oP:B,&!rlBL$'5634DE(#+ 5)f URV +\O4+,*A$6+,(#5˜56+Ar†256DE(#+,56
DE(uv+,P!B,#BA5'!O‘7Zrst,rs+lO4&:56"9DE(ux()#+†[2!5634BA$6B X 2;#rl+L$m(#)+†"9$656(#*L$'(#59+ *,+AOCOC()OŠ!3456+%+A$wDE(#+
+,"9$-()#+ "6&:()" N [2!593CBL$'B¬ac+A56rsB,+·;#+ X 7YO‘ux34E$6+,59"6+,*L$'34&: ;#+·OCš"9$656(#*L$'(#59+ *,+,O4OC(#OC!3C59+ª;)+
Y
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2#3C"9&!$'59&:1e+¬3C#*AOC(#+ª;0!#"-(# 1#OC!1#59&?>9+,*L$'34a jd [email protected]+·1#OŠ2 1#56&?>9+A*A$634aq+,"p$ª+A a`!34$
OCšP:56!"6"9r†!)#3C+A##+·;#+A" ;#56&!34$'+A"†;uv(# +,"910!U*A+ ;#+ ;#3Crs+,)"63C&! 3 +L$-!;#rs+A$ W *,+·$'3G$'59+
()#+l"p$'59(#*A$6(#56+s*,+AOCOC()OŠ!3456+-^c;u°23COCO4+,(#59";0!)"Œ*A+l*,!"ŒO W "6+,"o[2!5634BA$6B,"q;#+ ­ "9&:I$
$6&!$/2OC+,rs+,I$<&:59;#&:#)B,+,"s+A$<ac&:59rl+AI$l()#+ [K56!3C+ ™0OG$'562$'34&:{fL7Z1e&!(#59$'!I$<OŠ *,&:)3CDE(#+ux+A
2;#rl+L$w1#!"m(##+†1e&:()5w!(K$/!I$ UZV +†a`!3G$ŒDE(#+†*,+,OCªac&:#*L$'34&:##+s;0!#"w*,+‹*,!"wO W +,"p$%;#&:)*
2"6"6+ d+LJK*,+A1)$'34&:##+AO !5X!3CO4OC+,()56",7L"63!#&:(#""9&:rlrs+,"$'!I$X2$6$'!*/.#B OŠM;)BA™0#3G$'34&:+LJK1#O43C*A34$'+
;)+†*,+L$6$6+†"9$656(#*L$'(#59+‹mU *!A uv+,"p$%DE(#+†P!5 !*A+ W +AOCO4+l#&!(#"m!O4OC&:)"q1 &:()[!&:W3C5qBA$'!=#O43C5oDE(#+†*A+,59$'!3C)"
*LhR*,OC+A"_;#+ CH (X (α )) "6&:I$œz}’)Ž}“cF| ((€R‘ U
1
2
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2
X (α2 )
X(0,0) = Γ2 (H)
X(1,0)
X(2,0)
X(1,1) = Γ2 (V6 )
X(3,0)
X(2,1)
X(4,0)
X(3,1)
X(2,2) = Γ2 (V5 )
X(5,0)
X(4,1)
X(3,2)
X(5,1)
X(4,2)
X(3,3) = Γ2 (V4 )
X(5,2)
X(4,3)
X(5,3)
X(4,4) = Γ2 (V3 )
X(5,4)
X(5,5)
U ³ ˜!593CBL$'B,"T;)+ ­
;#+
Γ2 (H)
U ³R$'59(#*A$6(#56+q*,+AOCOC()OŠ!3456+œ;#+ X (α )
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& :()– $6+Aac&:34",7K*,+,OC%56+A[R34+,I$ W *,&:rs10!59+,5 20 3CI$'+A56"6+A*A$63C&:#"TW 1 &:(#5S56B,+AOCO4+,rsX+,I$(αT;#)3GFHBA56+,)*,3C+A5MXOC+A" U
BAOCBArl+AE$6"Y+L$S59+,*A&:#"9$634$6(#+,5Y(##+_™0O4$65'2$63C&: U!– (#"9"63—7E#&!(#"S1#59BAacBA56&:#"Y+LJK1#O4&:34$6+,5d(#†1 +,(l1#OC()"
O4+,"m59B,"9(#O4$'2$'"m;)+ ±xµONK* :)³,´M2™0;#+ "6i[!&:3C5w;#Q,"%rl!3CI$6+,0!I$,7Z;#+†*,&:rm=)3C+,;)+†$'+A56rs+,"mOC
™#O4$'562$'34&:‹"9+,5's*,&:)"9$'3G$'(#BA+o+A$œ;#+oDE(#+,O4OC+n;#34rl+A#"634&:-"6+,59&:I$œ*,+,"_$'+A56rs+," U A &:(#5œ*,+AOŠ%#&:()"
2OCOC&!#"M56+L[:+A#3C5_"6()5œO4+,"T"phK"p$'QArl+A"‰;#+o5'2*,3C)+," U
&:(#"56+,1)56+,)&:#"OC+,"-#&2$/2$63C&:#" ;)+›OC"9+,*L$'3C&! ³„;#( */.#!1#34$656+ !5 7T#&!(#"
;)B,"634§ P:#&:)"˜;)&:#*_(#sP:56&:()1e+T"9+,rs3 N "93Crs1#OC+S!;?>9&:34I$d+A$˜;#B,U 1)OC&ih:B UE§ &:(#"d*/.#&!3CU|"634"6A "9&:#"dG(#<$'&:59+
rl?JK3Crl!O T 23C#"93XDE(uv(# "9&:(#" N P:56&!(#1e+Œ;#+ œ® B OC+q*,&:I$6+,0!I$ UV l;#BA*,&:rs1 &:"63G$'3C&!
;)+ X­ ;#+ G ^c[:&!3C5T10!5_+gJ)+Arl1)OC+<±v¸S&:5,7 .1 U m³ : ´ fS+A"9$\!O4&:56"
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;)+ G ! "6"9&K*A3CBq!(-"6&!(#" N +,#"9+,rm=#O4+ Θ #; + Σ(T, b) 703COe[K34+,I$\!OC&!56"
w∈W
Θ
PΘ =
a
BwP
w∈WΘ
& W +A"9$ ¬OC+k"6&:()" N P!56&:(#1 +„;#+ W +A#P:+A#;#56B›10!5-OC+A"ª59B0+gJK3C&:#" s !"9"6&R*,34B,+,"¬!(KJ
BAOCBArl+AE$6" α ;#+nO—uv+,)"6+,rw=#OC+ Σ(T, b)\Θ U#§ &:(#"[email protected][!&:#"_+,+AFe+A$_;#B > W "63CP!0!OCB\;0!#"_O4+*/.0 N
1)34$'59+ DE(#+‰O4+_P:56&:()1e+ W BA$'!34$˜+A#P:+,);#56B\10!5˜OC+,"d59B0+gJ)34&:#" s !"6"9&K*A3CBA+," W (#‹BAOCBArl+AE$
;#+ Σ(T, b) U A !5‰*,&:#"9B,DE(#+AE$,71e&!(#5\$6&:()$nBAOCB,rs+,I$ w ;#+ W 7 3COX+LJK3C"p$'+m;)+,"n56B0+LJK34&:#"
α
$'+AOCO4+,"XDR()+ w = s ·. . .·s U ‰+S1#OC(#"A721#!56rs3KO4+,"Z;#B,*A&:rl1 &:"934$'34&:#"€1 &:"6"93C=#O4+,",7
s ,...,s
34OK+A<+LJK3C"p$'+M(#)+TDR()3#+,"p$]rl34#3Crl!O4+]+,%$'+A56rs+M;#+M1#59&K;)(#34$8;#+M56B 0+LJK34&:#" UI” +A$6$6+T;#B,*A&:rs1e& N
"934$63C&:qrl34#3Crl!O4+b+,"p$]!O4&:56"Z!1#1 +,O4B,+ ‚ žR:F| #š]| @“ Ž}“`F| (¬z}ž ‚ ~Z“ Žis^c[!&:3C5X10!5X+LJK+Arl1#O4+±v¸S&:(7
*/. U 9,³ U <m1 U u³ <!´Šf UK© ‹1e+A()$_!34#"63 W O‘u°23C;#+_;uv(##+‰$6+,O4OC+œ;#B,*A&:rl1 &:"934$'34&:s56B,;)(#34$6+\;#BL™0#345
()#+ac&!#*A$63C&: l : W → DE(#3œ!"6"9&K*A3C+ W (# B,O4B,rs+,I$ w = s · . . . · s ;#+ W 7]"6
cF| (Xy ~€R~XzE7b*2ux+A"9$ N W?N ;)3C56+†OC+‹#&!rm=#56+ rs3C)3Crm(#r;)+ 56B #+LJK3C&:)"sDE(#3S3CI$'+A59[R3C+A##+,I$s;0!)"
"6¬;#BA*,&:rs1e&!"634$63C&:k59B,;#(#3G$'+27Z"9&:34$ l(w) = t ;02#"w#&2$'56+l+LJK+,rs1#O4+ UZ© •1e&!"6+†BAPI!O4+,rs+,I$
&:(#"_;#BL™0#34"6"6&!#"!O4&:56"SO—uv+,#"9+,rm=#O4+o"6(#3G[!2E$
l(id) = 0 U{§
Θ
α
α
α1
αt
α1
αt
α1
αt
W Θ = {w ∈ W | l(wsα ) = l(w) + 1, ∀α ∈ Σ(T, b)\Θ}.
\+qO W 7#O‘u°!1)1#OC34*@2$63C&:
W Θ × WΘ → W
(w, w 0 )
7→ ww 0
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Wα
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[email protected]\_ mb^α[email protected] b^Wi\_DZ\[email protected] [email protected]&if_
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α
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W
W
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X (α2 )
+,"p$M=#3 >9+,*L$'34[!+Œ^c[:&!3C5 S10!5˜+LJK+,rs1#O4+Œ±vµ NR*u:K³:7E1#56&:1 U ³ U ³œ1 U <!´Šfb+A$Y&:‹o;#+œ1#OC()" l(ww ) =
1e&!(#5S$'&!()$M*A&:(#1#O4+ (w, w ) ;#+ W × W U ‰Q,"TO4&:56"A7K&!‹&:=)$63C+,I$TOCŒ;#BA*,&:r N
l(w) + l(w )
1 &:"63G$'3C&! "6(#3G[!2E$6+
0
0
0
XΘ =
a
Θ
BwP/P =
Θ
a
Cw .
_[2!5634BA$'B BA$'!I$](##+Y*,+,O4OC(#O4+d;#+ ­ l;)+ X 7?10!5X*,&:#"9B,DE(#+AE$˜34"6&:rs&:561).#+
(W V+,"61#!*,+ @C?l)+¬DE(#3œ;#+¬1#O4(#",7˜+,"p$‹;#+ª;#3Crs+,)*,3C&! l(w)
#
^`*Aa U ±vµ NR*u:)³:7d1#59&:1 U ³ U 1 U
"6rl!##34+,##+27€O—ux!;#.#BA56+,)*,+ C ;uv(##+
$'+AOC2}O4+n´ f *,U +,\O4OCQ,(#"wO4+O4&:+,56"p"A$œ7(#*,#+&:qrs"6rs&:(#+s" ;#[2!!#"59q3CBAOC$6+sBn*@;#!&:"qI$œ;uvOŠ(#%#+*,†OC!P!"65'"9!+o"9rl
&R;#(#O4&%B,DE(#34[2!O4+,#*A+o5'2$63C&:#)+,OCO4+\a`!3G$
10!5p$'34+];ux()q"phR"9$'QArl+˜;#+]P:BAN #B,562$'+A(#56"XOC34=#56+A";#(oP:59&:(#1 +];#+T«­X®¯ ;)+ X (α ) ^c*Aa U ±vµ NRu* :K³:7
” &:5 U 1#³ U1#2mO43C1 DEU(#&:)"2?Y´Šrlf U !3CI$6+,0!I$Y*,+A"S56BA"6(#OG$/2$6" #&!$'59+‰P:59&:(#1 + ^`!;?>9&:34E$,7E"6+,rs3 "634rl1#O4+M+A$
;#BA1#–OC&ih:[email protected];#+%$ hR1 + G U K&:„P:56&:()1e+<;#+ W ¬+,"[email protected]€#&:(#"qOC+<G56!1#1 +,OC&!#",7+A#P:N+,);#56Bl10!5
+A$ 7#O4+,"T56B 0+LJK34&:#"œ!"6"9&K*A3CBA+,"_!(KJ‹;#+,(RJ5'!*A3C#+A"M"634rl1#O4+,"M;)+ uv+,"p$œ(#-P:56&:()1e+
s
;uv&:56;)56+ s 12 1#()3C"6DE(uv3COX!P!34$œ"634rl1#O4+,rs+,I$T$'56!#"934$'3G[:+Arl+AE$\"6()5‰O—uv+,)"6+,rwG=#U{OC+q” ;#+,"\5'!*A3C#+A"\;)(
"9hR"p$'Q,rs+·;#+5'2*,3C)+,"s;#+ G ^ [:&:345sO‘ux34OCO4(#"9$65'2$63C&: U ;#("phK"p$'QArl+ª;#+¬56!*,34#+,"s;#+ G !(
*/.0!1)34$'59+ f U wux!()$'[email protected] s · s +A$ s · s "9&:I$_;#+,(RJ-56&!$'2$'34&:#"Y;ux&!56;#59+ 6 56+AOC3CBA+,"
+,I$656+q+,OCO4+,"T1025TOŠm56+AOŠ2$63C&:
w∈W Θ
w∈W Θ
w
Θ
w
2
2
α1
α2
α1
α2
α2
α1
(sα1 · sα2 )i = (sα2 · sα1 )6−i
1 &:(#58$6&:()$]34#;#3C*A+ i ;)+MO—uv+,#"9+,rm=)OC+ {0, . . . , 6} ^c*,+T"9&:I$];#+,"]59&!$/?$'3C&!#"8;ux!#P!OC+,"83CI[:+A56"6+A"/f U
‰QA"wOC&!56",7)&:(#"w1 &:()[!&:#"Œ;#&:##+A5ŒOC;)B,*,&!rl1 &:"63G$'34&: 56BA;#(#34$6+l;)+s$6&:(#"qOC+,"qB,O4B,rs+,I$'"q;#+
7 "'@[:&:345
W W
s α1
s α2
s α1 s α2
s α2 s α1
s α1 s α2 s α1
s α2 s α1 s α2
(sα1 sα2 )2
(sα2 sα1 )2
s α2 s α1 s α2 s α1 s α2 s α1 s α2 s α1 s α2 s α1
+A$œ+A)™0
(sα1 sα2 )3 = (sα2 sα1 )3 = −id
(sα1 sα2 )6 = (sα2 sα1 )6 = id
&!(#"[email protected][!&:#"SBAPI!O4+,rs+,I$˜;#&:##BœOŠn;#BA*,&:rs1 &:"63G$'3C&!s;)+\O‘uxBAOCB,rs+,I$d#+A()$'59+ ;#+ 7IrltArl+
6"§ 3Z*,+mHux+A"9$n102""'†;)B,*,&!rl1 &:"63G$'34&:ªrs3C)3Crl!OC+n1#()3C"6DE(#+ l(id) = 0 UH© ¬#&id!$'+m!W(·10!"9"'!P!+
DE(#+
sα1 sα2 sα1 = s3α1 +α2 ,
sα2 sα1 sα2 = sα1 +α2 ,
sα2 sα1 sα2 sα1 sα2 = s3α1 +2α2 ,
sα1 sα2 sα1 sα2 sα1 = s2α1 +α2 .
$WS]mi\_ i\_DbuXQjmX 1 [l/i c X^_b^[email protected][ ‚[email protected] ]^Z\l Y`a[4gƒWb ‚_Db iflaWbdl8bu‚_eZ\Yf_Sjm_gD_D`a`8_jm_ mg Fj^[[email protected]`a[
jm€@b^l8iflaWb j^_eY Y&[email protected] Z\[email protected]]=_ Y ]@[Z}5[ [IW`al [email protected]_eY
l8`gƒWbu‚ la_Db i$j^_Z\_DS]^` [gD_DZ ]=[Z$YfWb gƒWS]^`a€DS_ebi}[laZf_
j^[[email protected]
j^[[email protected] `4. [Z&i\lagD`8_WZ\l [email protected][` uS[laYPgDΘ_D` [b^_hg @[b __Db4Zfla_Db `a_kZ\€eYfX^Θ`8i\[[email protected][` ]
Σ(T, b)
U ³R$'59(#*A$6(#56+q*,+AOCOC()OŠ!3456+œ;#+
0
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\ +qO W 7##&!(#"œi[!&:#"
W = {s , (s ) = id}
+L$œ!3C)"63—7R#&:()"M$'56&!()[:&:)"œDE(#+ l(ws
7)"63H+A$_"6+A(#OC+Arl+AI$_"63 w m(#)+nB,*,5934$6(#56+
) = l(w) + 1
59B,;#()34$'+nDE(#3€"9+o$6+,59rl34#+n10!5 s U O"Aux+A#"6()34$‰!OC&:59"MDE(#+
α2
α1
α1
2
α1
α2
W α2 = {id, sα2 , sα1 sα2 , sα2 sα1 sα2 , (sα1 sα2 )4 , sα2 sα1 sα2 sα1 sα2 },
DE(#3H+,"9$M(#-+,#"9+,rm=#O4+ W 6 BAOCB,rs+,I$6" U#§ &:(#"T"6i[!&:#"T;#&!#*nDE(#+n#&:(#"S;#+A[!&:#"_56+A*/.#+,59*/.#+,5 6
BAOCBArl+AE$6"_+A$_DE(#+qOC+,"T*A+,O4OC(#O4+,"_!"9"6&R*,3CBA+,"_"6&!E$œ;#+o;#34rl+A#"634&:#" 0 7 1 7 2 7 3 7 4 +A$ 5 U
_+A[!+,#&:)"br†23CI$'+A0!I$ #&!"Z*A&:#"634;#B,562$'34&:#"€1#59B,*,BA;#+,I$6+," !#"ZOC+d*@2"Z;uv(##+˜P:56!"6"
rl!#)3C+,)#+lO4+l*/.)&:3GJ„;#(š;#5'W !1 [email protected](•;#+†;)B,10!5p$m34rl1 &:59$6+%1e+A(•1 U &:(#5qOŠª;)BA™0#3G$'34&:›;)+†"6+A"N
[2!593CBL$'B,"];)+ ­ [email protected]:rst,rs+M"630*A+,O4OC+," N *A3)*/.0!)P:+,I$Y"6(#3G[!2E$bOC+T;#56!1e+,!(7:O4+,(#5
*AOŠ!"9"6+o;0!)"\O‘u°!)#[email protected]!(-;#+%«‰­X®¯ 56+A"9$'+qDE(0!I$ W +AOCO4+!7#OC<rst,rs+ U jd56+L[2!#*/.#+27e*A+q*/.)&:3GJ
+A"9$Œ1#5634rl&!56;#3C!O8"93YO‘ux&:k[:+A()$m&!=)$'+A#3C5o(##+<[R5'!34+<"9$'59(#*A$6(#56+l*,+AOCOC()OŠ!3456+m1 &:(#5 X (α ) U j]
O‘ux&R*,*A(#56+A#*,+27!1 &:(#5X*,&:)"9$'59(#3C59+S*A+M;#5'21e+,!(m#&:()"810!5p$'&:)"8;uv(##+M=#!"6+M)&:56rl!OC+\^c*Aa U ^p³ U :Ifpf
;)+n#&!$'59+o!OCP!Q,=#59+n;ux&R*A$6&:#34&:#"M;#BA1#OC&ih!B,+ O +A$_)&:(#"Trm(#)3C"6"9&:#" H ;uv(##+o=0!"9+oDR()3",uv+,
;)B,;#(#3G$ U{” &!#*,59QA$'+Arl+AI$‰#&!(#"œi[!&:#"
2
H = Vect{e1 , f2 , f3 , e = e0 − f0 , e3 , e2 , f1 }
& d+,*L$ S +,"p$TOC+ kN +,"61#!*,+[!+,*A$6&:5634+,Oe+,#P:+A#;#59Bn10!5SO4+,"YB,OCBArl+AI$'"M;#+‰O—uv+,#"9+,rm=#O4+ S U \Q,"
O4&:56"A7#OC+n;)5'!1 [email protected]!(-DE(#+q#&:(#"_+Aª;#BA;#(#34"6&:#"_+A"9$\OC+"9(#34[2!I$
V7
∪
V6
∪
V5
∪
V4
∪
V3
∪
V2
∪
V1
∪
V0
= H
=
=
=
=
=
=
d+A*A$ {f , f , e, e , e , f }
d+A*A$ {f , e, e , e , f }
d+A*A$ {e, e , e , f }
d+A*A$ {e , e , f }
d+A*A$ {e , f }
d+A*A$ {f }
2
3
3
3
3
2
2
1
3
2
3
2
1
2
1
1
1
1
= {0}.
01 !59$63C5_;#+Œ*,+w;)5'!1 [email protected]!(H7 )&:(#"‰*A&:#"p$'56()3C"6&!#"‰;#&!#*wO4+,"_[!2563CBL$'BA"‰;#+ ­ X
;)+ Γ (H) @[!2E$m;ux+A1#56+A#;#56+ O‘ux34I$'+,59"6+A*A$'34&:@[:+A* X (α ) U OY+,"9$%$6&:()$%;Hu°!= &:59;=#3C+A
BL[R3C;#+AE$œDE(#+
2
2
X (α2 ) = X(0,0) ∩ X (α2 ).
λ
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
:
X (α2 )
]j #"6()34$'+27€1e&:()5$'&:()$6+ kN 2OCP:QA=#56+ R 7O4+ RN rs&K;)(#OC+mO43C=#59+ P +A#P:+A#;#56Bs10!5oO4+,"[:+A*A$6+,(#59"
+A$ f +,"p$\()ªBAOCB,rs+,I$_;#+ X (α )(R) 1#(#34"6DE(#+o10!5T;)BA™0#3G$'34&:-*,+o"6&:I$œ;#+A(KJ-B,O4B,rs+,I$'"
e
3C"9&!$'59&:1 +,"-;#+k$'5'2*,+kE(#OCO4+„+L$DR()+š;ux!1#59Q,"OC+ OC+Arlrs+ 7 e f = 0 Uœ© 5-*,+L$¬BAOCB,rs+,I$
ux!1#10259$'34+,I$œ10!" W X (R) U A !5_*A&:#"6BADE(#+,I$)&:(#"œ@[:&:#"
1
1
2
2
1 2
(1,0)
X(0,0) ∩ X (α2 ) 6= X(1,0) ∩ X (α2 ).
‰ +rltArl+27bOC+ RN rl&R;#(#O4+‹OC34=#56+ P +,#P:+A#;#59Bª1025sO4+,"m[!+,*A$6+,(#59" e +A$ f +,"p$l1 &:(#5<O4+,"
rst,rs+,"M56!3C"9&:#"SDE(#+n1)56B,*AB,;#+Arlrs+,I$T() B,OCBArl+AI$T;#+ X (α )(R) U ‰+n1#O4(#",7R3COe!1#1#!59$63C+,I$
W X rl!3C"S1#OC(#" W X (R) U0– 34#"63‘7)O W +A#*,&!56+
2
1
3
2
(1,0)
(3,1)
X(1,0) ∩ X (α2 ) 6= X(3,1) ∩ X (α2 ).
t,rs+,"d5'!34"6&:)",7EOC+
+A$ &:(?>9&:+,(#"p5˜$_1 =#&:3C+A(#-(5˜OC#+,-B"˜rs
,OCBArl+AI$T;#+
1#O4(#" W
e3
X(4,2) (R)
70;)&:#*o1#OC()" W
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+A$_10!5_*A&:#"6BADE(#+,I$
R
P3
(X(3,1) ∩ X (α2 ))(R)
(X(4,2) ∩ X (α2 ))(R)
2
X(3,1) ∩ X (α2 ) 6= X(4,2) ∩ X (α2 ).
j]*,&:#"93C;#BA5'!I$%+A#"6()34$'+†OC+ rl&R;#(#O4+lO43C=#59+ +,#P!+,#;#59B 10!5mO4+,"w[!+,*L$'+,()56" f +L$ e 7
#&:()"w*,&!#"9$'2$'&:)"wDE(#+‹O W +A#*,R&:N 59+!7X3COd",ux!P:3G$q;ux()P•B,O4B,rs+,I$m;#+ (X ∩ X (α ))(R)
DE(#3
uv+,"p$‰1#O4(#"T;0!)" X ∩ X (α ))(R) +L$_;#+oO W #&:()"\@[!&:#"
4
3
(4,2)
(5,4)
2
2
2
X(4,2) ∩ X (α2 ) 6= X(4,5) ∩ X (α2 ).
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#&:()"%@[:&:#"wO W +A#*,&!56+‹(#BAOCBArl+AE$w;#+ (X ∩ X (α ))(R) DE(#3Yux!1#10!5p$'34+,I$m1#O4(#" W
+A$œ+,*,&:)"6B,DE(#+A#*,+w#&:(#"œ@[:&!#"
(X
∩ X (α ))(R)
5
(5,4)
(5,5)
3
1
2
2
X(5,4) ∩ X (α2 ) 6= X(5,5) ∩ X (α2 ).
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X (α2 )
X(0,0) ⊃ X(1,0) ⊃ X(3,1) ⊃ X(4,2) ⊃ X(5,4) ⊃ X(5,5) ,
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I¶
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X5 (R) = X (α2 )(R) ∩ Γ2 (H)(R)
= X (α2 )(R)
X4 (R) = X (α2 )(R) ∩ X(1,0) (R)
= {P ∈ X (α2 )(R) | rg(P
X3 (R) = X (α2 )(R) ∩ X(3,1) (R)
= {P ∈ X (α2 )(R) | rg(P
X2 (R) = X (α2 )(R) ∩ X(4,2) (R)
= {P ∈ X (α2 )(R) | rg(P
X1 (R) = X (α2 )(R) ∩ X(4,2) (R)
= {P ∈ X (α2 )(R) | rg(P
X0 (R) = X (α2 )(R) ∩ X(5,5) (R)
= {P ∈ X (α2 )(R) | rg(P
X (α2 )
∩ (V7 )R ) > 2, rg(P ∩ (V5 )R ) > 1}
∩ (V6 )R ) > 2, rg(P ∩ (V3 )R ) > 1}
∩ (V5 )R ) > 2, rg(P ∩ (V2 )R ) > 1}
∩ (V3 )R ) > 2, rg(P ∩ (V1 )R ) > 1}
∩ (V2 )R ) > 2, rg(P ∩ (V1 )R ) > 1}
V"9+,+<*L$'OC3C+A&!*Aš;$6+,#(#+5o56+Ar†256DE(#@[:+,56+A*†DEu(#v3C+srs#&1e:&!()59$'"w+<!DE()(#56+A3C&!OCOC#"+<n[21#!(š5634BA59$6+,Blrs*A1#&:OŠrl2*,1)+,565 3C"9X+<"9$'593C*L∩$'+,Xrs(α+,I$w) +,10I$!'5o59+ O—uv3CI$'+A5 N
+L$ X 7]*,X&:rs(αrl+†) 10!5%+LJK+Arl1#O4+ X & X +L$s*,+!7]"6!#"<DE(#+*,+,OC #+-*/.0!#PX:+-OC
™#O4$'562$'34&: U O +Al+A"9$Y=#3C+Al+AI$'+,);#(‹;#+œrltArl+T1 &:(#5]$6&:()$'+œOC+A"d*,+,O4OC(#O4+,"]&s1#O4(#"634+,(#59"˜*/.)&:3GJ
BL$/!34+,I$n1 &:"6"93C=#O4+," U A 25n!3CO4OC+A(#56"A7e34OZ+A"9$o=#34+, *,OC!3C5œDE(#+%*,+A"o"6&:(#" N [2!593CBA$6B,""9&:I$nac+,59rlBA+,",7
34O"6( ?<$‰1e&:()5\"Aux+A *,&:I[2!3C)*,56+q;#+Œ56B,1 BA$6+,5\1#56+A"6DE(#+mrs&!$œ1e&!(#5\rs&!$œ*,+ŒDE(#+w#&!(#"@[:&!#"
;)34$Y1e&:()5S;#BArl&:I$656+,5SDR()+OC+A" X BA$'!3C+AE$Yac+,59rlBA+,"M;02#" Γ (H) +A‹()$'34OC34"'!I$YO—ux!56P:()rl+AE$
1 &:(#5TOCm;#BA™##34$63C&:-;#+A"œ"9&:(#" N ac&:#*L$'+A(#56" U 7ª!3CI$'+A0!I$œDE(#+o#&:(#"œ@[:&:)"\(#)+o™#O4$'562$'34&:7R3CO
)&:(#"\a`2()$_[:B,5934™0+A5‰DE(#+ŒOC+,"œ;#3GFHBA56+,)*,+,"‰"6&:I$\=#3C+A¬;#+A"‰*,+AOCO4(#OC+A",7{*!uv+,"p$ N¨W}N ;#3456+oDE(ux34O€",ux!P:34$
=)3C+,q;uv+,"910!*,+A"b= ?s#+A"[email protected]:!I$8O4+,"€;#3Crs+,)"63C&!#"X2$6$6+,#;#()+,",7?2™#Œ;[email protected][:&:345€+AFe+,*A$634[!+,rs+,I$b(#)+
;)B,*,&!rl1 &:"63G$'34&:†*A+,OCO4(#OŠ23C56+ 'U A &:(#5T*A&:rsrl+A#*,+A5,7))&:(#"T3CI$656&R;#(#3C"9&:#"YOŠw#&2$/2$63C&:‹"9(#34[2!I$6+
1 &:(#5T;)B,"634P:#+A5œO4+,"T;#3GFHBA56+A#*,+A"
(3,1)
2
2
(1,0)
(3,1)
(4,7)
(2,1)
λ
2
∀i ∈ {0, . . . , 5}
X(i\i−1) = Xi \Xi−1
@[!+,*MOC\*,&!E[!+,I$'34&: X = ∅ U OK)+S#&:()"b56+A"9$'+Yr†23CI$'+A0!I$81)OC(#"ZDR(Hu W BL$/!=)OC3C5€O4+Y59B,"6()O4$/?$
"9(#34[2!I$
d'# #š ∀i ∈ {0, . . . , 5}
−1
!#"%$ &$(')& *,+-% i .
/ + %0($1 . 2#"n"6&:·ac&!#*A$63C&:#)+,rs+,[email protected] OŠ†1)56+,(K[:+%+,"p$nOŠlrltArl+Œ1e&:()5\$'&!()$'+A"OC+A"
*A+,O4OC(#O4+," Ue§ &!(#"n!O4OC&:#"œ10!5‰*,&:#"9B,DE(#+AE$q#&!(#"OC34rl3G$'+A5 W ;#BA$'!3CO4OC+,5œ*,+AOCO4+w;#+ X ' 7
X(i\i−1) '
i
i
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4
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#³
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1 &:(#5€OC+A"Z2()$'59+,"X#&:(#"X#&:(#"X*,&!E$6+,I$'+A56&:)"8;#+˜;#&:#)+,5XOC+,"€"phK"p$'QArl+A"8;uvB,DE(02$63C&:)"M^c56B,"9&:OC()"/f
*,&!5656+A"61 &:#;02E$6" U
"9+;#&:)#+n;#&:#* R (#)+ kN !OCP:QA=#56+\+A$_#&!(#"T!O4OC&:#"YBA$'!=#OC345YDE(uv3COe+LJK3C"p$'+n(#)+=#3 >9+A* N
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OC 1#59&?>9+,*L$'3C&! *@!#&!#3CDE(#+ ‰&:)#&:#" #&:()" (# 1 &:3CI$
;#+ X jU :‰VQA"wO4&:56"A7XV;)/V3C56+<DE(#+ P +A"9$w(#šBAOCB,rs+,I$Œ;#+ U X (R) N +A"9$wB,DE(#P34[2!O4+,I$RN W ;#3456+
DE(#+ ?;ux()#+T10!5p$bO—ux!1#1)OC3C*,2$'34&: j BL$'+A#;#(#+ W R +L$]56+A"9$656+,34I$'+ W P 7 (j ) 7!+,"p$]"6()5c>9+,*L$'3G[:+
;ux!()$656+<10!5p$ ker(j ) ⊂ (V ) +A$q+,K™0 (V ) → (V /V ) 56+,"p$'59+,3CI$6+ W ker(j )
+,"p$†() 34"6&:rs&:591#.#3C"9rl+ U j] *,&:#"9B,DE(#+A#*,+27S#&:()"s1 &:"9&:#" i : (V /V ) ,→ P O—ux!1#1#O43C*@ N
$'34&: 3C);#(#34$6+n10!5TO‘ux34"6&:rs&:561).#3C"9rl+ (V /V ) → ker(j ) U ²8340!O4+,rs+,[email protected]Šm"934$'(#2$'34&: "6+
56BA"6(#rs+ W OCm"6(#3G$'+n+gJ#2*A$'+q*,&!(#59$6+q"9(#34[2!I$6+
4
(4\3)
7
7
6
(4\3)
(4\3)
R |P
R |P
5 R
5 R
5
4 R
5
∼
5
4 R
4 R
R |P
i
R |P
(jR )|P
0 → (V5 /V4 )R ,→ P (V7 /V6 )R → 0.
O+,;#BA*,&:(#O4+oDR()+
P ' (V5 /V4 )R ⊕ (V7 /V6 )R
& O—uv3C"9&:rs&:561#.)3C"6rs+s;#B,1 +,#;;#+-OC¬;#&:#)B,+-;uv(##+ "6+A*A$63C&:;#+
‰+-*,+A$<34"6&:rs&:5
1#.#34"6rs+S#&!(#"b;#BA;#(#3C"9&:#"bDE(#+ P +A"9$]+Ama`!34$Z(#mrs&R;#(#OC+˜OC34=#56+˜+A$810(j!5Z)*,&:#"U 9B,DE(#+AE$˜#&!(#"b!O NN
OC&!#"Z1 &:()[!&:3C5€56!3C"9&:##+A5 W 10!5p$'3C5;)+,"8=#!"6+A"8;#+A"8+,"910!*A+,"Z[!+,*L$'&:593C+,O4" V U2” &:rsrs+ (V /V )
+,"p$S+A#P:+A#;#56B\10!5 7I*,OC!"6"9+\;)+ rl&R;#(#O4&
+A$
1#!5 7E*,OC!"6"9+œ;)+ rs&
;#(#O4& (V ) 7#(##+oa`IeH,&!P!B,#BA563CDE(#e +o;#+o59+,rs&:(VI$'+,)5_*A+,"_(VB,O4B,rs/V+,I$)'"_+,"p$œ;#f+Œ1)56+,);#56+ f N
R |P
i
1
1
6 R
5
4 R
7
3
6 R
3
4 R
+A$
m = m(a, b, c, d, g, h) = e1 + a · f2 + b · f3 + c · e + d · e3 + g · e2 + h · f1
n = n(w, x, y, z) = f3 + w · e + x · e3 + y · e2 + z · f1
& a 7 b 7 c 7 d 7 g 7 h 7 w 7 x 7 y +L$ z "9&:I$\;)+,"œ"6*@2OŠ!3456+," U ” +,"œ;#+,(RJB,OCBArl+AI$'"Tac&:56rs+,I$œ=#34+,
(# rl&R;#(#O4+‹OC34=#56+l;#+-5'!)P 2 U j] a`[email protected]*,+-rs&K;)(#OC+‹+A"9$sOC+†rltArl+ "93T#&:(#"%59+,rs1#OŠIHA&:#"
10!5 m − b · n +L$q!3C#"938#&!(#"nB,O43Crs3C#&:)"\(#k"9*@!OC!3C59+w+A$o59B,;#()3C"6&!#"nOC+m1)56&:=#O4Q,rs+ U j]
m
*,&!#"6BADR()+,#*A++L$mDE(#3G$6$'+ W 59+,#&:rsrs+,5oOC+A"w"6*,!OŠ23C56+A"Œ59+,"p$/!I$'"A78#&!(#"w1 &:()[!&:#"Œ$65'@[2!3CO4OC+A5
@[:+A*
m = m(a, c, d, g, h) = e1 + a · f2 + c · e + d · e3 + g · e2 + h · f1
!(lOC3C+A(l;)+\OCq1#59B,*AB,;#+AE$6+n;#BA™##34$63C&:s;#+ 2#"YO‘ux+A#"6+Arm=#O4+\;#+œ$'&:()"YO4+,"˜rl&R;#(#O4+,"dDE(ux34O
+,"p$M1 &:"6"93C=#O4+_;ux+A#P:+A#;#56+A5 W 10!59$63C5d;#+ mm+AU$ n +A†a`!34"'!I$d[2!5634+,5dOŠn[2!OC+A(#5d;#+,"Y"6*@2OŠ!3456+,"A7
#&:()"b#&:(#"Z3CI$'BA56+A"6"6&!#"d!(m"9&:(#" N +,#"9+,rm=)OC+S;)+,"brs&K;)(#OC+A"ZDE(#3)!1#10!5p$'34+,##+AI$ W X (R) U
O +,"9$Y*,OC!3C5d1025Y;#BL™0#3G$'3C&!lDE(#+ m +A$ n "9&:I$S;02#" H U O{#+\#&:(#"˜56+A"9$'+\DE(u W [:B,5934™0+A5SDE(#+
+A$
m · n = 0 m = n = 0U
(4\3)
2
2
U ³R$'59(#*A$6(#56+q*,+AOCOC()OŠ!3456+œ;#+
< j]+AFe+,*A$6(0!I$\OC+,"T*,!OC*A(#OC"A7##&:()"œ&!=)$'+A#&:#"


0=




0=





0=



0 =
m · n = 0 ⇐⇒

0=





0=




0 =



0=
hw − cz − xg + yd

xc − dw + az

h=


yc + h − gw



d =
cw − z − d
⇐⇒ x =

cw − ya


y=



x + aw
a =
−w − a
−y − c
X (α2 )
c2 + gw
cw − z
w2
−c
−w
*A+MDE(#3R#&:(#"bOC!3C"9"6+˜;#B¨> W OC+,"X[2!563C!=#OC+A" c 7 g 7 w +A$ z OC34=#56+A" U?V +S*,!OC*A(#OK"9(#34[2!I$8#&:(#"brs&:I$'59+
m2 = 0 ⇐⇒ 0 = c2 − h − ag ⇐⇒ h = c2 − ag
*A+_DR()3#+,"p$˜$6563G[K3C!OC+Arl+AI$8[:BA563G™[email protected][:+,*œOC+A"]1#56BA*,BA;#+,I$'+A"S59+,OŠ?$'3C&!#" U j])™07IO4+T;#+,59#3C+A5]*@!O4*,(#O
)&:(#"_;#&:#)+
n = 0 ⇐⇒ 0 = w − x ⇐⇒ x = w
DE(#3+A"9$œO W +A#*,&:59+o(##+o56+AOŠ2$63C&: ;#B > W 1)56B,"9+,I$'+q;0!)"œO4+n1#56+Arl34+,5T*,!OC*A(#O U
_B,*A3C1#59&KDE(#+Arl+AE$,7Y"93_#&:(#"l*,&:#"93C;#BA56&:)"†(#rs&R;#(#OC+-O43C=#59+ P +A#P:+A#;#56Bª10!5 m =
+L$
[email protected]+,"*,!OC*A(#OC"1)56B,*AB,;#+AI$'"Xrl&:I$656+,I$
m(−w, c, cw−z, g, c +gw) n = n(w, w , −c, z)
;)3C56+A*A$6+,rs+,I$œDR()+ m = n = m · n = 0 +A$\10!5_*A&:#"6BADE(#+,I$nDE(#+ P ∈ X (R) U{§ &!(#"
@[!&:#"s;#&:#*rl34"%+, =#3 >9+A*A$'34&: X (R) +A$lO—uv+,)"6+,rw=#OC+-;#+A"lrs&K;)(#OC+A"%+,#P:+A#;#59B,"†1#!5
+A$
DE(#3{+A"9$Y+,†=)3 >9+,*L$'34&:si[!+,*
m = m(−w, c, cw − z, g, c + gw) n = n(w, w , −c, z)
&:rlrs+o!#)&:#*,B270#&:(#"œ@[:&!#"‰23C#"93BA$'!=#OC3HDR()+
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2
2
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2
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4
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"phR"9$'QArl+A"\;HuxBADR(#2$'34&:#"_+A$œOC+A(#56"T"9&:OC(K$'3C&!#" U
j] *A+kDR()3q*A&:#*A+,56)+ X 7T(# rl&R;#(#O4+ P +,"p$ª() B,O4B,rs+,I$-;#+ X (R) "63n+L$
"9+,(#O4+,rs+,I$œ"63
P ' (V /V ) ⊕ (V /V ) .
+” A*,3€#&:()"œ;)&:##+o;#&:)*q*A&:rsrl+P!B,#BA5'2$6+,(#59"\59B,;#(#3G$'"
(5\4)
(5\4)
7
+L$
6 R
6
5 R
m = m(b, c, d, g, h) = e1 + b · f3 + c · e + d · e3 + g · e2 + h · f1
n = n(v, w, x, y, z) = f2 + v · f3 + w · e + x · e3 + y · e2 + z · f1
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
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X (α2 )
[:BA563G™{!I$


0=




0=





0=



0 =
m · n = 0 ⇐⇒

0=





0=





0=



0=
−gw − bz + yc + vh

hw − cz − xg + yd

g=


cw − z − g − vd



y =
bw − vc − y
⇐⇒ h =

xc − h − dw


x=



cw − xb
b =
b−w
x−c
cw − z − vd
w 2 − vc
c2 − dw
c
w
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m2 = 0 ⇐⇒ 0 = c2 − h − bd ⇐⇒ h = c2 − bd
+A$
n = 0 ⇐⇒ 0 = w − y − vx ⇐⇒ y = w − vx
ED (#3S"9&:I$m;#B > W ;#+A"m56BA"6(#OG$/2$6"w3Crs1#O43CDE(#B,"o1025ŒO4+,"wBADE(02$'34&:#"Œ1#56BA*,B,;)+,I$'+A" U ‰QA"mOC&:59",7€+,
*,&!#*,O4(0!I$œ*,&:rsrl+‰1#56BA*,B,;)+,rsrl+AE$,7{#&:(#"œ@[:&:)"œ™0#!OC+Arl+AE$_1)56&:()[!BŒDE(#+
2
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2
2
X(5\4) '
X(3\2)
7#(#rs&K;#()OC+ P 21#10!5p$'3C+AI$ W
5
.
X(5\4) (R)
"63+A$œ"9+,(#O4+,rs+,I$œ"63
P ' (V6 /V5 )R ⊕ (V3 /V2 )R .
” +,*A3€#&:(#"_;#&:)#+o*,&:rsrl+nP!B,#BA5'2$6+,(#59"œ59B,;#()34$'"
A+ $
[:BA563G™{!I$
m = m(b, c, g, h) = f2 + b · f3 + c · e + g · e2 + h · f1
n = n(y, z) = e3 + y · e2 + z · f1

0=



0 =
m · n = 0 ⇐⇒

0=



0=

−bz + yc

g=
z2

−cz − g
⇐⇒ c = −z

−b − y

b = −y
c+z
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m2 = 0 ⇐⇒ 0 = c2 − g ⇐⇒ g = c2
<1<
U ³R$'59(#*A$6(#56+q*,+AOCOC()OŠ!3456+œ;#+ X (α )
DE(#3]+A"9$q;#B > W 34rl1#O43CDE(#BA+w10!5oO4+,"nB,DE(0?$'3C&!#"o1#56BA*,B,;)+,I$'+A"w+A$q;#+<1#O4(#",7 n = 0 +A"9$o[R5'!34+
DE(#+AOCO4+,"˜DR()+‰"6&!3C+,I$YOC+A"˜[2!O4+,(#59"Y;#+ y +L$ z U ‰QA"MOC&:59",7I#&!(#"[email protected][!&:#"S™0#!OC+Arl+AE$˜1#59&:()[!B‰DE(#+
2
2
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"9+,(#O4+,rs+,I$œ"63
3
X(3\2) '
X(2\1)
7T(# rl&R;#(#O4+
.
P
+,"p$ª() B,O4B,rs+,I$-;#+
X(3\2) (R)
"63n+L$
P ' (V5 /V4 )R ⊕ (V2 /V1 )R .
” +A*,3€#&:()"œ;)&:##+o*,&!rlrs+P:BA#B,562$'+A(#56"œ56BA;#(#34$6"
m = m(c, d, h) = f3 + c · e + d · e3 + h · f1
+L$
n = n(z) = e2 + z · f1
[!B,5934™{!I$
(
(
d = z2
0 = −cz + d
⇐⇒
m · n = 0 ⇐⇒
c= z
0 = −z + c
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m2 = 0 ⇐⇒ 0 = c2 − d ⇐⇒ d = c2
DE(#3e+,"p$M;#B > W (##+BADE(02$'34&:‹*A&:#E(#++L$M;#+\1#OC(#"A7 n = 0 +,"[email protected] W +,#*A&:56+27K[R5'!34+‰DE(#+AOCOC+œDE(#+
"9&:34$_OCw[2!OC+A(#5T;#+ z U A 25œ*A&:#"9B,DE(#+,I$,7e#&!(#"œi[!&:#"œ=#3C+A
2
A
&:(#5
2
X(2\1) '
X(1\0)
70(#ªrs&R;#(#OC+ P !1#10259$'34+,I$ W
.
X(1\0) (R)
"93X+L$_"6+A(#OC+Arl+AE$œ"93
P ' (V3 /V2 )R ⊕ (V1 )R .
” +A*,3€#&:()"œ;)&:##+o*,&!rlrs+P:BA#B,562$'+A(#56"œ56BA;#(#34$6"
+L$
m = m(h) = e3 + h · f1
+L$‰;#!#"_*,+Œ*@!"A7 O4+,"_B,DE(0?$'3C&!#" m · n = 0 7 m = 0 +A$ n = 0 "6&:I$œ[:BA5634™#B,+,"œDE(#+,O4OC+oDE(#+
"9&:34$_OCw[2!OC+A(#5T;#+ h U A !5_*,&!#"6BADR()+,[email protected] O W +A#*,&:59+o#&:(#"_$'59&:()[:&!#"_DR()+
n = f1
2
X(1\0) '
2
1
.
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X (α2 )
jdK™07XOC+,"Œrl&R;#(#O4+," DR()3M!1#1#!59$63C+,)#+,I$
"6&!E$m*,+,(RJ•DE(#3
"6&!E$œBAPI!(KJ W (V ) +A$œ;#PQ,"_O4&:56"A7#3CO"Au°2P:34$_=#34+,-;W uv(#X1e&:34I$(R)
œ=?s=#+ XU0– (R)
3C#"93—7
(0\−1)
2 R
X(0\−1) '
0
.
i
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0
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"6+Arm=#O4+ {0, . . . , 5} 7
X(i\i−1) '
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X (α )
(#q"9hR"9$6Q,rs+Y;)+˜P!B,#BA5'2$6+,(#59"ZO43C=#59+,";#(qP:59&:(#1 + CH (X (α )) U}V +d1 &:3CI$O4+]1#OC(#"€34rl1 &:59$'!I$
;#+%*A+A$6$6+%*,&:)"9$'59(#*A$63C&:H7&:()$656+mO4+wa`!34$‰DE(#+%O—uv&:k‹(##+%"p$'59(#*A$6(#56+<*,+AOCO4(#OŠ!3456+o+LJK1#O43C*A34$'+Œ+A$
P:BA&:rlBL$'593CDE(#+n;#+ X (α ) 70+A"9$œDE(#+
2
∗
2
2
X4 = X (α2 ) ∩ X(1,0) .
]j ;ux!()$'59+,"œ$'+,59rl+A",7eOC+q1#56+Arl34+,5_$'+,59rl+q#&:ª$'563G[R3Š!O;#+wOŠ%™0O4$65'2$63C&:70*A+,O4(#3ZDE(#3X[2l*,&!5 N
56+A"61 &:#;#59+%!(ª*AhR*,O4+m;#+w*A&K;)3Crs+,#"93C&:(#·;#+ (X(α )) +A"9$nBAPI!O W O—uv3CI$6+,56"9+,*L$'3C&!ª;#+
OŠ‰[2!593CBL$'B X (α ) +L$d;#(lP!B,#BA5'2$6+,(#5]+Al*A&K;#34rlCH
+A#"634&:m(#<;#+_OC‰P:5'2"6"6rl!#)3C+,)#+ UE” +Ta`!3G$
#&:()"œ1 +,59rl+L$6$65'<!(*/.0!1#3G$'56+n"9(#34[2!I$œ;uvBA$'!=#OC345MOŠm562$'34&:0!O434$6B‰;#+o*,+o*LhK*AOC+ U
∗
2
2
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X (α2)
&:(#"q!O4OC&:)"r†23CI$'+A0!I$o1#56BA*,34"6+,5qOC+A"n*/.#&:"9+,"w*A&:#*A+,56#!I$wO4+,"oP:56&!(#1e+A"o;#+«‰­X®¯·7
§
;#&:I$œ#&:()"\@[!&:#"œ;#B¨> W 10!59OCB"6!#"œOC+,"T;)BA™0#345T$'&!$'!OC+Arl+AE$ U
&:(#"\!1#1 +,O4OC&:)"T;#&:#* ()#+Œ[2!593CBA$6Bq!OCP:BA=#5634DE(#+nOC34"6"6+o+L$‰*A&:rl1)OCQA$6+ U · €Ic ‚ 
:| ‚ § “f#›E(X@“c|F( "6(#5 +,"pX$\()#+o"6&:rsrl+ac&!56rs+,OCO4+\™0)3C+
i
X
X
nl {Yl }
& mO4+," Y "9&:I$];#+,"8"6&:(#" N [2!593CBA$6B,"Xac+,56rsB,+A"8;#+ X ;#+Y*,&R;#3Crs+,#"93C&: i +L$bO4+," n ;#+,"Z#&:rm=#59+,"
+,I$63C+,59" U2V +Sy{z}|e~€] ‚ R‰ €IcR ‚ <:| ‚ “f#›E(X@“c|F( i "9(#5 X 7}#&2$'B Z (X) +A"9$8OC+˜P:56&:()1e+
OC34=#56+Œ"6(#5dG +,#P:+A#;#59B%10!5O4+,""9&:(#" N [2!5634BA$'BA"‰ac+,59rlBA+,"n;#+%*A&K;)3Crs+,#"93C&: i ;#+ X U (##+
"6&!(#" N [!2563CBL$'B Y ;)+ X &:›a`234$o*,&!5656+A"61 &:#;#59+†O4+s*LhR*,OC+ {Y } ;0!#" Z (X) U \+,(KJ›*AhR*AOC+,"
+A$ ;)+‰*,&R;#34rl+A#"634&: ;#+ "9&:I$Mz}’)Ž}“c|F((€R‘c'#š'(eŽož'%~X“ ’0`'(eŽiœ"Aux34O +gJK3C"9$6+
{Y } {Z}
(#-*AhR*AOC+ {W } ;#+n*A&K;#34rl+A#"634&:i i ;0X!#" X × ;#&:I$TO—uv3CI$6+,56"9+,*L$'3C&!  +,"9$MZz}|{Zz} œ@[:+,*
l
j
l
i
i
W
. _eY&iV jml8Z\_if_D`[email protected]_
p
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0
WX
1
]
1
codim(W ∩ X × {p}) = codim(W ) + codim(X × {p})
<
X × {0}
+A$
X × {1}
U – ##+,!(;#+%«­X®¯ ;#+
7#+L$_$'+,OHDR()+
X (α2 )
Y = W ∩ X × {1}
+L$
© o#&!$6+
*,+L$6$'+]59+,OC2$'34&:;uvB,DE(#3G[!2OC+,)*,+]+A$€&!q1 &:"9+]1e&:()5H$'&:(K$ i ;0!#"O‘ux+A#"6+Arm=#OC+
Z = W ∩ X × {0}.
∼rat
{0, . . . , dim X}
CHi (X) = Z i (X)/ ∼rat
*A+‰DE(#3e;#BA™0)3eO4+ i {y z}| ~X] ‚ ·ƒ†œ‡ˆ ;#+ X RU V +\y{z?| ~€] ‚  ƒ†_‡€ˆ Ži|0Ži’0 ‚ 
y{z}’ ‚ ~Xž¬Z’#z‹`’„:| ‚ f“ #›E(X@“c|F(›+,"p$\!OC&:59"
ème
X
X
CH∗ (X) = ⊕dim
CHi (X).
i=0
bO +A"9$o1 &:"6"93C=#O4+m;#+%59B,1 BA$'+A5n+LJ)!*L$'+,rs+,I$oOC‹rltArl+Œ*,&:#"p$'59(#*A$63C&:„2™0 ;)+%*,&:#"p$'59(#3C59+
7EOC+œy{z?| ~€] ‚ ·ƒ†_‡€ˆ ‚  X y{z}’ ‚ ~€ž-Z’)z<`’ ‚ “&#š'(Z,“`|F( UEV &!56"6DE(#+ X +,"p$
CH (X)
345659B,;#(#*L$'34=#OC+27)&:ª<!OC&:59"
CH (X) = CH
(X)
1 &:(#5œ$'&:(K$ i ;#+wO—uv+,#"9+,rm=#O4+ {0, . . . , dim X} U ¸S34+,·DE(#+m*A+,OC†"6&:3G$(#k2=#(#";#+wOŠ!#P:!P:+!7
)&:(#"n10!59OC+A56&:#"‰1#OC(#"‰"6&:()[!+,I$Œ;#+%*AhR*,O4+%DE(#+%;#+%*,OC!"6"9+%;#+%*AhR*AOC+," U€V ‹OC3C"9"63G$'Bw;)+ X a`!34$
DE(#+wO—uv3CI$'+A56"6+A*A$63C&:;#+,"‰*AhR*,O4+,"<^c*,&:rs1)$'[email protected][!+,*%rm(#OG$'341#OC34*,34$6Bifd;#BA™0)[email protected] 1 &:(#5_$'&:()$ i +L$ i
;)+qO‘ux+A#"6+Arm=#O4+ {0, . . . , dim X} 70(#1#56&R;#(#3G$
∗
i
dim X− i
0
0
CHi (X) × CHi (X) → CHi+i (X)
); &:#0!I$ W CH (X) (##+q"9$656(#*L$'(#59+o;u°2##[email protected](¬P:56!;#(#B U
+d=#()$X;#+d*,+L$6$6+Y"9+,*A$63C&:q+,"p$X;uxBL$/!=)[email protected];0!)"XO4+]*@!"€;#BA1#OC&ih!B!7}OŠT"9$656(#*L$'(#59+Y;)+˜O‘u°2##[email protected](
;)+‰«‰V ­Z®¯ ;#+ X (α ) U1A &:(#5b*A+,OŠK72#&!(#"d!OCO4&:#"X;#BA$6+,59rl34#+,5ZDR()+,OCO4+,"Z56+,OC2$'34&:#"X[:B,5934™0+AI$]OC+A"
P!B,#BA5'2$6+,(#59"M;#+ CH (X (α )) 7R+,†",ux!3C;02E$Y;#+\*,+,O4OC+A"Y;#+ CH (X (α , α )) UKV Œ*A&:#"9$656(#* N
$63C&:„;#+sOŠ-"9$656(#*L$'(#59+†*,+AOCO4(#OŠ!3456+%;#+ X (α ) BA$/2=#OC34+s;#!#"wOC "6+,*L$'34&:š1#59B,*AB,;#+AE$6+!7b& šOC+
59B,"9(#O4$'2$n1#O4(#"P:BA#B,56!O8;#+-±vµ NRu* :K³:7 ” &:5 U ³ U 2†1 U 2?´‘7H#&!(#"n1e+A56rs+A$9$'+,I$n;Hu°= ?s59rl+A5DE(#+
O‘u°2##[email protected]( (X (α )) 1 &:"6"9Q,;#+S(#w(#)3CDE(#+˜P:B,#BA5'2$6+,(#5Z10!5X*,&R;#3Crs+,)"63C&! U}” +SDE(uv3COE#&!(#"
a`2()$qr†!34I$CH
'+,#!I$o*,&:rs1#56+A#;#59+!7*!uv+,"p$w*A&:rsrl+AE$q*A+,"qP:B,#BA5'2$6+,(#59"Œ"9+<rm(#O4$63C1#O43C+AE$‰+,I$'59+
+A(KJ U A &:()5T*,+,OC)7))&:(#"œ!OCO4&:#"Y()$'34OC3C"9+,5YOC+‰a`!3G$MDE(#+ X (α , α ) 1 +,()$_tL$'56+[R(#+o*A&:rlrs+(#)+
™#=#5'2$63C&:l1#56&}>9+,*A$634[!+n!( N ;#+,"9"6(#"T;#+ X (α ) !(‹rs&ih:+, ;)+O—ux!1#1#O43C*,2$'34&:%"6(#3G[2!I$'+;#BL™0#34+
1 &:(#5M$6&:()$6+ kN !O4P:Q,=)56+ R 10!5
∗
2
∗
∗
2
2
1
∗
2
1
2
2
ml
i + i0 > dim X
X (α1 , α2 ) −→ X (α2 )
(D, P ) 7→ P.
`a_hZf€ Y&[email protected]`/i}[;i_eY&ib [email protected]`q_ƒi[]^]@[Zfifla_Db i V
dim X
CH
(X)
]
2
0
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
<
X (α2 )
jd+AFe+A$,70"63#&!(#"\;)BA™0#34"6"9&:#"œ(# >Xz?žªŽiz?“ “c’0 !( ;#+A"6"6()"\;#+
+,1 &:"'!I$
)
1 &:(#5w$6&:()$'+ kN 2OCP:QA=#56+ R 78OCª™0=#59+ !( N ;#+,"9"6(#"<; uv(#BJ,O4B,rs+,N I$ P ;#+ XX(α(α)(R)
*A&:rlrs+
BA$'!I$ P OC()3 N rst,rs+!7!;#Q,"YOC&:59"Y*A&:rlrs+ P +,"9$S(#‹rs&K;)(#OC+_1#59&?>9+,*L$'34a¨7E#&!(#"S1 &:()[!&:#"S*A&:#"93 N
;#BA56+,5 (P ) O‘ux+A"6102*,+¬1)56&?>9+A*A$'3Gao!"9"6&R*,3CBDE(#3œ+,"9$,7Y56!1#1 +,O4&:#"sOC+!7]O‘ux+A#"6+Arm=#O4+ª;#+A"†"9&:(#" N
rs&K;#()OC+,"T1)56&?>9+A*A$'3Gac" N ;#+ P ;#+o56!#P 1 $'+,O4"TDE(#+ P/N "6&:3G$_B,PI2OC+,rs+,I$T1)56&?>9+A*A$'3Ga U0§ &!(#"
‰+‰O 7E1 &:(#5Y$6&:()$6+ kN !OCP!Q,=#59+\DE(#3
;#BL™0#3C"9"6&:)"_!3C#"93{OC+\™#=#56B\1#56&?>9+A*A$634a "9(#5
+,"p$‰()ª2##[email protected](ªO4&K*,!O K 7##&!(#"œi[!&:#" (J ) X (α ) U W
+L$
(J )(K) = {(D, P ) | P ∈ X (α )(K) D ⊂ P }.
5S"93
+A$S+,‹$'+A0!I$M*A&:rs1)$'+\;#(†a`!3G$YDE(#+‰OC+,"Yrl&R;#(#O4+,"˜1#56&}>9+,*A$634ac"Y"6&:I$MO43C=#59+,"˜"6(#5
OC© +A"S2#D#[email protected]⊂(KJsPOC&R*@!(RJH7:$6&:(#"dOC+A"YBAOCBArl+AE$6"d;#+ D "6&!E$Y;#&:)*\;#+_$'56!*,+œ+A$Y;#+_*@25656BœE(#OC"A7E10!5
*,&!#"6BADR()+,I$ P +A"9$œ(#B,O4B,rs+,I$œ;#+ X (α )(K) ;ux&
2
2
2
2
1
(J )(K) = X (α1 , α2 )(K)
A+ $23C#"93OC+A"‹;#+,(RJ ac&:#*L$'+,()56"-"6&:I$B,P:!(KJ U j] a`!34$,7 X (α , α ) +,"p$-OC+·™0=)56B 1#59&?>9+,*L$'34a
!"9"6&R*,3CBk2( ™#=#56B [:+A*A$6&:5634+,O$'5934[R3Š2O +A$-;#+k1#O4(#",7M*A+A$9$'+ ™0=#562$'34&: 1#56&}>9+,*A$634[!+„34#;#(#3G$
(##+!1#1#O43C*@?$'3C&!sa`!34"'!I$Y;#+ CH (X (αJ , α )) (# CH (X (α ))N rs&K;)(#OC+_OC34=#56+œ;#+56!#P 2
^`* U a U ±v²S³:7 . U U =0f¨´Šf U
jdš[email protected]#O4(#"ŒP!B,#BA5'!O4+,rs+,I$wO4&:56"9DE(#+†O‘ux&:›¬(#„™0=#59B†[:+A*A$'&!563C+AO ;#+l5'!#P !(
;#+A"6"6()"b;uv(##+d[!2563CBL$'B]O43C"6"9+ X 7i&:q1e&:"9+ (E) OC+]™0=#59Bd1#56&?>9+A*A$634a02"6"6&R*,34BdE+A$ p : (E) r→ XN
OŠ%1#59&?>9+,*L$'34&:-"6(#5 X U{© ª%O—uv3C"6&!rl&:591#.#34"6rs+‰;ux!##+,!(KJ-P!5'!;#()B,"œ"6(#3G[!2E$
1
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1
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2
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CH∗ ( (E)) ' CH∗ (X)[ξ]/ ξ r − c1 (E)ξ r−1 + . . . (−1)r cr (E)
& „1e&!(#5$'&:(K$ i ;)+%O—uv+,#"9+,rm=#O4+ {1, . . . , r} 7 c (E) +,"p$q(#›BAOCBArl+AE$o;)+ CH (X) 21#1e+AOCB
Ic’0@, ‚  ƒ‹T¤H¥;#+ E +L$ ξ +,"p$MOC+œ*AhR*AOC+‰;)+‰*,&R;#34rl+A#"634&: 1 !"6"9&K*A3CB\!(†™0=#59B‰+A
i
;#59&:34$6+ O (1) U
+A"_*,OŠ2"6"6+A"_;#+%«­ „;#(-™0=#59Bq"9&:I$œ10!5T;#BL™0#34$63C&: O4+,"_*,&R+?s*A3C+,I$6"œ;)+qO‘uxBADE(02$'34&:
V
;#+q5'!)P DE(#+o[:B,5934™0+qO—uvB,O4B,rs+,I$ ‰!#"œ#&!$'59+Œ*,!",7{*A+A$6$6+wBADR(#2$'34&:ª+A"9$\;#+q;#+,P:59B 2 +A$
#&:()"\2OCOC&!r#"S[:&!3C5TDE(uv3CO"6+A5'<3CE()$63CξOC+\U ;#+qOŠm;#BL$'+,59rl34#+,5 U#– 34#"63H#&:(#"œ1e&!"6&:#"
∗
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CH∗ (X (α1 , α2 )) ' CH∗ (X (α2 ))[ξ]/ ξ 2 − c1 ξ + c2 .
+A$6$6+Œ"p$'59(#*A$6(#56+q;#+ (X (α ))N rs&K;#()OC+rl!3CI$6+,0!I$_+gJ)1 &:"9B,+!7 #&:()"‰!O4OC&:)"MO—uv()$'3 N
OC34"6+A5s” 1 &:(#5<*,!OC*A(#OC+A5s;)+ªCH
;#+A(KJa`IHA&:#"s;#34FeB,59+,I$'+A"†;#+A"l34E[2!593Š!I$6"s;)+ CH (X (α , α )) U
j]·*,&!rl1025'!I$œOC+A"œ59B,"9(#O4$'2$'"œ;#+q*,+,"\;#+,(RJ¬*,!OC*A(#OC"_#&!(#"\&!=)$'34+,#;#59&:#"\!OC&!56"_OŠm$/2=#OC+o;#+
rm(#OG$'341#OC34*@2$63C&:¬^`10259$'34+,O4OC+ifd;#+,"œP:BA#B,562$'+A(#56"_;#+ CH (X (α )) U
jd<1#59+,rs3C+A5bOC34+,(723CO))&:(#"ba`!()$b5'!1#1 +,O4+,5]DE(#+AOCDE(#+,"d59B,"6()O4$/?$'"]*,&!#E(#"d"6(#5dOŠ\"9$'59(#*A$6(#56+
;#+ CH (X (α , α )) U
∗
2
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1
2
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1
2
U – ##+,!(;#+%«­X®¯ ;#+
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X (α2 )
CH∗ (X (α1, α2))
€Vj]W•1+,##*59A+,&:rs56+23C+A7{5q#&O4:3C(#+A("œ7!OŠO4 OC&:#""9$6S56(#()*L$'$'34OC(#3C"959+,+‹5M;#O4+,+ "T#&2$/2$63C&:#"T;)+qOCm"6+,+Aš$*A$''34!&:I$w DEU ³ (#U +lP:56&!(#1e+s+,"p$w;)B¨>
*A&:#E(#+ +L$%¬;#B > W BA$6B‹BA$6(#;#3CBA+!7Z#&!$'+ArlCH
rs+,I$w(X;0(α!#" , α±87¬))!5 7 V +,r U 2 U ³ U ³,´ U K34#&:7X#&:()W "
1 &:()[!&:#"o$'&:(}>9&:(#56"o59+A$'59&:()[!+,5q*,+,"q56BA"6(#OG$/2$6"o+,š+gJK1#OC&:3G$/!I$,7*,&:rsrs+%#&:(#"qO—[email protected][:&:#"o;#B > W
a`[email protected]—ux!59$63C*AOC+¬;)+±xOµ NK* :)³,´ U !#" $6&:(#"-OC+A" *@!"A7T&: $'56&!()[:+kDE(#+ CH (X (α , α )) +,"p$
() P:59&:(#1 +ªO43C=#59+‹"6()5 G +,#P!+,#;#59Bª1#!5<;#&:( A+ªBAOCB,rs+,I$6",7d(# 1e&:()5s*/.#!DE(#+¬56!*,34#+-;#(
P!56&:(#1 +œ;#+ w;)+ G 7E#&!$656+œP:56&:()1e+w^‘!;}>9&:[email protected]!"6+,rs3 N "634rl1#O4+M+A$/fd;)B,1#O4&}h!B‰;#+_$ hR1e+ G U
j]·[email protected] #&:(#"‰@[:&:#"\;#B¨> [K(¬DR(Hu */.#!DE(#+mB,O4B,rs+,I$\;#+ &:a`!34$œ*A&:5659+,"61 &:#;)56+m(#)+
*A+,O4OC(#O4+];#+d;#3Crs+,#"93C&:oBAPI!OCW+ W OŠTOC&!#PW :(#+A(#5€;#+d*,+A$ZB,O4B,rs+,[email protected]+A#"6()34$'+˜O—uv+,)"6+,rw=#OC+d*,&!#"9$634$'()B
;)+¬O—ux!;#.)B,56+A#*,+ª;#+ª$6&:(#"l*,+,"†B,OCBArl+AI$'"lac&!56rs+ª(# "9hR"9$6Q,rs+·;#+ªP:B,#BA5'2$6+,(#59"‹OC34=#56+A"l;)+
&!(#5ª"93Crs1#OC3G™0+,5†OC+,"-#&2$/2$63C&:#"A7T#&:(#"56+A1#56&R;#(#34"6&:)"*A3 ;#+,"9"6&:(#"OC+A"
CH (X (α , α )) O
BAOCBArl+AE$6"S;#+ W iU [!A +,* W * ,!$6B‰OŠq#&!$'2$'34&:†2;#&:1)$6B,+1 &:(#5YOC+œP:B,#BA5'2$6+,(#5Y*,&:5956N +A"61 &:#;0!I$
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2
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2
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h5
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h3
h2
h1
g6
1
!
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!
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2
s α2
s α1 s α2
sα2 sα1 sα2 = sα1 +α2
s α1 s α2 s α1 s α2
sα2 sα1 sα2 sα1 sα2 = s3α1 +2α2
(sα1 sα2 )3 = (sα2 sα1 )3 = id
g5
g4
g3
g2
g1
g0
!
!
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s α1
s α2 s α1
sα1 sα2 sα1 = s3α1 +α2
s α2 s α1 s α2 s α1
sα1 sα2 sα1 sα2 sα1 = s2α1 +α2
(sα1 sα2 )3 = (sα2 sα1 )3 = −id
uv3C#;)3C*,+S"6(#1 B,593C+,()5];#+T*/.0!*A(#l;)+,"dB,O4B,rs+,I$'"]*A&:5659+,"61 &:#; "'n*A&K;#34rl+A#"634&: j]s"9+,*A&:#;
4OV 3C+A(73CO]+gJK3C"9$6+†(#)+sac&:59rm(#O4+s;)(#+ W «‰­ =#3C+A›DE(#W +l;#B,rs&:I$'59B,+s+A$Œ1#U (#=#O43CB,+<1025
"! l;02#"‰±v‰+Ar <)7 9 < &:5 ´—7EDE(#+\1e&:()5˜)&!$'59+\10259$Y#&:(#"Y!1#1 +,O4OC+,59&:#"d3C*A3#ac&:5
rw(#OC+\;#+w«‰­ š+L$TDE(#3e[2m” #&:U ()"M1e+A56rs+A$9$'56+\;#+*,!OC*A(#OC+A5SOC+œ1#59&K;#()34$M;)+uv3Crs1e&:5p$'+ N
DE(#+AOS*A&:(#1#O4+l;uvB,O4B,rs+,I$'"Œ;#&:I$mO‘ux()•!(›rl&:34#"q+,"9$w;#+†*A&K;)3Crs+,#"93C&: 1 UZ” +A$9$'+†ac&!56rm(#O4+!7
59+,1#59&K;)(#34$6+oi[!+,*qOC+A"T#&!$/?$'3C&!#"T;#+oO—ux!59$63C*AOC+‰&!563CP!3C0!O‘7K+A"9$
#|{@z #k~Zc ‚ šƒ†T¤%$ R'&(&)¤ )
X
[Xw0 sα ] · [Xw ] =
< β ∨ , $α > [Xwsβ ]
8j JK1#OC34*,3G$'&:#"qrl!3CI$'+A0!I$Œ*,+A$9$'+†ac&!56rm(#O4+ &:()$m;Hu°!= &:59;7 ;#B,"93CP:#+lO—uvB,O4B,rs+,I$w;)+
O4&:#P:()+,(#5Trl?JK3Crl!OC+œ;#+ W 7 w +A"9$œ(#B,OCBArl+AU I$ _DR()+,OC*A&:#DE(#+q;#w+ W +L$ R ;#BA"634P:#+oO—uv+, N
"9+,rm=)OC+·;#+A" 5'!*A3C#+A"‹1 &:"63G$'34[!+,"‹1#!5‹5'21#1e&!59$-!( */.)&:3GJ;uv(# "6&:(#" N P:59&:(#1 +·;#+ \® ;)+ G 7{DR()+w#&:()",7 #&:(#"‰i[!&:#"\#&!$'B ∆(T, b) U V l)&!$/2$63C&: X ;#B,"93CP:#+27e*A&:rsrl+o#&:()"
O‘u°@[!&:#"w;)B¨> W +gJK1#OC34DR()B!7XOŠ [!2563CBL$'B<;#+ ­ *,&!5656+A"61 &:#;02E$ W OC*A+,OCO4(#OC+<;#BL™0#34+
1#!5]OŠ56!*,34#+ w ;0!)"˜OC‰;#BA*,&:rs1e&!"634$63C&:%;)+ X­ -;)+ G/B U ‰+_O W 7!OC#&!$/?$'3C&! [X ]
;)B,"634P:#+_"'o*,OC!"6"9+\;#!#" CH (X (α , α )) U wu°2()$'59+\[email protected] α +A"9$˜(##+œ5'2*,3C)+T"63Crs1#O4+!7I!34#"63
+L$]10!5X*,&:)"6B,DE(#+AI$ [X ] +,"p$];#+Y*,&R;#3Crs+,#"93C&: l(w )−l(w s ) = l(s ) = 1 U
l(s ) = 1
β∈R+ , l(wsβ )=l(w)−1
0
+
+
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α
w
w
1
w0 sα
2
0
0 α
α
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1 &:(#5M$6&:()$ β ;)+ Σ(T, b) 7
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< β ∨ , $α >
E = X(T ) ⊗
β ∨ ($α )
&
$α
+,"9$SO—uvB,O4B,rs+,I$
β ∨ ($α ) = δαβ
&
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K3#&:()"_$65'!;)(#3C"9&:#"T$'&:(K$_*,+o*,3;02#"\)&!$'59+n*,&:I$'+gJR$'+!7{#&!(#"œi[!&:#"_$'&:(K$\;Hu°!= &:59;
δαβ
$α1 = $1 = 2α1 + α2 ,
$α2 = $2 = 3α1 + 2α2 ,
,* &!rlrs+n&:1e+A()$23C"6BArl+AI$_OC+o*@!O4*,(#O4+,5T+A¬()$63COC34"'!I$TO4+a`!34$_DE(#+ α ($ ) = δ +A$\OC+a`!3G$
DE(#+ α (α ) = −3 7 α (α ) = −1 ^c*,&:rsrs+Œ#&!(#"O—[email protected][:&:)"‰;#B > W *@!O4*,(#O4BŒ1 &:(#5œa`!=#593CDE(#+,5
OC+<;#3C!P:56!rlrs+m;#+ ;#+ G +,›"6+,*L$'34&: U ³}f+A$ α (α ) = 2 1#()3C"6DE(#+ s (α ) =
•56+L$'56&!()[:+l10!5m23COCO4+,(#59"n*,+A"m56BA"6(#OG$/2$6"Œ;02#" ±v¸S&:(7h9 < U ³ 1 U
−α = α − α (α )α UZ©
1 ³A´ j]#"9(#34$6+!70#&!(#"‰@[!&:#"_;#B¨> [R(ªDE(#+
U
W
∨
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2
2
∨
i
i
i
∨
i
j
ij
1
∨
i
i
αi
i
i
∆(T, b)+ = {α1 , α2 , α1 + α2 , 2α1 + α2 , 3α1 + α2 , 3α1 + 2α2 }
+A$œDE(#+qOC+A(#5TB,*A5634$6(#56+n59B,;#()34$'+o+A"9$
sα1 +α2 = sα2 sα1 sα2 ,
s3α1 +α2 = sα1 sα2 sα1 ,
s2α1 +α2 = sα1 sα2 sα1 sα2 sα1 ,
s3α1 +2α2 = sα2 sα1 sα2 sα1 sα2 .
j])™07{*A&:rlrs+
7#&:ª<;)&:#*
w0 = (sα1 sα2 )3 = (sα2 sα1 )3
Xw0 sα1 = Xsα2 sα1 sα2 sα1 sα2 = Xs3α1 +2α2 = Xs$2 = h1
+A$
Xw0 sα2 = Xsα1 sα2 sα1 sα2 sα1 = Xs2α1 +α2 = Xs$1 = g 1 .
d&}h!&:#"‹rl!34E$6+,02E$l*,&!rlrs+,I$†!1#1#O43CDE(#+,5s*,+L$6$6+¬ac&:56rw(#OC+27˜1 &:(#5s*,+AOŠš)&:(#"‹2OCOC&!#"
#; +L[:&:345w*@2OC*,()OC+,5 β ($ ) +L$ β ($ ) 1e&!(#5Œ$6&:()$'+A"wO4+,"w56!*,34#+,"Œ1e&!"634$634[!+," β ;)(•P:56&!(#1e+ U
j]·+AFe+A$,7{*,&:rsrl+oO‘u°!1)1#OC34*@2$63C&: β 7→ β Hux+A"9$1#!"\O43C#B,!3C59+!7K*,&:##!34$656+ α +L$ α ux+A"9$
10!"o"9( ?s"'!I$ U Ob+A"9$o$'&!()$'+Lac&:3C"= &:k;#+<#&!$6+,5nDE(#+s10!5n;#BL™0#3G$'3C&! &:„ $6&:()$o;#+%rst,rs+
1 &:(#5M$6&:()$'+o56!*,34#+ β U
β (β) = 2
&:(#"œ!O4OC&:)"T;#&:#*q10!5_+LJK+Arl1#O4+o*@!O4*,(#O4+,5
‰!#"œOŠ%=0!"9+ {α , α } 7
OC+A"_3C§ I[:&:O4()$'34&:#" s +A$ s &:I$\1 &:(#5Trl2$6563C*A+ (α + α ) ($ ) U
∨
∨
1
2
∨
∨
1
∨
2
∨
1
α1
+A$
2
∨
α2
−1 3
M (sα1 ) = M1 =
0 1
1
0
M (sα2 ) = M2 =
.
1 −1
1
1
2
!¶
U – ##+,!(;#+%«­X®¯ ;#+
2
\Q,"œOC&:59",7#*A&:rlrs+
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sα1 +α2 = sα2 sα1 sα2
)&:(#"œ@[:&:#"
2 −3
M (sα1 +α2 ) = M2 M1 M2 =
1 −2
X (α2 )
2 −3
2
sα1 +α2 ($1 ) =
·
1 −2
1
1
.
=
0
A
!5'!O4OCQAOC+,rs+,I$,7)#&:()"œ"[email protected][:&:#"œDE(#+q10!5T;#BL™0#34$63C&:
sα1 +α2 ($1 ) = $1 − (α1 + α2 )∨ ($1 )(α1 + α2 )
= 2α1 + α2 − (α1 + α2 )∨ ($1 )(α1 + α2 )
= (2 − (α1 + α2 )∨ ($1 ))α1 + (1 − (α1 + α2 )∨ ($1 ))α2
*A+oDR()310!5_3C;#+AI$'34™#*@2$63C&:[email protected][:+A*ŒO4+n56BA"6(#OG$/2$_1#59B,*AB,;#+AE$\#&:()"œ;)&:##+
(α1 + α2 )∨ ($1 ) = 1.
]j 1#56&R*,BA;0!I$œ;#+oOŠmrst,rs+‰a`IHA&:ª1 &:(#5M$6&:()$6+,"_OC+A"T5'!*A3C#+A",70#&:()"M$'56&!()[:&:)"œ™0#!OC+Arl+AE$
DE(#+
α1∨ ($1 ) = 1,
α1∨ ($2 ) = 0,
α2∨ ($1 ) = 0,
α2∨ ($2 ) = 1,
(α1 + α2 )∨ ($1 ) = 1,
(α1 + α2 )∨ ($2 ) = 3,
(2α1 + α2 )∨ ($1 ) = 2,
(2α1 + α2 )∨ ($2 ) = 3,
(3α1 + α2 )∨ ($1 ) = 1,
(3α1 + α2 )∨ ($2 ) = 1,
(3α1 + 2α2 )∨ ($1 ) = 1,
(3α1 + 2α2 )∨ ($2 ) = 2.
w ux!()[email protected]#&:(#"<!O4OC&:)"[email protected][:&:345m= +,"6&!3C;#+ ;#BL$'+,59rl34#+,5wDR()+,OCO4+,"%"9&:I$<OC+,"w5'!*A3C#+A"
1 &:"934$'3G[:+A" β $'+AOCOC+A"‰DE(#+ l(ws ) = l(w) − 1 +L$n*,+27H1 &:(#5‰$6&:(#"‰O4+," w 1e&!"6"634=#OC+A",7 W "'@[!&:3C5
O4+,"M;)3GJ†5'!*A3C#+A"M;#BA™##3C"9"'!I$œOC+A" g +A$TO4+," h U A !5S+LJK+,rs1#OC+27#1 &:(#5M*,!OC*A(#OC+A5 (h ) 7)34OH#&:()"
a`2()$T$'59&:()[:+A5_OC+,"_56!*,34#+," β $6+,O4OC+,"SDR()+
β
i
i
1 2
l(s3α1 +2α2 sβ ) = 4.
OS+,"9$s=#34+,BA[R34;#+,I$<DE(#+-#&:(#"<1 &:()[!&:#"<;uv&:56+A"s+L$<;#B¨> W BA*@!5p$'+,5
β = 3α1 + 2α2
U j]
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K³
2
X (α2 )
*@2OC*,()OŠ!I$T$'&!(#"_OC+,"_2()$'59+,"_1#56&R;#(#3G$'"T1 &:"6"93C=#O4+,"T#&!(#"_&:=)$'+A#&:#"
l(s3α1 +2α2 sα1 ) = l(sα2 sα1 sα2 sα1 sα2 · sα1 )
= l((sα2 sα1 )3 ) = 6
l(s3α1 +2α2 sα2 ) = l(sα2 sα1 sα2 sα1 sα2 · sα2 )
= l((sα2 sα1 )2 ) = 4
l(s3α1 +2α2 sα1 +α2 ) = l(sα2 sα1 sα2 sα1 sα2 · sα2 sα1 sα2 )
= l(sα2 sα1 ) = 2
l(s3α1 +2α2 s2α1 +α2 ) = l(sα2 sα1 sα2 sα1 sα2 · sα1 sα2 sα1 sα2 sα1 )
= l((sα2 sα1 )5 ) = l(sα1 sα2 ) = 2
l(s3α1 +2α2 s3α1 +α2 ) = l(sα2 sα1 sα2 sα1 sα2 · sα1 sα2 sα1 )
= l((sα2 sα1 )4 ) = l((sα1 sα2 )2 ) = 4.
– 3C)"63—7K#&:(#"_&:=K$'+,)&:#"
(
β = α2
l(s3α1 +2α2 sβ ) = 4 ⇐⇒
β = 3α1 + α2 .
j] 1#56&R*,BA;0!I$_;#+OCŒrst,rs+\a`IHA&:‹1 &:(#5S$6&:(#"MO4+,"S!()$'59+,"MBAOCBArl+AE$6"M#&:()"S$'59&:()[:&!#"TDE(#+
l((sα1 sα2 )2 sβ ) = 3 ⇐⇒
(
β = α2
β = 2α1 + α2
l(sα1 +α2 sβ ) = 2 ⇐⇒
(
β = α2
β = 3α1 + 2α2
l(sα1 sα2 sβ ) = 1 ⇐⇒
(
β = α2
β = α1 + α2
l(sα2 sβ ) = 0 ⇐⇒ β = α2
U – ##+,!(;#+%«­X®¯ ;#+
2 +L$
X (α2 )
(
β = α1
l(s2α1 +α2 sβ ) = 4 ⇐⇒
β = α1 + α2
(
β = α1
l((sα1 sα2 ) sβ ) = 3 ⇐⇒
β = 3α1 + 2α2
2
(
β = α1
l(s3α1 +α2 sβ ) = 2 ⇐⇒
β = 2α1 + α2
(
β = α1
l(sα1 sα2 sβ ) = 1 ⇐⇒
β = 3α1 + α2
l(sα1 sβ ) = 0 ⇐⇒ β = α1
+A"%*@!O4*,(#O4"w1#59B,OC34rl340!3456+,"nBL$/!=)OC3C"A7€#&:(#"m1 &:()[!&:#"mrl!34E$6+,02E$w+AFe+,*L$'(#+A5m$'&:(#"ŒOC+A"
*,!OC*A” (#OC"TO43C=#59+,rs+,I$ U)§ &:()"\2OCOC&!#"M*,&!rlrs+,#*A+,5T1025œ*,!OC*A(#OC+A5MOC+A"_1#(#3C"9"'!)*,+,"œ;#+ g
1 (g 1 )2 = Xw0 sα2 · Xs2α1 +α2
= α1∨ ($2 ) X(sα1 sα2 )2 + (α1 + α2 )∨ ($2 ) X(sα2 sα1 )2
= 0 + 3 X(sα2 sα1 )2 = 3g 2
i
h
(g 1 )3 = 3g 1 g 2 = 3 Xw0 sα2 · Xs(α2 sα1 )2
= 3(α1∨ ($2 ) Xsα2 sα1 sα2 + (3α1 + 2α2 )∨ ($2 ) Xsα1 sα2 sα1 )
= 3(0 + 2 Xs3α1 +α2 ) = 6g 3
(g 1 )4 = 3g 1 g 3 = 6 Xw0 sα2 · Xs3α1 +α2
= 6(α1∨ ($2 ) Xsα1 sα2 + (2α1 + α2 )∨ ($2 ) Xsα2 sα1 )
= 6(0 + 3 Xsα2 sα1 ) = 18g 4
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
2
X (α2 )
(g 1 )5 = 18g 1 g 4 = 18 Xw0 sα2 · Xsα2 sα1 (g 1 )5
= 18(α1∨ ($2 ) Xsα2 + (3α1 + α2 )∨ ($2 ) Xsα1 )
= 18(0 + Xsα1 ) = 18g 5
(g 1 )6 = 18g 1 g 5 = 18 Xw0 sα2 · Xsα1
= 18(α1∨ ($2 ) [Xid ])
= 18(0) = 0.
§ &:(#"_*,!OC*A(#OC&:)"Mr†23CI$'+A0!I$_OC+A"œ1)(#3C"9"'!#*A+,"œ;#+ h U
1
(h1 )2 = Xw0 sα1 · Xs3α2 +2α1
= α2∨ ($1 ) X(sα2 sα1 )2 + (3α1 + α2 )∨ ($1 ) X(sα1 sα2 )2
= 0 + X(sα1 sα2 )2 = h2
i
h
(h ) = h h = Xw0 sα1 · Xs(α1 sα2 )2
= α2∨ ($1 ) Xsα1 sα2 sα1 + (2α1 + α2 )∨ ($1 ) Xsα2 sα1 sα2
= 0 + 2 Xsα1 +α2 = 2h3
1 3
1 2
(h1 )4 = 2h1 h3 = 2 Xw0 sα1 · Xsα1 +α2
= 2(α2∨ ($1 ) Xsα2 sα1 + (3α1 + 2α2 )∨ ($1 ) Xsα1 sα2 )
= 2(0 + Xsα1 sα2 ) = 2h4
(h1 )5 = 2h1 h4 = 2 Xw0 sα1 · Xsα1 sα2
= 2(α1∨ ($2 ) Xsα1 + (α1 + α2 )∨ ($2 ) Xsα2 )
= 2(0 + Xsα2 ) = 2h5
(h1 )6 = 2h1 h5 = 2 Xw0 sα1 · Xsα2
= 2(α2∨ ($1 ) [Xid ]) = 2(0) = 0.
&:(#"†!OCO4&:#"%rl!3CI$6+,0!I$l*@!O4*,(#O4+,5sOC+1#56&R;#(#3G$s;#+ g @[!+,*·O4+," h USV +OC+,*L$'+A(#5†59+ N
§
rl!56DE(#+A5'<DR()+q;#!#"_OC+n*,!"\1#!59$63C*,()OC3C+A5S;#+ g h 7##&!(#"‰@[!&:#"_OC+n*/.#&!3GJ-;#+o1#5934[R3CO4B,P:34+,5YOC+
1#59&K;#()34$b+A%()$'34OC34"'!I$XOŠ\ac&!56rm(#[email protected][!+,* g &:(<@[:+A* h U *A3—7?#&:(#"]@[!&:#"]*/.#&:34"63K;ux(K$'3CO43C"9+,5XOŠ
ac&:59rm(#OC+n;)+<«‰­ @[:+A* g r†23C"MO4+nOC+,*L$'+A(#5_+,"9$\OC34=#56+‰;#+n[:B,5934™0+A5_DR()+ŒO4+n56B,"9(#O4$'2$
+,"p$‰O4+rst,rs+n+,()$'34OC3C"6!I$ h U
V +A"_*@!O4*,(#O4"T#&:(#"_;#&:)#+,I$
1
1 1
1
1
1
1
i
U – ##+,!(;#+%«­X®¯ ;#+
2=<
X (α2 )
g 1 h1 = Xw0 sα2 · Xs3α1 +2α2
= α2∨ ($2 ) X(sα2 sα1 )2 + (3α1 + α2 )∨ ($2 ) X(sα1 sα2 )2
= X(sα2 sα1 )2 + X(sα1 sα2 )2 = g 2 + h2
g 1 h2 = Xw0 sα2 · X(sα1 sα2 )2
= α2∨ ($2 ) Xsα1 sα2 sα1 + (2α1 + α2 )∨ ($2 ) Xsα2 sα1 sα2
= Xs3α1 +α2 + 3 Xsα1 +α2 = g 3 + 3h3
g 1 h3 = Xw0 sα2 · Xsα1 +α2
= α2∨ ($2 ) Xsα2 sα1 + (3α1 + 2α2 )∨ ($2 ) Xsα1 sα2
= Xsα2 sα1 + 2 Xsα1 sα2 = g 4 + 2h4
g 1 h4 = Xw0 sα2 · Xsα1 sα2 = α2∨ ($2 ) Xsα1 + (α1 + α2 )∨ ($2 ) Xsα2
= Xsα1 + 3 Xsα2 = g 5 + 3h5
g 1 h5 = Xw0 sα2 · Xsα2 = α2∨ ($2 ) [Xid ] = [Xid ] = g 6
j])™0H7Y#&!(#"lHu°@[:&!#"†1#O4(#"sr†!34I$'+,#!I$lDE(u *,!OC*A(#OC+A5%OC+-1#59&K;#()34$s;#+
BAOCBArl+AE$6" g U *,+A*,3#&!(#"\;)&:##+oOC+A"T56BA"6(#OG$/2$6"œ"9(#34[2W !I$6"
i
h1
@[:+,*ªOC+A"
h1 g 2 = Xw0 sα1 · X(sα2 sα1 )2
= α1∨ ($1 ) Xsα2 sα1 sα2 + (3α1 + 2α2 )∨ ($1 ) Xsα1 sα2 sα1
= Xsα1 +α2 + Xs3α1 +α2 = h3 + g 3
h1 g 3 = Xw0 sα1 · Xs3α1 +α2 = α1∨ ($1 ) Xsα1 sα2 + (2α1 + α2 )∨ ($1 ) Xsα2 sα1
= Xsα1 sα2 + 2 Xsα2 sα1 = h4 + 2g 4
h 1 g 4 = X w 0 s α1 · X s α2 s α1
= α1∨ ($1 ) Xsα2 + (3α1 + α2 )∨ ($1 ) Xsα1 = Xsα2 + Xsα1 = h5 + g 5
h1 g 5 = Xw0 sα1 · Xsα1 = α1∨ ($1 ) [Xid ] = [Xid ] = g 6
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
;#+
212
X (α2 )
jd56BA"6(#rsB!70OCw$/!=)OC+n;#+orm(#OG$'341#OC34*@2$63C&:s;#+o#&!$656+o"9hR"9$6Q,rs+Œ;)+qP!B,#BA5'2$6+,(#59"m^cOC3C=)56+,"'f
+A"9$
CH (X (α , α ))
∗
1
(g 1 )2
(g 1 )3
(g 1 )4
(g 1 )5
(g 1 )6
2
= 3g 2
= 6g 3
= 18g 4
= 18g 5
=0
(h1 )2
(h1 )3
(h1 )4
(h1 )5
(h1 )6
= h2
= 2h3
= 2h4
= 2h5
=0
h1 g 2
h1 g 3
h1 g 4
h1 g 5
h1 g 1
= g 3 + h3
= 2g 4 + h4
= g 5 + h5
= g6
= h2 + g 2
g 1 h2
g 1 h3
g 1 h4
g 1 h5
= g 3 + 3h3
= g 4 + 2h4
= g 5 + 3h5
= g6
&:rlrs+%#&:()"ŒO‘u°@[:&!#"o1#56BA*,B,;)+,rsrl+AE$m!##&!#*,Bs#&:(#"Œ!O4OC&:#"‰r†!34I$'+,#!I$Œ*,!OC*A(#OC+A5
”
;#+o;#+A(KJ-a`IH,&!#"_;#34FeB,59+,I$'+A"œ*A+,5p$/!34#"_3CI[2!563C!I$'"M;#+ CH (X (α , α )) U
∗
"(( 1
2
CH∗ (X (α1, α2))
&:(#"Z;#B,"93CP:)&:#"8$6&:()$b;Hu°!= &:59;%10!5 OC+Y"6&:(#" !)#[email protected]!(%;)+ (X (α , α )) +,#P:+A#;#59B
10!5o§ O4+,"oB,O4B,rs+,I$'"q;#(„P:56&:()1e+ CH (X (αA , α )) 7N *!uv+,"9$ N¨W?N ;#3C59+%CH
10!5qOC+A"q1)(#3C"9"'!#*A+,"q;#+ g
+A$ª;#+ h U\§ &:(#"*,&:)"63C;)Q,56&!#"¬+A#"6(#3G$'+ A OC+„P:56&!(#1e+„!= B,O43C+A O43C=#59+ +,#P:+A#;#59Bš"6()5 G
10!5oO4+,"oB,O4B,rs+,I$'"q;#+s*,&R;#3Crs+,#"93C&: ;#+ 7*2ux+A"9$ ;#3C59+s+A#P:+A#;#56Bs10!5oO4+,"o1e&!O4hR ,:rs+,"
.#&:rs&:P:QA#+,"X;#+Y;#+,P:59B i +, g +A$ h =U A i !5X*,&:A#"9B,DE(#+AE$N¨W?,N7:;ux!1#59Q,"8OC_$'!=#O4+˜;)+Yrm()O4$'341#OC34*@2$63C&:
1#59B,*,BA;#+,rsrs+,I$‰*,!OC*A(#OCBA+!7#)&:(#"_&:=)$6+,#&:)"œDE(#+
∗
1
1
1
2
1
2
1
i
1
1
A1 = gr(g 1 , h1 )
A2 = gr((g 1 )2 , g 1 h1 , (h1 )2 )
= gr(3g 2 , g 2 + h2 , h2 )
A3 = gr((g 1 )3 , (g 1 )2 h1 , g 1 (h1 )2 , (h1 )3 )
= gr(6g 3 , 3(g 3 + h3 ), g 3 + 3h3 , 2h3 )
A4 = gr((g 1 )4 , (g 1 )3 h1 , (g 1 )2 (h1 )2 , g 1 (h1 )3 , (h1 )4 )
= gr(18g 4 , 12g 4 + 6h4 , 6(g 4 + h4 ), 2g 4 + 4h4 , 2h4 )
A5 = gr((g 1 )5 , (g 1 )4 h1 , (g 1 )3 (h1 )2 , (g 1 )2 (h1 )3 , g 1 (h1 )4 , (h1 )5 )
= gr(18g 5 , 18(g 5 + h5 ), 12g 5 + 18h5 , 6g 5 + 12h5 , 2g 5 + 6h5 , 2h5 )
A6 = gr((g 1 )6 , (g 1 )5 h1 , (g 1 )4 (h1 )2 , (g 1 )3 (h1 )3 , (g 1 )2 (h1 )4 , g 1 (h1 )5 , (h1 )6 )
= gr(0, 18g 6, 18g 6 , 12g 6 , 6g 6 , 2g 6 , 0)
gr(E)
G
E
Ai
CHi (X (α1 , α2 )) = gr(g i , hi )
i
{1, . . . , 6} A
i
{1, . . . , 6}
Ai
CH∗ (X (α1 , α2 ))
(CHi (X (α1 , α2 )) : Ai )
& O W +,)*,&:59+!7‰*A&:rsrl+„;0!#"¬OŠ "9+,*L$'3C&! U ³:7\)&:(#"·;#BA"634P:#&:#"¬10!5
OC+›P:56&!(#1e+
!= B,O43C+, O43C=#59+š+A#P:+,);#56B10!5 O4+,"·BAOCBArl+AE$6" ;#+ "9(#5 U Om+,"p$ $'&:()$k;Hu°!= &:59; *,OŠ23C5
DE(#+ +,"p$‹(#"6&:(#" N P:59&:(#1 +ª;)+
1 &:(#5<$'&!()$ ;#+¬O‘ux+A N
"6+Arm=#O4+
!
Œ
5
A
*
:
&
#"
9
,
B
E
D
#
(
,
+
I$
,
8
7
#&
:
)
(
Œ
"
1
:
&
)
(
!
[
:
&
#"
,
X
7
1
:
&
#
(
o
5
'
$
:
&
)
(
$
;#+
7*@!O N
U
*,()OC+,5‰O—uv3C#;#34*,+q;#+ ;0!#"
DE(#+%#&!(#"#&!$6&:#"
U
2
U – ##+,!(;#+%«­X®¯ ;#+ X (α )
+s"6&!E$w*,+A"Œ#&!rm=#56+A"Œ+AI$'3C+A56"qDE(#3Y*A&:#"p$'34$6(#+,I$ŒOC+A"Œ34E[2!593Š!I$6"n56+,*/.)+,56*/.)B," O˜+A"9$Œ*,OC!3C5
DE” (#+ A = CH (X (α , α )) U wu°2()$'59+1#!59$,7 h +A"9$s;0!#" A +L$ g + h U !()"6"63‘7b;#&:)*
21#10!5p$'3C+AI$ W A +A$ 1#!5¬"9(#34$6+š#&:()"·@[:&:)" B,PI!O4+,rs+,I$ A = CH (X (α , α )) U
g
&:(#5TO4+,"_!()$656+A",70#&:()"\2OCOC&!#"S1#59&K*AB,;#+A5_10!5MP:BA#B,562$'+A(#56"_+L$\59+,OC2$'34&:#" +oP:56&:()1e+DE(#&
$6A 3C+AE$ CH (X (α , α )/A +A"9$¬+,#P:+A#;#59Bš10!5;#+A(KJ B,OCBArl+AI$'" g +A$ UKhV DE(#3n[:BA5634™#+,I$N
j]a`23C"'2E$mOŠ¬;#3GFHBA56+A#*,+-;#+A"%;#+,(KJ
+A$
2h = 0 = 6g
;)+,56)3CQ,59+,"o56+AOŠ2$63C&:)"3g#&:(#"+$6563h&:(K[:=&:#"o0 ;#=&!#*g<DR+()+ 3h2gU = 0 710!5n*A&:#"9B,DE(#+,I$wOC+,"nBAOCBArl+AE$6"
+A$ h "6&:I$o;uv&:59;#56+<;#+A(KJk+A$q*A&:rlrs+ 3g + 3h = 0 7H3COZ[R34+,I$oDE(#+ g = −h = h
g
1)(#3C"9DR()+ h +,"9$Œ(#„B,O4B,rs+,I$q;ux&!56;#59+s;)+,(KJ UA !5q"6()34$'+27#&:(#"o1 &:()[!&:#"q3C;#+AE$634™0+A5 g +L$
+A$\!34#"63
h
G
G
CH (X (α , α ))/A ' /2 .
&:(#5 CH (X (α , α ))/A 7€O—uv(#š;)+,"wP!B,#BA5'2$6+,(#59",7 h 7X+,"p$m;uv&:59;#56+s;#+,(RJ•+L$w1025Œ"9(#34$6+
A
O4+,"_!(K$'56+A"_56+,OC2$'34&:#"T;#+L[R3C+,)#+,I$
2
1
1
1
2
2
2
2
2
3
3
1
2
2
2
3
2
3
2
3
3
3
3
1
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
4
1
1
3
2
4
2
4
18g 4 = 12g 4 = 6g 4 = 2g 4 = 0.
\&:#*_OC+A"b;#+,(RJ<P:B,)B,5'?$'+,()56"˜"6&:I$d;uv&:56;)56+T;#+A(KJ<+A$˜#+T"9&:I$d10!"dOC34B,"b+AI$'56+_+A(KJe7I10!5b"9(#34$6+
G
G
CH4 (X (α1 , α2 ))/A4 ' ( /2 )2 .
]j 56+La`!3C"6!I$_rl&!$M1e&:()5œrs&!$TO4+nrltArl+n56!3C"9&:##+Arl+AI$_1e&:()5 CH (X (α , α ))/A 70#&:()"
$656&:(K[:&:#"_DE(#+
G
G
CH (X (α , α ))/A ' ( /2 ) .
j]l*,+_DE(#3{*,&:#*A+,59#+ (X (α , α ))/A 7I3CO#1 &:"6"9Q,;#+œ(#l"6+,()O P!B,#BA5'2$6+,(#5˜;uv&:56;#59+_;#+,(KJ
+L$œ10!5T*,&!#"6BADR()+,I$#&CH
!(#"œi[!&:#"
5
5
5
1
1
2
5
2
1
2
5
2
5
CH6 (X (α1 , α2 ))/A6 '
j]56B,"9(#rsB!70OC+A"_3CI[2!563C!I$'"M59+,*/.#+A56*/.#BA""6&:I$œ;#&:)*
G
G
/2 .
(CH1 (X (α1 , α2 )) : A1 ) = 1,
(CH2 (X (α1 , α2 )) : A2 ) = 1,
(CH3 (X (α1 , α2 )) : A3 ) = 2,
(CH4 (X (α1 , α2 )) : A4 ) = 4,
(CH5 (X (α1 , α2 )) : A5 ) = 4,
(CH6 (X (α1 , α2 )) : A6 ) = 2.
#O )+T#&:(#"]59+,"9$6+_1#OC(#"]rl!34E$6+,02E$8DE(u W *,!OC*A(#OC+A5 W #&:()[[email protected]!(s*,+A"d3C#;#34*,+A"b+,<()$63COC34"'!I$
OC‰"9$656(#*L$'(#59+T;#+ CH (X (α ))N rs&R;#(#OC+˜;#+ CH (X (α , α )) +A$ W *,&:rs10!59+,58O4+,"b59B,"9(#O4$'2$'" U
∗
2
∗
1
2
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
2
X (α2 )
CH∗(X (α2))
:& (#5_*A&:rlrs+,)*,+,5A70#&:(#"œ5'21#1e+AOC&:)"TDR()+ CH (X (α )) 1 &:"9"6Q,;)+w(#"6+A(#O€P:B,#BA5'2$6+,(#5
10!5_*A&K;#34rl+A#"634&: DE(#+o#&:(#"œ#&!$6&:#" h 1 &:(#5 i 10256*,&!(#5'!I$œO—uv+,)"6+,rw=#OC+ {0, . . . , 5} U j] N
"6()34$'+27e)&:(#"‰"[email protected][:&:#"‰DR()+ CH (X (α , α )) +,"p$n(# CH(X (α ))N rs&R;#(#OC+nO43C=#59+o;#+w56!#P 2
rl!3C"_1#O4(#"\1)56B,*A3C"9+,rs+,I$*A+,OŠs"634P:#34™#+ŒDE(u 3CO€!;)rl+L$‰1 &:(#5œ=0!"9+wO—uv+,)"6+,rw=#OC+ {1, ζ} & 1
;#BA"63CP!#+oO—uvB,OCBArl+AI$_#+,()$656+o+A$ ζ +,"p$‰(#B,O4B,rs+,I$_DE(#3[:BA563G™0+
A
∗
2
i
2
∗
1
2
2
ζ 2 − c1 h12 ζ + c2 h22 = 0
+A$˜& c ;)B,"634P:#+_OŠ i *,OŠ2"6"6+œ;#+n«‰­ ª;#(<™0=#59Bœ[!+,*L$'&:593C+,O{2"6"6&R*,34Bœ!(s™#=#56B_1#59&?>9+,*L$'34a
!( N ;#+,"9"6(#"S;#+ X (α ) UE– 3C)"63—7I1 &:(#5 i W [2!OC+A(#56"˜;0!)"YO‘ux+A#"6+Arm=#OC+ {1, . . . , 5} 7EO4+,"dP:56&!(#1e+A"
*,&:5956+,"91e&!#;#+,I$ W gr(h , ζh ) +L$ª1 &:(#5 O4+," 34#;#3C*A+," 0 +L$ 6 7M#&:(#"
CH (X (α , α ))
@[:&:)" (X (α , α )) ' CH (X (α )) 7€+A$ CH (X (α , α )) ' CH (X (α ))ζ U§ &:(#"
#&:()"œ;)CH
&:##&:)"\+A#"6(#3G$'+oDE(02$656+o#&:rm=)56+,"_+AE$63C+A56"_1e&!"634$634ac" l, m, n +A$ p $'+AOC"TDE(#+
ème
i
i
2
1
0
2
1
2
(h12 )2 = lh22 ,
Ai
i−1
2
6
i
2
0
2
(h12 )3 = lmh32 ,
1
2
(h12 )4 = lmnh42 ,
5
2
(h12 )5 = lmnph52 .
”+gJ)+,1)*,563K3CrsBA$'B,!"_I$b*A+A1 $6&:$6+n"6B2ac7!&:#&3C":()*,"b3H!+,-aO4OC&:c#"&:X#*A*,$6&!3Crl&: rs;)+Y+ 1)56B,*AB,;#+A+Arl$ rs+,I$8*,&:#"93C;#&:(#BA565_+A*,5M&:72rsO4+,rs"Z"6+,&:#*(#A"+,N 5,P:7 59&:(#1 +,"
N
(h )
ζ(h ) U A
1 i
2
#&:()"\@[!&:#"œ;#&:#*o=#34+,
j]#"6()34$'+27
1 i−1
2
A1 = gr(h12 , ζ)
(CH1 (X (α1 , α2 )) : A1 ) = 1.
A2 = gr((h12 )2 , h12 ζ, ζ 2 )
= gr(lh22 , h12 ζ, c1 h12 ζ − c2 h22 )
= gr(lh22 , c2 h22 , h12 ζ)
+A$œ*A&:rlrs+
#&:()"b+,%;#BA;#(#34"6&:#"]DE(#+SOC+,"8#&:(CH
rm=#59+,(X" l (α+L$ c, α"9&:))I$]: 1#A59+,)rs=3C+,591"8+AE$656+A"]+,(KJ UE” +M59B,"9(#O4$'2$b"9+,5'
;ux!3CO4OC+A(#56"œ$'56QA"oOŠ!59P:+,rs+,I$n+gJ)1)OC&:3G$'Bm;02#"oOC+,"‰*@!O4*,(#O4"n"6(#3G[!2E$6" U€” &!#*,+A560!I$qOC+%P!56&:(#1 +
7)#&:(#"\@[:&:#"
A
2
1
2
2
2
3
;uv&
A3 = gr((h12 )3 , (h12 )2 ζ, h12 ζ 2 , ζ 3 )
= gr((lmh32 , lh22 ζ, lc1 h22 ζ − mc2 h32 , (lc21 − c2 )h22 ζ − mc1 c2 h32 )
= gr(mh32 , h22 ζ)
(CH3 (X (α1 , α2 )) : A3 ) = m = 2.
U – ##+,!(;#+%«­X®¯ ;#+
2I0
§ &:(#"_*@2OC*,()OC&:#"Trl!34E$6+,02E$ A 7
X (α2 )
4
23C#"93
A4 = gr((h12 )4 , (h12 )3 ζ, (h12 )2 ζ 3 , h12 ζ 3 , ζ 4 )
= gr(2lnh42 , 2lh32 ζ, 2lc1 h32 ζ − 2nc2 h42 , 2(lc21 − c2 )h32 ζ − 2nc1 c2 h42 ,
2n 2
(lc1 − c2 )c2 h42 )
2c1 (lc21 − 2c2 )h32 ζ −
l
2n 2 4
4
3
= gr(2nh2 , 2h2 ζ,
c h)
l 2 2
(CH4 (X (α1 , α2 )) : A4 ) = 2 pgcd(2n,
2n
c2 ) = 4
l
2n
c
2n 'A
l 2
pgcd(n, nl c2 ) = 1
& pgcd(2n, c ) ;#B,"93CP:#+\OC+œ1#OC()"YP:56!#;‹*A&:rsrm(#srm(#O4$63C1#O4+_;#+
*A&:rsrl+ pgcd(2n, c ) = 2 7€#&!(#"[email protected][!&:#"m#BA*,+A"6"'23C56+Arl+AI$
&:(#"_*,!OC*A(#OC&:)"Tr†!34I$'+,#!I$
l = n U{§
2n
l 2
2n
l 2
+L$ U !5Y"6(#3G$'+27
L+ $%!34#"63
A5 = gr((h12 )5 , (h12 )4 ζ, (h12 )3 ζ 2 , (h12 )2 ζ 3 , h12 ζ 4 , ζ 5 )
= gr(2l2 ph52 , 2l2 h42 ζ, 2l2 c1 h42 ζ − 2lpc2 h52 , 2l(lc21 − c2 )h42 ζ − 2lpc1 c2 h52 ,
2lc1 (lc21 − 2c2 )h42 ζ − 2pc2 (lc21 − c2 )h52 ,
(2l2 c41 − 6lc21 c2 + 2c22 )h42 ζ − 2pc1 c2 (lc21 − 2c2 )h52 )
= gr(2ph52 , 2h42 ζ)
2 3C#"93
;Hux& p = 1 U j])™07{+,*@2OC*,(CH
()OŠ!I$ (X (α , α )) : A ) = 4p = 4
5
1
2
5
A6 = gr((h12 )6 , (h12 )5 ζ, (h12 )4 ζ 2 , (h12 )3 ζ 3 , (h12 )2 ζ 4 , h12 ζ 5 , ζ 6 )
= gr(0, 2l2 h52 ζ, 2l2 c1 h52 ζ, 2l(lc21 − c2 )h52 ζ, 2lc1 (lc21 − 2c2 )h52 ζ,
(2l2 c41 − 6lc21 c2 + 2c22 )h52 ζ, (2l2 c51 − 8lc31 c2 + 6c1 c22 )h52 ζ)
= gr(2h52 ζ),
)&:(#"T$'59&:()[:&!#"\=)3C+,-DE(#+
(CH6 (X (α1 , α2 )) : A6 ) = 2.
²Z3C0!O4+,rs+,[email protected]Šw$'!=#OC+n;)+qrw(#O4$63C1#O43C*@?$'3C&!·^[email protected]˜;#+
(h12 )2
(h12 )3
(h12 )4
(h12 )5
= lh22 ,
= 2lh32 ,
= 2l2 h42 ,
= 2l2 h52 .
CH∗ (X (α2 ))
+,"9$
‰BA*,&:rs1e&!"634$63C&: *A+,O4OC(#OC!3C59+‰;#+
2I:
X (α2 )
7¬!O4P:56Bl$'&:(#"%)&:"%+AFe&:59$,78O4+‹#&:rm=)56+
59+,"p$'+ ;#&:)*Œ>9(#"6DE(#+-O 3C#;#BL$'+A56rs3C#B &!(#59$'[email protected]
#&:()"l1 &:()[!&:#"<"634P:0!O4+,5mDE(#+O—uv3Crl!P:+ll;#+ h ;0!#" CH (X (αW , α )) 1#!5%O—uxU!1#A 1#O43C*,2$'34&:
7Z!1#1)OC3C*,2$'34&:„3C#;#()34$'+l10!5ŒOŠ™#=#5'2$63C&:›1#56&}>9+,* N
pr : CH (X (α ) → CH (X (α , α ))
$'3G[:+270+,"9$œ(#)+q*A&:rm=#340!34"6&: ;#+ g +A$ h U j]ª;ux!()$656+n$'+A56rs+,"A7
∗
∗
∗
2
∗
1
2
1
1
1
2
2
1
pr ∗ (h12 ) = ag 1 + bh1
@[:+A* +A$ ;0!)" Gq7 1#56+Arl34+,59"‰+,I$656+m+A(KJ·1)(#3C"9DR()+%O—uvB,OCBArl+AI$
1e+A()$ntL$'56+Œ*/.#&:34"63
*,&!rlrsa+n(#;b #+,"_P:BA#B,562$'+A(#56"œ;#+ CH (X (α , α )) U ‰+o1#OC(#"A7##&:pr()"‰([email protected][:&!)#"
∗
1
1
1
2
2
lpr ∗ (h22 ) = (pr ∗ (h12 ))2 = (3a2 + 2ab)g 1 + (b2 + 2ab)h2 ,
*,+SDR()3K34rl1 &:"6+˜!(RJw)&:rm=#59+," +L$ ;uvtA$656+S1#!3C59"Z"93 O—uv+,"[email protected]?*,+YDR()3K+AE$656+A5'!3G$8+Am*,&:I$65'!;#34*
$'34&: @[:+A*nOC+œa`!34$SDR()+ a +A$ b "6a&:I$_b1#59+,rs3C+A56"S+AI$'56+n+Al (KJ U A !5M*,&!#"6BADR()+,[email protected] l +,"9$M#B,*A+,"6"6!3 NN
56+Arl+AI$S34rl1023C5]+A$Y*,+_a`!3G$Y)&:(#"˜"6+,56Œ$'59Q,"˜()$'34OC+_10!5˜OŠn"9(#34$6+\1 &:(#5˜BA$'!=#OC345œ^`OCq562$'34&:0!O434$'B
;#[email protected]—uv3C"6&!rl&:591#.#34"6rs+˜rs&!$634[R3CDE(#+ U j]<+AFe+A$,7!=#34+,%DE(#[email protected]!I$d102"8$'&!$'!OC+Arl+AI$b;#BL$'+,59rl34#B
OŠœ"9$656(#*L$'(#59+M;#+ CH (X (α )) 7?#&:(#"X[:+,5956&:)"]DR()+MOC+A"Z34)ac&:56rl2$63C&:#"€;#&!E$8#&:(#"Z;#3C"91e&!"6&:#"
rl!3CI$'+A0!I$n"9+,56&!E$o"9( ?s"'!I$6+,"q1e&:()5BA$'!=#OC345‰O‘ux34"6&:rs&:561).#3C"9rl+ors&!$'3G[K34DE(#+w+AE$656+ X (α )
+A$ X (α ) +,-$'&!()$'+nP:BA#B,56!OC3G$'B U
&:(#"ma`23C"6&!#"m56+Ar†256DE(#+,5m!(OC+A*A$'+A(#5%DE(#+ OŠ·DE(02;#5634DR()+
·+,O4OC+ !()"6"63S;0!#"
OC+w*@§!"n;#BA1#OC&ih!B!7(# "9+,(#O]P:BA#B,562$'+A(#5n10!5‰*,&R;#3Crs+,#"93C&:¬DR()+s)&X:(#(α"o!)O4OC&:#"œ#&!$6B h 7H& i
;#BA*,563G$˜O—uv+,#"9+,rm=#O4+ {0, . . . , 5} U &:()$6+Aac&:34"]OŠ‰$/2=#OC+T;#+_rm()O4$'341#OC34*@2$63C&:q;#+_"6+A"YP!B,#BA5'2$6+,(#59"
+,"p$
∗
2
1
2
1
i
1
(h11 )2
(h11 )3
(h11 )4
(h11 )5
= h21 ,
= 2h31 ,
= 2h41 ,
= 2h51 .
25œ*A&:#"9B,DE(#+,I$,7H3COe+,"p$‰1 &:"6"93C=#O4+nDE(#+ (X (α )) +A$ CH (X (α )) "6&!3C+,I$_34"6&:rs&:561).#+,"
6"A 3 l = 1 U (#&:3TDE(uv3COM+A "9&:34$,78#&:()"lCH
2OCOC&!#"w[:&!3C5%DE(#+-OC+A"srs&!$634ac"m2"6"6&R*,34B," W *,+A"l;#+A(KJ
[2!5634BA$6B,"n"6&!E$q34"6&:rs&:561).#+,"DE(#+<O4+,"o!#)[email protected]!(KJ OC+m"9&:3C+AE$q&:( )&: U€V +%OC+A*A$'+A(#5n"6&!(#.0!3G$/!I$
1#O4(#";#+m;)BA$/23COC"\"6(#5‰OŠsa`=H,&: ;#&!E$&: &:=)$63C+,I$‰*,+,"n59+,OC2$'34&:#"œ+,I$'59+%OC+,"\P:B,)B,5'?$'+,()56"n;#+
OŠ%DE(0!;)563CDE(#+ X (α ) 1 &:(#595')7#1#!5_+LJK+,rs1#OC+27#"6+q5'21#1e&!59$'+A5 W ±xµo!5 ::¶K7 9 U ³A´ U
∗
1
1
∗
2
¶
U – ##+,!(;#+%«­X®¯ ;#+
X (α2 )
‹ ª
!#"$'&!()$€*,+d*/.0!1)34$'59+!7 k ;#BA"634P:#+d#&!$656+d*,&:591#"€;#+d=0!"9+˜;)+˜*,!5'2*A$'BA5634"9$'34DE(#+];#34FeB,59+,I$'+
;#+o;#+A(KJ-+A$ (##+o;#+n"6+A"\*AO-,!$6(#56+A"_!OCP!B,=#593CDE(#+," U#V +,"T[2!593CBA$6B," X (α ) +L$ X (α ) "9+,59&:[email protected]
"6()34[2!I$SO4+_*,&:I$'+gJR$'+!7R;#BA1#OC&ih:BA+,"Y&:(l)&:†"9(#5 k U ‰+‰1)OC(#"A7I;0!#"˜*,+œ*/.0!1#3G$'59+!7 Hom(X, Y )
;#BA"63CP!#+,56s1025 CH (X × Y ) `ªy z}| ~€] ‚ Rs:|{z?z}R,]F| ( ‚ ’ (X:R ‚  ‚ Ey z}ž U
!#"l(##+¬1)56+,rs3CQA56+-10!5p$'34+!7d#&:(#"l3CI$'59&K;)(#3C59&:#"<OŠ„:’)ŽižEy{|{z2“c ‚ R:|{z}z}KA]$| ( )
‚ ’ (Z:R‹DE(#3˜+,"p$mOŠ*@2$6B,P:&!563C+srs&!$'3G[K34DE(#+si[!+,*‹OC!DE(#+,O4OC+<#&:()"m!O4OC&:#"n$65'@[2!3CO4OC+,5 U€” +A$9$'+
"6+A*A$63C&:kuv+,"p$Œ+A„!(#*,()„*,!"n&:593CP:340!OC+q+A$oOC†1#593C#*A3C10!O4+wrs&!$'3G[22$'34&:¬;#+%*A+A$o+gJ)1 &:"9Bs+A"9$
;uv3CI$'59&K;)(#3C59+ #&:" #&!$'2$'34&:#" U j]#"6(#3G$'+27T;0!#"-(##+k"9+,*,&!#;#+„10!59$63C+27M#&:(#"-BL$/!=#O43C"9"6&:#"A7
1 &:(#5qOŠ‹[2!5634BA$'B X (α ) 7()š$'.#BA&:56QArl+<;uv3C"9&:rs&:561#.)3C"6rs+<!02OC&:P:()+ W *,+AOC(#3dDE(uxªBL$/!=#O43
\® š1 &:(#5œOC+A"\DE(0!;)563CDE(#+A" j])™07 +,¬$656&:34"634Q,rs+n10!59$63C+270#&:(#"\BA$/2=#OC34"6"6&!#"\;#!#"œOC+
*@2"Y;#BA1#OC&ih!B!7E(#s3C"6&!rl&:591#.#34"6rs+MU rl&2$'34[R34DR()+T+,I$'59+ X (α ) +A$ X (α ) @[2!I$Y;)+\1#59&:()[!+,5˜OŠ
5'?$'3C&!0!OC3G$'BT;#+\OŠo*,&:5956+A"61 &:#;0!)*,[email protected]!O43C"'2E$˜*A+A$M34"6&:rs&:591#.#3C"9rl+27I5'2$63C&:#!OC3G$'BMDE(#3 BA$'!=#OC3G$
!34#"63O‘ux34"6&:rs&:591#.#3C"9rl+\+,-$'&:(K$'+oP:B,)B,5'2OC34$6B U
&:(#5<(# +LJK1 &:"6B*,&!rl1#O4+A$<"9(#5<OC+,"%*A&:5659+,"91e&:);0!#*A+,",7˜#&:()"s*A&:#"9+,3CO4OC&:)"%OŠ OC+A*A$'()56+
;#+ A±x²Y³!7 [email protected]³ KU ³,´ U|A &!(#5-OŠ*@2$6B,P:&!563C+ ;)+,"*,&:5956+,"91e&!#;0!#*A+,"ª+, 1#!59$63C*,()OC3C+A5,7YOC+kOC+A*A$'+A(#5
1 &:(#5656"6+T59BAacB,59+,5 W 8± 7¬! 0?´ U jd<&:(K$'56+27:#&:()"]#&:(#"]"9&:rlrs+,"8B,PI!O4+,rs+,I$b$656Q,"]OC!56P!+,rs+,I$
3C)"61#3456Bn;#+s± œ&:" ::¶2´+L$Œ±xµo!56¶:¶?´ U
1
2
dim Y
2
1
" ! S
2
7 OC%*@2$6B,P:&:593C+o;#+A"
§[2!56&:34BA(#$'"œBA"Œ;#BAO43C"6"93C"6P!+,#&"A:7€#"*,_&:rs3C*A1#3OC+AQL$\$'+A1e"Œ&!rl(#5_!OC3C+o"n56)&+A:"9•$'+q#B;#A+q*,+,*,"9+q"'!*/34.056+A!rl1#3G+A$'E$59+Œw10*,&:2)5 #+Var(k)
LJK+," Ub§ &:(#"w*A&:#"93C;#BA56&:#"
kN
B,P:!OC+Arl+AE$bDR()+TO—uv+,#"9+,rm=#O4+S[R3C;#+ ∅ +,"9$b(#<B,OCBArl+AI$];#+ Var(k) U:§ &:(#"[email protected][:&!#"];#B¨> W "93CP:#!OCB
DE(#+˜1e&:()5 X ;0!)" Var(k) 7 CH (X)
;#BA"63CP!#+dO—ux!##+,!(Œ;)+_«‰­X®¯;#+ X +L$XDR()+!7}=#3C+AŒDE(#+
*,+AOŠ"9&:34$8(#<!=#(#"];)+MOŠ!)PI!P:+272#&!(#"b1#59BAacB,59&:#"]10256OC+A58;#+T*LhR*,OC+S1#OC()$ ,!$bDE(#+T;)+M*,OC!"6"9+,"];#+
*AhR*AOC+," U wu°2()$'59+q1#!59$,7##&:(#"œ!O4OC&:)"M56+L[:+,)3C5_3C*A3"6(#5TO‘u°2##[email protected](;)+<«‰­X®¯ CH (X) ;uv(##+
[2!5634BA$6B X 1 &:(#5T34E$656&R;#(#3456+OCm#&!$'34&:-;#+q:|{z?z}R,]F| ( ‚ ’ (X: U
∗
∗
U ³ ” &:5659+,"61 &:#;#!#*,+A"\+L$œrl&!$634ac"
– 3C#"93—7M(#7d)++Ak"9$l:10| !z}5<z}R;#,BA]™#|F#3( 4$6‚ 3C&:’ 7(X8:(# *;#L+ hK*AXOC+;0;0!!#"#"sJ O—Yux!#7T)+&@!( X +L$ Y "6&:I$;#+,"-„BAOC*AB,&:rsrs+,1eI$&6"
;)+ Var(k)
"934$63C&:;#+A"†*,&!5656+A"61 &:#;02#*,+A"‹"6+-a`!34$l!O4&:56"%;)+¬OŠ·a`IHA&:*AOŠ!CH
"9"634DR()(X+ª"9×(#34[2Y!)I$U˜'+šV ^c[!&:3C5<1025N
+gJK+,rs1#OC+<±v²S³:7{‰BL™ U ³ )U ³ U ³,´Šf
<
∗
1#5
CH∗ (X × Y ) × CH∗ (Y × Z) −→ CH∗ (X × Z)
(f, g)
7→ g ◦ f = ( 13 )∗ ((f × Z) · (X × g))
& · ;#B,"93CP:#+nOCmrm(#OG$'3C1)OC3C*,2$'34&:<;#+,"T*LhR*,OC+A"_;0!#" CH (X × Y × Z) +A$ (1#5
ac&!5 S!56;;#+oOŠm1#59&?>9+,*L$'34&:
1#5 : X × Y × Z −→ X × Z
∗
13 )∗
OC+‰1#(#"6. N
13
7→
(x, y, z)
(x, z).
K&!3C+,I$˜rl!3CI$'+A0!I$d;#+A"d[2!593CBL$'B," X +A$ Y ;0!)" Var(k) & Y +,"9$˜"9(#1#1 &:"6BA+‰*,&!##+LJK+
^c&:(;uv(##+‹a`IHA&:1#OC(#"%P!B,#BA5'!O4+ žE%H~Z“ ‚ “&#š'(X@“`|F((XR`` 9f ;#BL™0#3G$s!OC&:59"m;0!)"%(#
1)56+,rs3C+A5Y$6+,rs1#"M(##+:| z}z}R,]|F( ‚ ’ (X: ‚  X ‚ ’ (Z Y ‚  U]© ‚ Ey{z}ž p *,&:rsrl+\BA$'!I$T(#
*LhR*,OC+ .)&:rl&!P:Q,#+†;#+ CH (X × Y ) U]© BA$6+,#;+,)"6(#3G$'+ *,+L$6$6+ ;#BA™0)34$'34&:!(*,!"
;Hux(#)+w[2!5634BA$'B Y DE(#+AOC*,&!#DE(#+m;0!)" Var(k) +,¬;#B,"93CP:02E$10!5œ*,&:5956+,"91e&!#;0!#*A+%;#+w;)+ N
P!56B p OC+A"]*AhR*,O4+,"d.#&:rs&:P:QA#+,"d;#+ ⊕ CH (X × Y ) 7:& <OC+A" Y "9&:I$˜OC+,"]*A&:rs1e&:"6!I$'+A"
*A&:##+gJK+,"_;#+ Y U)V +OC+A*A$'+A(#5M#&2$'+,56mDE(#+n"63 X +A"9$TBAPI!O4+,rs+,I$TB,DE(#34;#3Crs+,)"63C&!##+,O4OC+27R&!-
2OC&:59"
⊕ CH
(X × Y ) = ⊕ CH
(X × Y )
& ·3C*A38!()"6"63‘7HO4+,"\[2!593CBL$'B," X ;#BA"63CP!#+,I$nO4+,"‰*A&:rs1e&:"6!I$'+A"*,&:)#+LJK+,"o;#+ X UHV +mO4+,*L$'+,()5
1)56+,);#5'PI!56;)+l;#+s#+l10!"q*,&:)ac&!#;#56+sOŠ-#&!$63C&:„;#+s*,&:5956+A"61 &:#;0!)*,+†;)+†;#+AP:56B p @[!+,*
O4+ ‚ Ey{z}ž;uv(# *LhK*AOC+27]#&!$'34&:DE(#3_"6+,56„BAPI!O4+,rs+,I$s()$'34OC3C"9B,+‹1#!5<OŠ "6(#3G$'+ jd +AFe+A$,7
() 0N *AhR*,O4+\;Hux(#)+\0N [2!5634BA$6B X +,"p$M(#lB,OCBArl+AI$ c = P n [P ] ;0!#" CH (X) +A$YU "6&:†;#+,P:59B
+A"9$\!O4&:56"
;#+,P (c) = X n [k(P ) : k].
V\+d+Aa OC+A*A³ $6+,<!(#´ 5X"6&:().0!34$'!I$€;#+˜1#OC(#"X!rs1#O4+,"3CKac&:56rl2$'34&:#""9(#5€*,+˜"6(?>9+L$81 &:(#595'T*,&!#"6(#OG$'+A5S±v²S³!7
_+U A[!U +,#&U :)"\rl!3CI$6+,0!I$ W OŠm#&!$63C&:†;#+n;#+,P!56Bn;uv(##+o*,&!5656+A"61 &:#;02#*,+ U{V &!56"6DE(#+nO‘ux&!
*A&:rs1e&:"9+-;#+,"s*,&:5956+A"61 &:#;0!)*,+,"A7YO4+,"<;#+AP:56BA"†",ux}>9&:(K$'+,I$·^`*A&:rlrs+ OC+-O4+,*A$6+,(#5<1 +,()$s1025
+gJK+,rs1#OC+qOC+n[:BA563G™0+,5œ;0!#"Œ±v²S³:7{j8J U ³ KU ³ U ³,´Šf U O€+,"p$23C#"93*,OŠ23C5TDE(#+wO‘ux+A#"6+Arm=#O4+q;)+,"‰*A&:5 N
59+,"91e&:);0!#*A+,"M;#+œ;#+AP:56B 0 +A"9$S"9$/2=#OC+_10!5˜*,&!rl1 &:"63G$'34&:<+A$Y&:†;#BL™0#3G$Y1#!5Y*A&:#"9B,DE(#+,I$TOC
:’KŽižEy |{z?“c ‚ Rl:| z}z}R,]F| ( ‚ ’ (X:R ‚  ‚ Ey z}ž 0 70)&!$'BA+ Corr (k) 70DR()3Z+A"9$()#+Œ*,2$'B N
P!&:5634+n’ ‚Z‚ “ Ž}“  s;#&:I$\OC+,"T&!=R>9+A$6"\"9&:I$\*A+,(KJª;#+ Var(k) +A$œOC+A"_P:56&:()1e+A"œ;)+qrs&:591#.#3C"9rl+A"
dim Y +p
i
i
dim Yi +p
dim Yi +p
i
i
j
i
dim Xj −p
j
j
∗
P
P
P
0
T m l
Wb ]@[Z\`a_DZ}[ mj _6gDWZ\Zf_ &Y ][email protected]@[[email protected]_YfX^Z
\i WXmigDWX^ZfiF]
{ t s v oIvEsun nLs4`8WX
Z\Y\c ^X _ifWXmif_ YSY&_ YgDWS]=W Yf[b if_eY l8Z\Z\€[email protected]ƒifl6[^`a_eYWb i
eS_6jml8[email protected]&laWb ]
b^Y_4‚=[ZfXla€ƒi\€O_ Y+ijml8if_
V ux34"6&:rs&:591#.#3C"9rl+‰rl&!$634[R3CDE(#+
"6&!E$]$6&:()$d"93Crs1#OC+Arl+AE$8OC+A"b*,&:5956+A"61 &:#;0!)*,+,"˜;#+T;#+AP:56B 0 U:V +TOC+A*A$6+,(#5b$656&:()[!+,56o;#!#"_±v²S³:7
j8J U ³ )U ³ U ³ ´;#+o1#OC()"\2rl1#O4+,"T59+,#"9+,3CP!#+,rs+,I$'"œ"6()5œOC%*@2$6B,P:&:593C+ Corr (k) U{§ &!(#"_"63CP!0 N
OC&!#"œ$6&:()$\;#+wrst,rs+qDR()+ id 7eO‘u°21#1#OC34*@2$63C&: 34;#+,I$634$'Bq;#+ X ;0!)" Corr (k) +,"p$;#&!##B,+
10!5YOŠq*,OŠ2"6"6+‰;#+‰O‘u°!1)1#OC34*@2$63C&:<;#3C!P:&:02OC+_"6(#5 X × X U wu°2()$'[email protected]#&:(#"M;)B,"634P:#+A56&:#"
10!5 End(X) OC+nP!56&:(#1 + Hom(X, X) ;#+,"œ*,&!5656+A"61 &:#;02#*,+A"\;#+ X ;0!#"_+AOCOC+ N rst,rs+ U
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d'# #š x $% X (α ) $ 2
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A
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1)56&:1 &:"63G$'34&: ;)+ \® b7)& )&:(#"M1#59+,#&!#" B = X (α ) +A$MDR()+O—uv&:‹1 +,()$T!1#1#O43CDE(#+,5
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2 x
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x
x
2
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2
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A+ $‹10!5‹*A&:#"6BADE(#+,I$OŠš*A&:5659+,"91e&:);0!#*A+ f" +A"9$ $6&!$/!O4+,rs+,I$l;#BA$6+,56rs3C)B,+·1025‹"6&: 2* N
$'34&:k"6(#5qOC+A"oP:56&:()1e+A" CH (X (α )) DE(#3d"6&:[email protected]€1 &:(#5n$'&!()$'+A"ŒO4+,"n[2!OC+A(#56"o;#+ i 1e&:"9"634=#OC+A",7
3C"9&:rs&:561#.)+," W G UR– 34#"63‘7 f" "'2$63C"pa`!34$ W O‘uxBADE(02$'34&: t − 1 = 0 1#(#34"6DE(#+*2ux+A"9$M()‹3C"9&:rl&!5 N
1#.#34"6rs+w;#Q+ G ;02#" Go7e*2ux+A"9$ N W?N ;)3C56+ŒDE(#+ f" = (∆ )" & ∆ ;#B,"93CP:)+wO—uvB,O4B,rs+,I$\#+,()$656+
;#+ End(X (α )) ^ W "'@[:&!3C5,7IOCn*,OŠ2"6"6+_;#+œO—ux!1#1)OC3C*,2$'34&:w;)3Š!P:&!0!OC+S;#+ X (α )) U wux!1#59Q,"˜OŠ
1#59&:1e&!"634$63C&: 7{#&:()"\+A·;#B,;)(#3C"9&:#"‰DE(#+ f − ∆ = g 7{& g +A"9$\(#·B,O4B,rs+,I$\#3CO41e&!$6+,I$
+A$œ+A*A&:#*AOC(#"93C&: f = ∆ + g +,"p$\1#!5T*,&:)"6B,DE(#+AI$(#!()$6&:rs&:561#.)3C"6rs+;#+ X (α ) U
‰+,1#()3C"]DE(#+TO—ux!()$6+,(#5bBA$/2=#OC3K*,+T$'.#BA&:56QArl+S;ux34"6&:rs&:591#.#3C"9rl+Y1e&!(#5bOC+S*@!"]10259$'34*,(#O43C+A5
;#+ X (α ) 7 C«‰­ Z® }® Y7 o+L$ ‹&:I$€P:B,)B,5'2OC3C"9B,"*A+]56B,"9(#O4$'2$
^`+L$_OC+$6.#B,&:59Q,rs+n;#+n#34OC1 &!$'+A#*,+ fS;0!#"TO4+,(#5T1#59B,1#()=#OC34*@2$63C&: ± ” 7¬¶ 7 . U KU <2´& -3CO4"
BA$'!=#O43C"6"9+,I$˜DE(ux34O0+,"p$Y[!B,563G™0Bœ1e&:()5d$6&:()$6+,"dOC+A"d[2!593CBL$'B,"d.#&!rl&:P!Q,#+A"Y1#59&?>9+,*L$'3G[:+,"Y!"9"6&R*,3CBA+,"
W (#-P:56&:()1e+q!O4P:B,=#593CDE(#+ U
i
2
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2
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"6&!3C+,I$ ;)B,1#O4&}h!B,+A" &:(#&: USA &!(#5‹*,&!rlrs+,#*A+,5A7Y#&!(#"‹#&:()"‹1#OŠ2*,+,59&:#"l;0!#"‹O4+¬*,!"‹&
OC+A"n[!2563CBL$'BA" X (α ) +L$ X (α ) "9&:I$w;)B,1#O4&}h!B,+A" UX§ &:()"q56!1#1 +,O4&:#"oDE(#+l*A+,OC-"93CP:)34™0+%DE(#+
O—ux!O4P:Q,=#59+;uv&K*L$'&:)3C&:#"T"9(#5_OŠ!DE(#+AOCO4+OC+nP:59&:(#1 +n;#+n$ hR1e+ G !P!34$TO‘ux+A"[email protected])&:(-+,#*A&:56+oDE(#+qOŠ
ac&:59rl+<DE(0!;#562$'34DE(#+ q ;#BL™0#34+l"9(#5ŒO‘u°2OCP:QA=#56+%;#+A"w&R*A$6&:#3C&!#"o+,"9$Œ3C"9&!$'59&:1 + U€§ &!(#"w2OCOC&!#"
;#&:)*\;02#"Y()†1#59+,rs3C+,5]$6+,rs1#"YBL$/!=)OC3C5]DE(uv3CO0+gJK3C"9$6+\()#+\*A&:5659+,"91e&:);0!#*A+‰;#+\;#+,P:59B 0 DE(#3
56B,!OC34"6+q(#·34"6&:rs&:561).#3C"9rl+o+,I$656+ X (α ) +A$ X (α ) U A (#34"‰;02#"(#·"9+,*A&:#;·$'+Arl1)",7 #&:(#"
!O4OC&:#" W O—ux!3C;)+q;)(ª$'.)B,&:59Q,rs+Œ;uv3C"9&:rs&:561#.)3C"6rs+nDE(#+w)&:(#"@[!&:#"‰BL$/!=)OC3;0!)"‰OŠ<"6+A*A$63C&:
1#59B,*,BA;#+,I$6+!71#59&:()[!+,5DE(#+Œ*,+A$9$'+m*A&:5659+,"91e&:);0!#*A+w+,"p$n5'?$'3C&!##+,O4OC+n*A+wDE(#3XBA$/2=#OC345's!OC&:59"
O—uv3C"9&:rs&:561#.)3C"6rs+‰+A$6&:()$6+nP:B,#BA5'!O434$6B U
1
2
2
1
2
+A#*,&!56+!7H*,&:rsrl+A H,&!#"10!5™KJ)+A5nDE(#+,O4DR()+,"o#&!$'2$'34&:#" U K&:3G$ j (#k+,I$'34+,50?$'(#59+,O
V
X
W
;#+lO—uv+,#"9+,rm=#O4+ {0, . . . , 5} 7€#&!(#"m!1)1e+AOC&:#" h OC+sP:B,#BA5'2$6+,(#5Œ;#(•P:59&:(#1 + CH (X (α ))
+A$ h *,+AOC(#3 ;#+ CH (X (α )) U)V +A"S56+AOŠ2$63C&:)"˜rw(#O4$63C1#O43C*@?$'34[!+,"]+AE$656+*A+,"YP:B,#BA5'2$6+,(#59"M"6&!E$
*,OC!"6"93CDE(#+A"8;0!)"ZO4+Y*,!"Z;)+SOŠTDE(0!;)563CDE(#+
+L$8;)+,1#(#34"ZOCœ"9&:(#" "6+A*A$63C&: ^`1#59+,"6DE(#+
$'&2$/!O4+,rs+,I$gfw;#BL$'+A56rs3C#BA+,"<;0!#"<O4+ *@!"<X;#(α+ X)(α ) U6A &:(#5<1#59&:()[:N +A5l*A+A$s3C"9&:U rsU &:561#.)3C"6rs+!7
#&:()"S!O4OC&:)"]+LJK.#3C= +,5˜;#+A(KJl*,&:5956+,"91e&!#;0!#*A+," f +A$ g 7 f ;0!)" Hom(X (α ), X (α )) +A$ g
j
2
j
j
1
j
1
2
1
2
1
2
0
U "6&:rs&:591#.#3C"9rl+nrs&!$634[R3CDE(#+
;#!#" Hom(X (α ), X (α )) $'+AOCO4+,"bDE(#+ g ◦ f = ∆ +L$ f ◦ g = ∆ & ∆ +A$ ∆ ;#B,"93CP:#+AI$
59+,"91e+A*A$'3G[:+Arl+AE$oOCl*AOŠ!"9"6+m;#+wO—ux!1#1#O43C*@?$'3C&!ª;#3C!P:&:#!OC+o;#+ X (α ) +A$ X (α ) U j] *,+%DE(#3
*A&:#*A+,56)+ ∆ 7#*A+A$6$6+Œ*A&:5659+,"91e&:);0!#*A+Œ+A"9$œ;#B¨> W *A&:#E(#+!7
<
2
1
1
2
1
1
2
2
1
∆1 =
5
X
hi1 × h5−i
1 .
]j 56+L[!2#*/.#+!7{34OH[2<)&:(#"Ta`!OCO4&:3C5˜+LJK1#O43C*A34$'+A5 ∆ U
j]šP!(#3C"9+%;#+<1#56BAOC3Crs3C#!3C59+!7{"634P:0!O4&:#"o!(„OC+A*A$'+A(#5oDE(#+!7*A&:rlrs+%OC+A"n[!2563CBL$'BA" X (α )
+L$ X (α ) &!E$‰$'&:()$6+,"O4+,";)+,(KJ o(##+m"p$'59(#*A$6(#56+%*A+,OCO4(#OŠ23C56+<7 X (α ) × X (α ) 7 X (α ) ×
7
+A$ X (α ) × X (α ) +A•&:I$mBAPI!O4+,rs+,I$w()#+†DE(#+‹O‘ux&!›;#BA;#(#3G$
X (α ) X (α ) × X (α )
;)+ *,+,O4OC+‹;)+ X (α ) +A$ X (α ) ^c[!&:3C5‹±xµo!56¶)³:7b\BA™ U KU ´Šf U]§ &:(#"<@[:&!#"<;u°23COCO4+,(#59"w;#B > W
rs&:I$'59Bm;0!)"nOŠ†1#59+,()[!+<;#( OC+Arlrs+ DE(#+%*A+A$6$6+%"9$656(#*L$'(#59+s"9+%;#B,*A563G[!234$\+,krm(##34"6"6!I$
O4+,"Z*,&:(#1)OC+," (i, j) ;#+MO‘ux+A#"6+Arm=#OC+ {0, . . . , 5}×{1, . . . , 5} ;ux()#+T56+AOŠ2$63C&:Œ;ux&!56;#59+M10!5p$'3C+AO U
” +A*,3€#&:()"œ1 +,59rl+L$œ!OC&!56"M;uvBA$'!=#O43C5TOC+n59B,"9(#O4$'2$_"6(#3G[2!I$
d'# #š , l $% {0, . . . , 5} &$" " $1 %
i=0
2
1
2
2
1
2
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1
2
1
1
2
2
⊕lm=0 CHm (X (αi )) ⊗ CH5−m (X (αj )) → CHl (X (αi ) × X (αj ))
(f ⊗ g)
7→
f ×g
i j "%$ {1, 2} + " +- .
/ + %0($1 . 2 #"wO4+lO4+,rsrl+ 7€#&:()"[email protected][:&:)"m;#B¨> W BA$'!=#OC3d*A+†56BA"6(#OG$/2$Œ;0!)"wOC+s*@2"
&
U &:()5œO4+,"_*@2"œ59+,"p$/[email protected]%;#B,rs&:#"p$'5'?$'3C&! +,"9$œ+gJ#2*A$'+Arl+AI$‰OCmrltArl+ U
+M59B,"9(#O4$'2$8+A"9$8+,wa`!3G$81)OC(#"XP:B,)B,5'2O—7}3COE+,"p$b+A%+AFe+A$8[!B,5934™0BY;#Q,"bDE(#+YOC+A"8;#+A(KJq[!2563CBL$'BA"
”
2;#rl+L$6$6+,I$œ(##+o"9$656(#*L$'(#59+Œ*A+,OCO4(#OŠ23C56+ U
‰+œ*,+œ;#+A56#34+,5dOC+Arlrs+!7:)&:(#"d;#B,;)(#3C"9&:#"Y!3C#"930DE(#+\O4+,"dP:BA#B,562$'+A(#56"˜;#+ CH (X (α ) ×
",uv+LJK1#593Crs+,I$T+, ac&:)*A$'34&:‹;#+n*A+,(KJ‹;#+ CH (X (α )) +A$ CH (X (α )) U j]-10!5p$'3 N
X (α ))
*A(#OC34+,5A7R)&:(#"S59+,rl!56DE(#&!#"SDE(uv+,‹$'!I$SDE(ux!##+,!(KJe7#34OC"d"6&:I$S"'!#"Y$'&:59"63C&!l+L$M56+L$'59&:()[:&!#"
;)+q*A+A$9$'+na`I H,&!ª(#56BA"6(#OG$/2$_;)B¨> W BA$'!=#OC3H;0!#"q±vµ NRu* :)³:7 ” &:5 U ³ U 2?´ U
+A*,3˜BA$/2E$Œ;#[email protected]#&:()"Œ"9&:rlrs+,"orl!3CI$6+,0!I$q+,›rl+A"6(#59+†;uv+,›;#B,;)(#3C59+lO‘ux+gJ)1)56+,"9"634&:
”
;)+ ∆ 7021#1#OC34*@2$63C&:†;#3Š!P!&:0!O4+;#+ X (α )
d'# #š i=j=2 A
∗
∗
j
1
∗
i
2
2
2
∆2 =
E[
5
X
hi2 × h5−i
2
i=0
Y&ifZ\[email protected]\X^Z\_gƒ_e`8`aX^` [laZf_kjm_
X (α1 )
^la_Db c X^_b^Wb4jm€egDZfl8if_lagDl=_ Y+i _Db1 [l8i gD`a[YfYflac X^__ƒi ^l8_eb gDWb^buX^_
V ux34"6&:rs&:591#.#3C"9rl+‰rl&!$634[R3CDE(#+
/ + %1($0 .œV ª"6+,()OC+†*/.)&:"6+ W a`23C56+s3C*A3—7€+,"p$%;#+‹1)56&:()[!+,5%DE(#+‹OCª*A&:5659+,"61 &:#;#!#*,+
P
!P:34$Œ$'5934[R3Š!O4+,rs+,I$m"9(#5%OC+A"%P:B,)B,5'?$'+,()56"<;#+ O—ux!#)[email protected]!( End(X (α )) =
h ×h
CH (X (α ) × X (α )) U
wu°21#56QA"œO4+nOC+Arlrs+ 2m#&:(#"\@[:&:)"
(
"63 i = j
deg(h · h )(h × h )
(h × h ) ◦ (h × h ) =
"634#&:
0
& deg(h · h ) ;#B,"93CP:#+_O4+œ;)+,P:59Bœ;)( 0N *AhR*,O4+ h · h U ‰Q,"˜OC&:59",7:;#!#"˜OC+_*@!"d& i = j 7
#&:()"˜;)+A[:&!#"d;#&:#*_*@2OC*,()OC+,58OC+M1#59&K;)(#34$ h · h 1 &:(#58$6&:()$'+A"dOC+,"b[2!O4+,(#59"b;#+ j 1 &:"9"63C=)OC+,"A7
"'@[!&:3C5 7 +L$ OZ+,"p$n$'&:()$;u°2=e&:59; *,OC!3C5œ+, [:+A59$6(„;)+%*,+mDE(#+%#&!(#"[email protected][:&:#"o*,!OC*A(#OCB
DEW (#+ 0 1 2 U
h ·h =h ·h =h
+A$œDE(#+
h ·h =h ·h = h .
!#"_OC+n*,!"_56+A"9$/2E$,70*!uv+,"p$ N¨W}N ;#3456+ j = 2 7##&:()"œ*,!OC*A(#OC&!#"M$'&:(K$_;u°2=e&:59;
5
i=0
5
5−i
2
i
2
2
2
2
i
2
j
2
5−i
2
5−i
2
5−j
2
5−j
2
i
2
5−i
2
i
2
i
2
5−i
2
i
2
0
2
5
2
5
2
0
2
5
2
1
2
4
2
4
2
1
2
5
2
j
2
5−i
2
2h22 · h32 = (h12 )2 · h32
= h12 · 2h42
= 2h52
+A$œ1025T*,&:#"9B,DE(#+AE$‰#&:(#"œ+,;#BA;#(#3C"9&:#"_DE(#+
h22 · h32 = h32 · h22 = h52 ,
+,-[:+A59$6(¬;#+oDE(#&:3#&!(#"_"6&:rsrl+A"T+,rl+A"6(#59+o;#+o*,&:#*AOC(#59+oDR()+
deg(h5−j
· hj2 ) = deg(h52 ) = 1
2
1 &:(#5T$'&:(K$‰34#;#3C*A+ j ;#+qO—uv+,#"9+,rm=#O4+ {0, . . . , 5} U j]¬+AFe+A$,7{1#(#3C"9DE(#+ X (α ) +,"9$\;#BA1#OC&ih:BA+!7
#&:()"oi[!&:#" h = [X ] +A$n34OZ+A"9$n*AOŠ!345\DE(#+ [k(X ) : k] = 1 U ‰+%O W 7 )&:(#"n+, ;)B,;#(#34"6&:)"
DE(#+q1e&:()5M$'&:()$_34#;#34*,+ j 7
2
5
2
0
(
0
5
X
j
5−j
j
5−j
hi2 × h5−i
2 ) ◦ (h2 × h2 ) = h2 × h2
+A$œ;#+oO‘ux(#)3C*,3G$'B;)+qO‘uxBAOCBArl+AE$M#+,()$656+q#&:(#"_1 &:()[!&:#"œ*,&:#*AOC(#59+nDE(#+
i=0
∆2 =
5
X
i=0
hi2 × h5−i
2 .
:
¶
U "6&:rs&:591#.#3C"9rl+nrs&!$634[R3CDE(#+
uvB,OCBArl+AI$n#+,(K$'56+<rl!3CI$6+,0!I$‰*,&:#E(7)&:(#"n!O4OC&:)"‰;#&:)*%+LJK1#OC34*,3G$'+,5\OŠ†*A&:5659+,"91e&: N
;#!#*V ,[email protected]!O43C"'2E$_O‘ux34"6&:rs&:591#.#3C"9rl+œrs&!$'3G[K34DE(#+;0!)"œO4+n*@!"_;#BA1#OC&ih!B U
qz}|{]|e,“cŽ}“c|F() <
$ #" $
J=
5
X
hi1 × h5−i
2
% + CH (X (α ) × X (α )) = Hom(X (α ), X (α )) $ + " +- + 0 0 X (α ) X (α ) , $ #" $ 1($% " J .
&!(#5Y*A+,OCo)&:(#"˜()$'34OC3C"9&:#"](#)+
/
&:(#"Y!OCO4&:#"b1#59&:()[!+,5SDE(#+
ac&!3C+"T+, )*%,&:059($+q1O4 +n OC+Aœ. rl§ rs+ 2m1e&!(#5T*@!O4*,(#O4+,5 J ◦ J = ∆ U'A
i=0
5
1
2
1
1
2
2
t
t
2
(
deg(h5−j
· hi1 )(hj2 × h5−i
j
5−j
1
2 )
(hi1 × h5−i
)
◦
(h
×
h
)
=
2
2
1
0
+L$œO W +,)*,&:59+q)&:(#"\i[!&:#"
"93 i = j
"93C#&!
deg(h5−j
· hi1 ) = deg(h51 ) = 1
1
1)(#3C"9DR()+
(h11 )2
(h11 )3
(h11 )4
(h11 )5
= h21
= 2h31
= 2h41
= 2h51
*A&:rsrl+OC+OC+A*A$'+A(#5s1e&:()565'›",uv+, !"9"6(#59+,5‹+A*,&!#"6(#OG$/!I$·±xµo!5 ::¶?´ UA !5†*A&:#"9B,DE(#+,I$,7M3CO
[R34+,I$_$'&:()$_#2$'(#59+,O4OC+,rs+,I$T+Aª;)BA[:+AOC&:1)10!I$‰DE(#+
t
J◦ J=
5
X
hi2 × h5−i
= ∆2 .
2
\ +rst,rs+!78+, *@!O4*,(#OC!I$ J ◦ J 7b&! $656&:()[!+-+, 1#56&R*,BA;0!I$s+LJ)!*A$6+,rs+,I$l;#+-OŠ·rst,rs+
rl!#34Q,59++A$œ+,()$63COC34"'!I$TO4+,"T59B,"6()O4$/?$'"_;#+oOŠm1#59+,()[!+Œ;)(ªO4+,rsrl+ :K7#DE(#+
i=0
t
t
–1)(#C3 #"3C"99DR3—()7 J+q*A56B,+A!$9$'OC34+q"6+!*,7i&:*A59&:56+,rs"91erl&!+˜#;!0#!)&#*:A+Œ#*,1 B2&:72"9()"6m3Q,;)C"9+Œ&:(#rl3&!56CI[1#!.#+,3456"6"9rs+ +brs;#&!59$'&:3G34[K$634+DE(#+A$ +d+,I$PI'!59+ (#*/X.)(α+ ) +A$ X (α )
W
W
U
uV vBA$/21e+ª"6(#3G[!2E$6+¬+A"9$‹;)&:#*¬;HuxBL$/!=#O43C5<DE(#+¬OC›*A&:5659+,"61 &:#;#!#*,+ J +A"9$ 562$'34&:##+AOCOC+ U
\+lO 7#&:()"Œ+gJ)1)OC3CDE(#+A56&:)"q1)OC(#"oOC&!3C +,›;#BA$'!3CO8*,&:rsrl+AI$Œ+A•!1)1#OC34DR(#!I$oOC+m$'.)B,&:59Q,rs+
;Hux34"6&:W rs&:561).#3C"9rl+l*,+,OC·#&:(#"%1 +,59rl+L$<;#+ *,&!#*,O4(#56+ DE(#+-O—uv3C"9&:rs&:561#.)3C"6rs+<rl&2$'34[R34DR()+‹+,"p$
2(#"6"[email protected]"9Bn;0!#"_O4+n*@!"œ!#34"6&!$656&:1 + U
J ◦ J = ∆1 .
1
2
V ux34"6&:rs&:591#.#3C"9rl+‰rl&!$634[R3CDE(#+
³
#1 59B,"9+,I$%OC+A"Œ[2!593CBA$6B," X (α ) +L$ X (α ) "6&:I$%"9(#1#1 &:"6BA+,"%2#3C"9&!$'59&:1e+A"‹^c3 U + U #&:
;#BA1#OC&ih:BA+,"'fY"6(#5YOC+œ*,&:591#"S;)+‰=0!"9+ k UR§ &:(#"S*A&:#"93C;#BA56&:#"S!OC&!56" (##+*AO-,!$6(#56+\!OCP!B,=#593CDE(#+
;#+ k ^c& 1#OC()"%"63Crs1#O4+,rs+,I$m(#)+ +LJR$'+A#"634&:DR(#!;#5'?$'3CDE(#+ ;)+ k "9(#5%OŠ2DR()+,OCO4+†O‘u°!O4P:Q,=)56+
;#+A"&R*A$6&:#34&:#" O +,"9$ ;#B,1#O4&ih:B,[email protected]‹+A$-OC+A"†[2!5634BA$6B," X (α ) = X (α ) ×$%'& ( K1e+A*
+A$ X (α ) = X (α ) ×$%'&)( R1e+A* DE(#3]"9&:[email protected]+AOCOC+A",7;)B,1#O4&}h!B,+A" U8” &:rlrs+m1#59B,*AB,;#+Ar N
rs+,[email protected]#&:()"S;#BA"63CP!#&:#"S!OC&:59"d10!5 h +A$ h OC+,"˜P:BA#B,562$'+A(#56"Y56+,"91e+A*A$634ac"S;)+ CH (X (α ) )
+A$ CH (X (α ) ) 1 &:(#5\$6&:()$ i ;0!)"O—uv+,#"9+,rm=)OC+ {0, . . . , 5} UH§ &:(#"‰!##&: H,&:#"‰r†!34I$'+ N
0!I$
qz?|{]| @“ Ž}“c|$( $ #" $
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2
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2
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2
J=
1
k
5
X
hi1 × h5−i
2
i=0
CH5 (X (α1 ) × X (α2 ) ) = Hom(X (α1 ) , X (α2 ) )
X (α1 )
($0 , .
/ + %1($0 . K(#5-OC+ *A&:561#" 7_OC+A"‹[!2563CBL$'BA"
+A$ X (α ) 9" &:I$¬;#BA1#OC&ih!B,+,"
+A$†! ;#rs+A$9$'+AE$s1025s*A&:#"9B,DE(#+,I$†(##+"9$'59(#*A$6(#56+*,+AOCO4(#OŠ!3456+ Ub§ !& (#"<5'!1#1 +,4O &:#"<;#&:)*ªDE(#+
2
;ux!1#59Q,"œOC+nOC+Arlrs+ 7
⊕ji=0 CHi (X (α1 ) ) ⊗ CHj−i (X (α2 ) ) ' CHj (X (α1 ) × X (α2 ) )
1 &:(#5S$6&:()$ j ;0!#"TO‘ux+A#"6+Arm=#OC+ {0, . . . , 5} U ‰QA"TOC&:59",7 CH (X (α ) × X (α ) ) +A"9$_*AOŠ!3 N
56+Arl+AI$w+A#P:+,);#56B‹1#!5oOC+,"qB,O4B,rs+,I$'"q;#+lOŠ ac&:56rs+ h × h UZ” +,"Œ1#56BAOC34rl340!3C59+,"‰™KJ)BA",7
#&:()"<!OCO4&:#"m2OC&:59"m1#56&R*,BA;#+,5%+A1#OC(#"93C+A(#56"%BL$/!1 +,"%1 &:(#5%1#59&:()[!+,5%OŠ·562$'34&:0!O434$6Bs;)+ J U
&!(#5_*,&:rsrl+A#*,+A5,70#&!(#"œ!OCO4&:#"M;#BArl&!E$656+A5TOC+,"_;#+A(KJ-O4+,rsrl+A"T"6(#3G[!2E$6"
A
dE# #š ! h $% CH (X (α ) ) $1 .
/ + %1($0 .\” +Œ59B,"6()O4$/?$\1 +,(K$‰tL$'56+o[R(¬*A&:rlrs+n(##+q*,&!#"6BADR()+,#*A+%;#+ŒOC%*,&:#"p$'59(#* N
$'34&:„;#+†OC"p$'56()*A$'()56+†*A+,O4OC(#OC!3C59+%;#+ X (α )SU j]•+AFe+A$,7ZO4&:56"9DE(#+l)&:(#"wO‘u°@[!&:#"q*@!O4*,(#O4B,+
^`*La U "6+,*L$'34&: U ³}fg7I#&:(#"˜@[:&!#"Srs&:I$'59BœDE(#+œOC+_1#59+,rs3C+A5b$'+,59rl+_;#+_OCn™#O4$'562$'34&:s!;#rs+A$9$/!3G$
1 &:(#5 RN 1 &:3CI$6"o^`& R +,"p$\()ªBAOCB,rs+,I$œ;#+ Alg(k)fTO‘ux+A#"6+Arm=#O4+
∗
i
1
j−i
2
1
2
1
2
1
2
2
X4 (R) = {P ∈ X (α2 )(R) | rg(P ∩ (V7 )R ) > 2, rg(P ∩ (V5 )R ) > 1}
A+ $lOŠk*,OC!"6"9+ª;)+ª*A+A$6$6+[2!593CBL$'B-rl&R;#(#O4&kB,DE(#34[2!O4+,#*A+ª562$'34&:##+AOCO4+‹+,"p$o>9(#"p$'+Arl+AE$†#&!$656+
B,O4B,rs+,I$ h U wux!()[email protected]:O4+TP:56&:()1e+T;#+n«‰­X®¯ ;#+_OŠP!5'!"9"6rl!##34+,##+ Γ (H) nBAPI N
OC+Arl+AI$8(#w"6+,()O)P:BA#B,562$'+A(#58+Am*,&R;#3Crs+,)"63C&! 1 7 W "'@[:&!3C58OC_[2!593CBL$'B˜;#+ ­ X
;#&:I$œ#&:()"œ56!1#1 +,O4&:#"TDE(#+qO—uv+,#"9+,rm=#O4+n;#+," RN 1 &:34E$6"M+,"p$\1#56BA*,3C"9+,rs+,I$
1
2
2
(1,0)
X(1,0) (R) = {P ∈ Γ2 (H)(R) | rg(P ∩ (V7 )R ) > 2, rg(P ∩ (V5 )R ) > 1}.
U "6&:rs&:591#.#3C"9rl+nrs&!$634[R3CDE(#+
1 &:(#5]$'&!()$'+‰!OCP:QA=#56+ R
(R)
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;)+ Alg(k)
;#+o1#=O4(#" X (α )(R) ∩ X
4
2
(1,0)
codimX (α2 ) X4 = codimΓ2 (H) X (α2 ) ∩ X(0,1)
2OC&:59",78O‘ux34r†2P:+†;#(P:BA#B,562$'+A(#5<;#+ (Γ(H) ) +,"p$s+LJ)!*A$6+,rs+,I$ h = [X ] U OT#&:()"
a`2()$rl!3CI$6+,0!I$\BA$'!=#OC345_OŠl562$'34&:0CH
!O434$6Bn;#+m*,+w*AhR*,O4+ U A &:(#5\*,+,OC)7 #&:(#"*A&:#"93C;#BA56&:#"\OC+A"
1)(#OCO N =0!,* +l;#+,"_+gJK$6+,#"93C&:)"\;#+A"_"6*@2OŠ!3456+,"
+A$
X (α ) −→ X (α )
Γ (H) −→ Γ (H)
*2ux+A"9$ N¨W?N ;#3C59+nOC+,"_21#1#OC34*@2$63C&:)"
1
2
1
2
K
2
res
/k :
2
4
2
CH1 (X (α2 )) −→ CH1 (X (α2 ) )
+L$
O{+A"9$Mrl!34E$6+,02E$˜+AE$6+,res#;)( DR:(HuxCH
(#)+(Γ=0!"9(H))
+‰#&:59−→
r†2OC+TCHuv+LJK(Γ3C"p$'(H)
+\10!"S)."9(#5˜OC+œ*,&:591#"S;#+œ=0!"9+ k 7
$6&:()$6+Aac&:34"*,&:rsrs+w#&:()"qO‘u°@[!&:#"n;#B > W "63CP!0!OCB27Huv3Crs1 &:59$6+wDE(#+,O4OC+m=#!"6+<1 +,56rs+A$;#+%;#BA™ N
)3C5‰$6&:()$'+A"nOC+A"[!2563CBL$'BA";#+ ­•;uv(##+%P:56!"6"9r†!)#3C+A##+%+A$oO4+,"n*AOŠ!"9"6+,"nrs&R;#(#OC&
BADE(#34[2!O4+,#*A+‹5'2$63C&:)#+,O4OC+<;#+ *,+A"m[2!5634BA$'BA"Œac&:59rl+AI$%(#"9hR"p$'Q,rs+ ;#+‹P!B,#BA5'2$6+,(#59"%OC3C=)56+,"
;)+ CH (Γ (H)) U ‰QA"\O4&:56"A7#3CO+A"9$œ=#34+,-*,OC!3C5MDE(#+
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CH∗ (Γ2 (H)) ' CH∗ (Γ2 (H) )
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CH1 (Γ2 (H) )
CH1 (X (α2 ) )
res
CH1 (Γ2 (H))
/k
CH1 (X (α2 ))
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#!OC3GV $'B;#+ J U
d'# #š c = h ×h +h ×h CH (X (α ) ×X (α ) ) ($1 .
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1
2
0
1
3
2
3
1
0
2
3
1
2
CH3 (X (α1 ) × X (α2 ) )
res
(idX (α1 ) ×p )∗
(X (α2 ) ) )
res
/k
CH3 (X (α1 ) × X (α2 ))
CH3 (X (α1 )
(idX (α1 ) ×p)∗
(X (α2 ) )/k(X (α2 ))
CH3 (X (α1 )k(X (α2 )) )
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&OC+A" 0Q,*/.#+A"‰.#&!563 A&:I$/!O4+,"M"9&:I$‰O4+,"T1#()OCO N =0!*,+E"T10!5T56!1#1 &:59$œ2(KJ-rs&:561#.)3C"6rs+,"T1)OŠ2$6"
idX (α1 ) × p : X (α1 )k(X (α2 )) −→ X (α1 ) × X (α2 )
+A$
idX (α1 ) × p : X (α1 )
−→ X (α1 ) × X (α2 )
& p ^`56+A"61 +,*L$'34[!+,rs+,I$ p fM+,"9$\OC+nrs&:591#.#3C"9rl+n1 &:34E$_P!B,#BA563CDE(#+q;#+ X (α ) ^`56+A"61 +,*L$'34[!+ N
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K1 +,*
p:
k(X (α )) −→ X (α )
+A$
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p :
(X (α ) ) −→ X (α ) .
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1#.#34"6rs+n1e&!3CI$_P:B,)B,5634DE(#+ p +L$_10!5
1)5 : X (α ) ×$ %&)( X (α ) −→ X (α )
OŠ„1#56&}>9+,*A$63C&: "6(#5lOC+"6+A*,&:#;a`2*A$'+A(#5 U ‰QA"†OC&!56",7d*A&:rlrs+;0!#"lOŠk;#B,rs&:#"p$'5'?$'3C&! ;#(
OC+Arlrs+ 7!#&!(#"]@[:&:)"b+,<DE(#+,O4DE(#+M"6&!59$'+ 3}>9&:()$6B|5Œ(#%1 &:3CI$85'2$63C&:#)+,O W X (α )×$%'&)(
j]a`!34$,7˜O‘ux+A"6102*,+¬$6&:1e&!OC&:P:34DE(#+-"6&:(#" N >6!*A+,I$ ;#+ X (α ) ×$ %&)( X (α ) ×
X (α ) U
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+A"9$_+gJ#2*A$'+Arl+AI$_OŠw™0=#59+o!( N ;#+,"9"6(#"T;#(-1 &:3CI$TP!B,#BA563CDE(#+n;)+ X (α ) 10!5
k(X (α ))
OC+ rs&:591#.#3C"9rl+†1#5 Ub– 3C#"93—78$6&:(?>9&:()56"%*,&!rlrs+‹;0!)"sOC 1#56+A()[:+;#(OC+Arlrs+ 7bOC·ac&:56rs+
DE(0!;#562$'34DE(#+ q +,"9$d$6&!$/!O4+,rs+,I$]34"6&!$656&:1 +T+A$˜10!5]*A&:#"9B,DE(#+,I$ X (α )
+,"p$˜;#B,1#O4&ih:B,+
;uv& OŠm5'?$'3C&!0!OC3G$'B\;#+ (h )
wu°2()$'59+„1#!59$,7TO—ux!1#1#O43C*@?$'3C&! (idU × p ) +A"9$"6()5c>9+,*L$'3G[:+ U j] [email protected]"93n#&:(#"
1#59+,#&:)"›(# P:BA#B,562$'+A(#5 g = [x] ;#+ CH (X (α )
& x +,"p$š(# B,O4B,rs+,I$
)
;#+ X (α )
^`3 + +,"9$\$'+AOZDE(#+
fn7H2OC&:59"œ*A+A$‰B,OCBArl+AI$ x 1 +,()$\+,¬10!59$63 N
6= 0
*,()OC3C+A5tA$656+m[R(„*,&:rsrlU +wU (#kB,OCBArl+AI$n;#p+ (x)
+A$o"6 *,OŠ2"6"6+ g = [x]
X (α ) × $%'&)( X (α )
;0!)" CH (X (α ) ×X (α ) ) +A"9$X$'+,O4OC+dDE(#+ (id ×p ) (g ) = g ^ [:&:345Z2(#"6"93H± ¨µo¶:¶)7
92R701#56&:1 2 ³A´ f
‰+†OCªU "6()U 5c>9+,*LU $'3G[K3G$'Bl;#+ (id × p ) 7X3CO˜;#B,*A&:(#OC+l;#&:#*lDE(ux34OY+gJK3C"9$6+‹(#•*LhR*,OC+ d
;0!)" CH (X (α ) × X (α ) ) DE(#3€+A"9$œ;#BA™##3"6(#5 k +A$_$6+,ODE(#+
(X (α2 ) )
2
2
2
2
2
1
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2
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k
1
2
2
1
k
2
2
X (α2 )
2
2
k
2
2
1
2
1
k
k
2
X (α2 )
2
2
2
1 k(X (α2 ))
3
1
(X (α2 ) )
X (α1 )
1
∗
∗
1
(X (α2 ) )
(X (α2 ) )
∗
3
1
1
2
X (α1 )
X (α1 )
1
k
∗
∗
2
(idX (α1 ) × p )∗ (d) = (h31 )
0
2
(X (α2 ) ) .
0
<
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"6(#5SOC+,"SB,OCBArl+AI$'"M+A#P:+,
;)5'!I$§ ]O4+YP:59&:(#1 + CH (X (α ) ×X (α ) ) ?™0(id
m;#+M;)BA$'+A56×rsp3C#+A)58(#)+M+LJK1#59+,"6"93C&:<P:BA#B,56!OC+ N
;)+ d U)– 3C)"63—7
(
"93 i = 3
(h )
(id
× p ) (h × h ) =
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0
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3
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2
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–
)+,OCO4+ \+lrst,rs+!7#&!(#"o"'@[:&:)",7X;u°21#56QA"ŒO4+sO4+,rsrl+†³i¶)7DE(#+sOC+<*AhR*,O4+ +,"p$w562$'34&:##+AO—7
;)&:#*SU "9&:m*,!5659B (h ) = lh O‘ux+A"9$8B,PI!O4+,rs+,I$ U wux!()$656+Y10!59$,7}10!58()m!59P:h(#rs+,I$b;#+˜$'56!#" N
ac+A59$,7 2h +A"9$\!(#"9"63€5'2$63C&:)#+,O U A OC(#"_1#59B,*A3C"[email protected]"63 ;)B,"634P:#+o(##+o+gJK$6+,#"93C&:-™0#34+o;#(
*A&:561)" k +L$ X (#)+ kN [2!593CBL$'B!7)!OC&!56"TO—uv+LJR$6+,#"93C&:-;#+A"\"9*@!OC!3C59+,"o^c*@!#&!#3CDE(#[email protected]
i=1
1
2
2
1 2
2
2
1
2
2
2
X" −→ X
34#;#(#3G$T10!5A7056+A"61 +,*A$634[!+,rs+,I$‰1)(#OCO N =0!*,+s+A$œ1#()"6. N ac&:5 S!56;H70OC+,"œ!1)1#OC34*@2$63C&:#"
56+A" " : CH (X) −→ CH (X" )
+L$
$'5 : CH (X" ) → CH (X)
;)&:I$\OCm*,&:rs1 &:"6B
$'5 ◦ 56+A" " : CH (X) −→ CH (X)
+A"9$SOŠŒrw(#O4$63C1#O43C*@?$'3C&!m;#+,"S*AhR*,O4+,"M1025 [ : k] = n 7ROC+œ;#+,P:59B;#+\O—uv+LJR$'+A#"634&:k^`*La U ±v²S³:7Rj8J U
³ ³ ´ f j]l&!()$'59+T+A$˜10!5d;#BL™0#34$63C&:%rst,rs+!7!(#l*LhR*,OC+_+A"9$˜5'2$63C&:)#+,O#"930+A$˜"6+A(#OC+Arl+AI$Y"9303CO
+A"9$8U ;0!#"U ZO—uv3Crl!P:+b;#+Y56+A" " U R3Rrl!34E$6+,02E$Z&:Œ"6(#1#1 &:"9+MDE(ux()mB,O4B,rs+,I$ [x] ;#+ CH (X" )
Hux+A"9$œ10!"_5'?$'3C&!##+,O!OC&:59"TO—uvB,O4B,rs+,I$ n · [x] 7{OC()3—7)O4+"6+A5' U j][email protected])&:(#"œ@[:&:#"
$'5 (n · x) = $'5 (56+A" " ($'5 (x)))
;Hux&
$'5 (n · x − 56+A" " ($'5 (x))) = 0
+L$\!3C#"93—7K"63O—ux!1#1)OC3C*,2$'34&:†;)+o$65'!)"9ac+,5p$_$'5T+,"p$œ3C?>9+A*A$'3G[:+270#&:(#"_+,;#BA;#(#3C"9&:#"_DE(#+
59+," $65
n · x = " ( (x)).
∗
/k
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/k
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25œ*A&:#"9B,DE(#+,I$,7 n · x +,"9$\;0!)"œO‘ux34r†2P:+‰;#+o59+," " +A$œ+,"p$\;)Q,"_OC&:59"T5'2$63C&:)#+,O U
A
!#"o#&!$656+<*@2"q;)+<™#P:(#56+27#&:()"Œ"6i[!&:#"qDE(#+<OC+,"[2!593CBL$'B," X (α ) +L$ X (α ) "9+l;)B N
1#O4&:3C+AE$\"6()5‰(##+Œ+LJR$'+A#"634&:·;#+Œ;#+,P!56B 2 +A$‰O—uv+LJK3C"p$'+,)*,+w;Hux(#)+w"9$656(#*L$'(#59+m*,+AOCO4(#OŠ!3456+n1 &:(#5
*,+A"<;#+,(KJš[2!5634BA$'BA"m#&:()"<!"6"9(#56+ DE(#+ O4+,"mP:59&:(#1 +,"
+A$ CH (X (α )" ) "6&:I$
(α ))
OC34=#56+A"S;#+\5'!)P 1 1 &:(#5˜$'&:()$ i ;02#" {0, . . . , 5} ^c+A$ jCH;02#"(X{1,
]
f
*,+DE(#3H!"6"9(#56+O‘ux34?>9+,* N
2}
$'3G[R34$'B\;#(‹$'56!#"pac+,59$ U)§ &:(#"T1 &:()[!&:#"M;#&!#*n!1#1)OC3CDE(#+A5SO4+‰5'23C"6&!##+,rs+,I$S1e&:()5S+, ;)B,;#(#3456+
*,&!rlrs+Œ)&:(#"‰O‘u°@[!&:#"‰;)34$œ1#OC()"‰.0!(K$‰DE(#+ 2h +,"p$n5'2$63C&:)#+,O U0§ &:()"@[:&:)";ux!()$'59+w10259$
;#B > W "634P:0!O4Bq^`*La U "9&:(#" N "6+A*A$63C&: U U fbDE(#+œOC+_#&:rw=#56+ l +,"p$M34rl1023C5 U!§ &:()"Y1 &:()[!&:#"˜;#&:#*
+,<;#BA;#(#3456+TDE(#+_OC+M*LhR*,OC+ h +,"p$dOC(#3 N rst,rs+Y562$'34&:##+AO U=A !5d!34OCO4+,(#59",7 h +A"9$˜!(#"9"630(#<*LhK*AOC+
5'?$'3C&!##+,O U j] [email protected])*A+n56BA"6(#OG$/2$T+A"9$T*AOŠ!"9"634DR()++A$_59+,OCQL[:+;)(rst,rs+256P:(#rs+,I$SDR()+n;0!#"
OC+œ*@2"M;#+ h U ‰&:#*\*,&:rsrs+\#&!(#"[email protected][:&!#"T!(#"9"63eDR()+ h = (h ) 7R*,+‰*AhR*,O4+ N *A3e+A"9$TO4(#3 !(#"9"63
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"9&:I$Y562$'34&:##+AOCO4+," UIA &:()5˜O4+,"]rst,rs+,"]56!3C"9&:#",7 h × h +A$ h × h "9&:I$˜B,PI!O4+,rs+,I$
h ×h
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/k
1
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2
i
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1
1
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2
2
1
2
1
1
2
1 2
1
0
1
2
2
0
1
1
1
2
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2h01 × h32 = (h01 × h12 ) · (h01 × h22 ),
C3 OH",uv+,#"9(#34$œDE(#+qOC+n*AhR*AOC+ 2h × h +,"p$œOC(#32(#"6"93X562$'34&:##+AO U
‰Q,"MO4&:56"A7K+A "6&:()"9$'[email protected]!I$_;#+A"Trm(#O4$63C1#O4+,"d;#+*A+,"T*LhK*AOC+A" W d 7K34Oe+A ;#B,*A&:(#O4+‰DE(#+n"6+AOC&:
OŠœ10!5934$'BY;#+ a 7?"6&:3G$ h ×h +A"9$]562$'34&:##+AOCO4+!7i"6&!34$ h ×h +h ×h O—uv+,"p$ U !#"b*A+M;#+A56#34+,5
*@2",7bOCª1#59+,()[!++A"9$m$6+,59rl34#B,+ U 2#"%O—ux!()$656+†*,!",7Z#&:(#"w;#+A[!&:#"<56+La`!3C59+l+gJ#2*A$'+Arl+AI$<OC+
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0
1
3
1
3
2
3
1
0
2
2
3
CH (X (α1 ) × X (α2 ) )
res
0
1
1
(X (α1 ) )
3
1
0
2
(q ×idX (α2 ) )∗
3
2
(X (α2 ) )
CH3 (X (α2 )
(X (α1 ) ) )
res
/k
CH3 (X (α1 ) × X (α2 ))
0
2
(q×idX (α2 )
)∗
(X (α1 ) )/k(X (α1 ))
CH3 (X (α2 )k(X (α1 )) )
&sOC+A" 0QA*/.#+,"Y.#&:593 ,&!E$'!OC+A"]"6&:I$Y3C*A30OC+,"d1#()OCO N =0!*,+E"d10!5d5'21#1e&!59$˜!(KJsrs&:591#.#3C"9rl+A"]1#OŠ?$'"
q × idX (α2 ) : X (α2 )k(X (α1 )) −→ X (α1 ) × X (α2 )
+A$
q × idX (α2 ) : X (α2 )
−→ X (α1 ) × X (α2 )
& q ^c56+,"91e+A*A$634[:+Arl+AI$ q fo+A"[email protected]*A3—7O4+lrs&:561).#3C"9rl+<1 &:3CI$qP:B,)B,5634DE(#+†;#+ X (α ) ^c56+,"91 U
f !5lOC+rltArl+ª!59P:(#rs+,I$†DE(#+¬1#56BA*,B,;)+,rsrl+AE$,7S#&!(#"†+A;#B,;)(#3C"9&:#"‹DE(#+ªOC+
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+,"9$_;)BA™0#3e"6(#5 k(X (α )) 7R+A$T;#+nOCŒ"9(#5c>9+A*A$'3G[R34$'B‰;#+ (q × id )
*,&!rm=#3C)B,[email protected][:+A*YO4+b*@!O4*,(#O!;#+d"6&:q!*L$'3C&!n"6(#5€OC+A"P:B,)B,5'?$'+,()56"€;#+ CH (X (α ) ×X (α ) ) 7
(X (α1 ) )
1
1
3
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1
∗
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2
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0
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d =
×
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+
2
X
bi hi1 × h3−i
+ bh31 × h02 ,
2
& b 7 b +A$ b "6&!E$œ;#+A"\BAOCBArl+AE$6"T;#+ qG 7#+L$œ$6+,O4OC+DE(#+
i=1
1
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(q × idX (α1 ) )∗ (d0 ) = (h32 )
(X (α1 ) ) .
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+A$
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1
1
0
2
3
1
2
2
2
1
1
2
3
2
0
1
c = h31 × h02 + h01 × h32
+A"9$œ5'?$'3C&!##+,O4OC+\*,&:rsrl+o2##&:#*AB U
!5_;#+,"œ!59P:(#rs+,I$'"_"634rl34OŠ!3456+A"S!(RJ-1#56BA*,BA;#+,I$'"A7e)&:(#"_1e&!()[:&:)"œ+Aª;)B,;#(#3456+oDE(#+oOC+A"
*LhR*,OCA +A" h × h +L$ h × h "6&:I$œ5'2$63C&:)#+,O4" U A !5_*,&:#"9B,DE(#+AE$,7eO4+,"T*A&:5659+,"61 &:#;#!#*,+A"
1
1
1
2
2
1
0
2
(h01 × h22 ) · c = h31 × h22 + h01 × h52 ,
(h11 × h12 ) · c = h41 × h12 + lh11 × h42 ,
(h21 × h02 ) · c = h51 × h02 + h21 × h32 ,
"9&:I$TB,P:!OC+Arl+AE$Y5'?$'3C&!##+,O4OC+A" U j])™07K+, ()$63COC34"'!I$ W #&!()[:+,!( O—ux!56P!(#rl+AI$S;#+\$'5'2#"[email protected]
)&:(#"d1e&!()[:&:)"Y+A†;#BA;#(#3C59+_DR()+\OC*,&:5956+,"91e&!#;0!#*A+ 2h × h = h × 2h +A"9$Y5'2$63C&:)#+,O4OC+
+L$œO W +,)*,&:59+!7{*,&:rsrs+ l +,"p$œ3Crs10!3C5A7
1
1
4
2
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7
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;)+q*A&:#"9B,hDE(#×+,)*h,+w+562h$'34&:×##+hAOCO4+ hU × h + h × h h × h + h × h
²8340!O4+,rs+,[email protected]@O4+ba`!34$€DE(#+ J "6&:3G$€5'2$63C&:)#+,O4OC+b"93CP:#3G™0+];#&:)*YDE(uv3COI+gJ)34"9$6+˜()#+Y*A&:5659+,"91e&: N
;#!#*,+ f ;0!#" Hom(X (α ), X (α )) $'+,O4OC+˜DE(#+ f = J U:§ &:(#"]@[:&!#"]10!5]23COCO4+,(#59"€1#56&:(K[:B
;#!#"sOŠ 1#56&!1e&:"934$63C&: <„DR()+ J BA$'!=#O434$<(# 3C"6&!rl&:591#.#34"6rs+‹+,I$'59+ X (α ) +L$ X (α )SU
\+OŠorltArl+_a`I HA&:7ROŠŒ*A&:5659+,"91e&:);0!#*A+‰$'56!#"61 &:"9B,+ J BA$'!=#OC3G$S()‹3C"9&:rl&!561#.#34"6rs+T+,I$'59+
+A$
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X (α )
1 &:#;#!#*,+‰5'X2$6(α3C&:#))+,OCO4+ UE– 34#"63‘7K;Hu°!1)56Q,"SOC+\$6.#B,&!56Q,rs+;uv3C"9&:rs&:561#.)3C"6rs+w^`*Aa U *,&:59&:OCOC!3C59+ fLN 7
O4+,"*A&:5659+,"91e&:);0!#*A+,"
;#+
+L$ f ◦ f ;#+ End(X (α )) "6&!E$;#+,"o!( N
(α ))
$6&:rs&:561#.)3C"6rs+,"ors&!$'3G[R3CfDE(#◦+,"of;#+ XEnd(X
A
+
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(α )
X (α )
[email protected]!O43C"9+%(#›34"6&:rs&:591#.#3C"9rl+wrl&!$634[R3CDE(#+w;#+ X (α ) ;02#" X (α ) +L$ f ()š3C"9&:rs&:561#.)3C"6rs+
3
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V ux34"6&:rs&:591#.#3C"9rl+‰rl&!$634[R3CDE(#+
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;0!#"
+ OC+A*A$'+A(#5m#&2$'+,56·DE(#+ +A$ )+ "6&:I$<10!"%)B
*,+A"6"6!3C59+,rs+,I$<O—Xuv3CI[(α:+A56)"9+‹O—uv(##+lX;#(α+ O—)ux!Ub()V $656+ U j]+AFe+A$,78*,&!rlrs+ ff ◦ f f− id = r +,"9$N
E(#OCO4+n"6(#5TO4+n*,&:591#" 7#*!uv+,"p$‰(#-BAOCB,rs+,I$_#34OC1 &!$'+AI$T+A$ f ◦ f = id + r +,"p$œ=#3C+A()
3C"9&:rs&:561#.)3C"6rs+‰;)&:I$_O—uv3CI[:+A56"6+n+A"9$_OCw"6&:rsrs+‰™0#34+ id − r + r − . . . O+A-+,"9$_=)3C+,
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P:BA=#5634DR()+s^`&:( $'.)B,&:593C+;#+A"_*@2$6B,P:&:593C+A"/fY+L$_;u°2OCP:QA=#56+*A&:rsrm()$/?$'34[!++A$_DE(#+AOCDE(#+A"œ)&!$'34&:#"
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V
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F (Y ) = Hom(Y, X)
1 &:(#5Y$6&:()$S&:=R>9+A$ ;#+ j])™07K&:‹;#BA"634P:#+1025 Ens) OŠo*@2$6B,P:&:593C+\;#+,"Sac&!#*A$6+,(#59"
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2$'BAP:&:5634+S;#+A"d+,#"9+,rm=#O4+,"]+L$d10!5h. =
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” +,"_;#34FeB,59+,I$'"œ1 &:3CI$'"TBL$/!I$œ1#59B,*,34"6BA",70#&:()"œBA#&: HA&:#"œOC+nO4+,rsrl+ U
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$ $ 1 $ J 1 $ " ,& F(C, Ens) .
X
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C
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.
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() R N rl&R;#(#O4+sO43C=#59+l;)+†5'2#P s +A$ ϕ : F → M O‘uxBA1#3Crs&:561).#3C"9rl+%DE(#3Y+,I[:&:34+‹OC+ i
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1 &:()[!&:#" ;#&!#*·B,*A563456+·O4+¬rs&K;#()OC+ F "9&:(#" OŠ„ac&:56rs+ M ⊕ ker ϕ & M +,"p$-(# R N
rs&R;#(#OC+\3C"9&:rl&!561#.#+ W M U A !5M"6()34$'+27 ℘F = ℘M ⊕ ℘ ker ϕ +A$_*,&!rlrs+ ker ϕ +A"9$_3C)*,OC(
;#!#" ℘F 7 F/℘F = M /℘M 7H;ux& ker ϕ/℘ ker ϕ = 0 7H1#!5"6()34$'+ ker ϕ = ℘ ker ϕ +A$
*A&:#"9B,DE(#+,)*,+¬;)( O4+,rsrl+†;#+ " d7 ker ϕ = 0 U ‰QA"sO4&:56"A7 F = M 7 ϕ +,"p$s(#
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rst,rs+w5'2#P‹DE(#+ M U– 3C#"93—7{3COX+LJK34"9$'+w(# 3C"9&:rl&!561#.#34"6rs+ ϕ : F → M U j]#"9(#34$6+!7e#&:(#"
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DE(uv3CO€+LJK3C"pW $'+Œ(#¬rs&:561).#3C"9rl+ ϕ : F M→ M $6+,OXDE(#U + ϕ = 1 &:(#5_† (#¬*,+,5p$/!34ªBAOCBArl+AE$ f
ux!1#10259$'+A0!I$Z10!" W ℘ U?” &:rsrs+ ϕ +,"p$8(#q3C"9&:rs&:561#.)3C"6rs+8O4&:56"9DE(ux&!ŒO4+]OC&R*@!O43C"6+Z10!556!1 N
1 &:59$ W OCw10!5p$'3C+‰rm(#O4$63C1#O43C*,2$'3G[:+_;#BA™##3C+n1025 ℘ 7)3COH+gJK3C"9$6+n(#-B,O4B,rs+,I$ f ux!1#1#!59$6+,0!I$
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W+A$w℘!34+A#"$‰63‘>97&!"6(0(#!5 I$%OC+†7€OCrl+A"ŒtArl21#+‹OC34e5 *@,:2OC$6+†3C&:1e#"&:()5 ψ Ub§ &:+A(#$ "<1 &:"6&!#"<"9&:!I$OC&[email protected]€56" 10!f5q*,=&!#"f9$656f(#f*L$'3Cf&!#"Œ^ frst,∈/rs+,℘"Af 7
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1 &:(#5Y$6&:()$S3C;#B,!Oe1)56+,rs3C+A5 ℘ ;#+ R $6+,OeDR()+ M ' F 1 &:(#5Y(#‹*A+,59$'!3C†rl&R;#(#O4+\O43C=#59+ F U
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$'+AOCO4+nDR()+q1 R&:()U$T§ $6&:()$œ3C);#3C*A+ i ;#+ {1, . . . , n} 7 M "6&:3G$_(#rl&R;#(#O4+OC34=#56+ U
&:(#"\BA$'!=#OC34"6"9&:#"‰rl!34E$6+,02E$ ⇒ ³ U &:()$‰;u°2=e&:59;7 #&:(#"‰!OCO4&:#"œ1#56&!()[:+A5DE(ux()
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B,O4B,rs+,I$_;#+ N fS+,"p$œ#B,*A+,"6"6!3C59+,rs+,I$œE(#O U j]-+AFe+A$,70"63 t +,"p$\E(#OH;0!#" N 7#*A+,OŠ%"93CP:#3Gac+
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$'&!()$ RN rs&R;#(#OC+ N DE(#3)+,"9$8$'+,OKDE(#+ N = 0 1e&:()58$'&:(K$]3C#;)3C*,+ i +,"p$](#%rs&K;)(#OC+YR()O U!” +,*,3
BA$'!=#O43—7€#&:(#"w#&:(#"m;)&:##&:)"mr†23CI$'+A0!I$w() RN rs&K;#()OC+<OC34=#56+¬^`;)+†5'2#P·"6()1eBA563C+A(#5m&:(
B,P:!O#!(%)&:rm=#59+M;#+MP:BA#B,562$'+A(#56"];)+ M f8+A$b*,&:#"p$'59(#3C"9&:#"](#%BA1#3Crs&:561).#3C"9rl+ ϕ : F → M
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1#.#34"6rs+n+,"p$\"9(#5c>9+A*A$634aZOC&R*@!O4+,rs+,I$M+A*/.#!DE(#+ r 7#O4+nrl&R;#(#O4+ M BL$/!I$_(#rl&R;#(#O4+‰1#59& N
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