52 VI. COHOMOLOGICA L INVARIANT S O F O
n
, SO,
Prove als o tha t th e restriction map
Inv(£6, Z/2Z) - Inv(G2, Z/2Z )
is an isomorphism. I n particular, Inv(2?6 , Z/2Z) ha s basis {1, ^3} over H(ko), wher e / 3 is
the mo d 2 component o f the Rost invarian t i n the case wher e EQ is simply connected .
22.10. T E S T I N G INVARIANT S O N SMAL L SUBGROUPS ? Th e reade r ma y hav e
observed tha t fo r severa l "large " group s G an d prime s p , on e can fin d a "small "
(e.g., finit e abelian ) subgrou p H o f G suc h tha t th e restriction ma p
Res : Inv(G, C) -+ Inv(tf, C)
is injective , fo r every C whic h i s an abelia n p- group. Fo r p = 2, this i s the case for
the followin g group s G , with H abelia n o f type ( 2 , . . . , 2):
G
On
50
n
_ i
G2
split F
4
split EQ
&n
isoikH
n
n-1
3
5
3
[n/2]
reference
elementary
elementary
18.4
22.5
22.9
24.11
When p 2 , there ar e similar example s wit h H abelia n o f type ( p , . . . ,p) , for
instance:
G p #
P G L
P
p Z/pZ x /xp
split F
4
3 Z/3 Z x /x
3
x /x3
split simpl y connecte d EQ 3 Z/3 Z x /x
3
x /x
3
x /z3
split £ 7 3 Z/3 Z x /x
3
x /x
3
x /i3
split E
8
5 Z/5 Z x /x
5
x /x5
[The firs t tw o are not difficult t o prove, b y methods simila r t o those use d i n the
present chapter . Th e surjectivity o f H 1(^1F4 x /x
3
) —• i 71 ( * , ^
6
) , se e e.g. [GaOl ,
§3], give s th e third. Th e last on e follows fro m [Ch94]. ]
Note tha t i n each case , H i s one of the subgroups use d b y Reichstein i n [R e 00]
for givin g a lowe r boun d o f the essentia l dimensio n o f G. I t woul d b e interestin g
to hav e mor e example s o f such pair s (G , H); on e should no t expect, however , tha t
one subgrou p H woul d alway s b e enough, no r that H woul d alway s b e elementar y
abelian.
23. Cohomologica l invariant s m o d 2 m
Let C b e a finit e r^
0
-module whos e orde r i s a powe r o f 2. I n thi s section , w e
compute Inv(Quad
n
,G).
Let u s say that a clas s z G Hl(k, Z/2Z ) i s liftable if , fo r every m 1, z belong s
to the image of the natura l ma p W(k, Z/2 m Z(z)) - W(k, Z/2Z) . Recal l tha t "(i)"
denotes th e i-th Tate twis t a s in 7.8. Note that :
I f i = 1, ever y z i s liftable , sinc e Z/2 m Z(z) i s just th e Galoi s modul e o f
2 m -th root s o f unity s o that th e map
H^k, Z/2 mZ(i)) - Hl{k, Z/2Z )
is non e othe r tha n th e projection k*/k* 2™— » /c*//c*2.
Th e cup product o f two liftable element s i s liftable .
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