36
V. RESTRICTIO N AN D CORESTRICTIO N O F INVARIANT S
decomposition o f T' .G a s T' .G
\\T'.H's~lH.
Henc e th e torso r T decompose s
as th e disjoin t unio n o f th e torsor s T
s
= T
1 .H's-1
^; le t Y
s
= T s/H b e th e bas e
of T
s
. I t i s no t difficul t t o se e tha t T
s
i s deduce d fro m T' b y first restrictin g th e
group actio n fro m H r t o H'
s
, an d the n takin g th e correspondin g iJ-torso r vi a th e
embedding (j)
s
: H's - H o f Remar k 14.7. Thi s show s tha t a(T
s
) = a'
s
{T').
We have y = ] J Fs, henc e H(Y, C) = [ J H(YS,C). B y wha t w e have just seen ,
the component s o f a(T) ar e the a'
s
(T'). Th e corestrictio n o f a(T) i s the su m o f th e
corestrictions o f its components . Thi s give s
Corr(a(T))
= ^ C o r ^ ( a ; ( T O ) ,
i.e,c(T') = Ecs(T') .
REMARK
14.9. Th e decompositio n o f T a s \\T
S
i n the proo f o f Prop. 14.8 is a
general fact abou t torsor s which is true over any base (no t necessaril y the spectru m
of a field).
15. Stabilit y
DEFINITION
15.1 . W e keep the notatio n o f 13.2 and 14.1. Let a be a n elemen t
of Inv(iJ, C). A s in [CE56 , XII.9] , w e say that a is stable (wit h respec t t o G) i f it
has th e followin g property :
For every H
r
C H and every s G G such that sH
fs~x
C H, the two
maps i, j : H' » H defined by
( 1 5 ' 2 ) '( \ A U \ - 1
i(x) = x and j{x) sxs
are such that i*(a) = j*(a) in lnv(H
,
J
C).
Note th e specia l cas e o f (15.2) wher e H' H. I n tha t case , s belong s t o th e
normalizer N o f H i n G , an d th e conditio n i*(a) = j*(a ) mean s tha t a i s fixed
under th e actio n o f N/H o n Inv(iJ , G) define d i n 13.2.
PROPOSITION
15.3. (1) Every element of the image of the restriction map
Inv(G, G) - Inv(# , G) is stable,
(2) Let a be a stable element of Inv(H,C). Then
Res(Cor(a)) = n-a where n = ( G : H).
PROOF.
Assertio n (1) follow s fro m 13.1 . Assertion (2 ) follow s fro m th e doubl e
coset formul a 14.8 applied t o H' = H. Indeed , Res(Cor(a) ) i s the su m of the /
s
(a),
and th e stabilit y o f a implies that f s(a) = n
s
-a where n
s
= (H : Hs); on e then use s
the fac t tha t n i s the su m o f the n s, a s one sees by writing G/H a s a disjoint unio n
of if-orbit s indexe d b y S.
The mai n consequenc e o f the propositio n is :
COROLLARY
15.4. If (G : H) and \C\ are relatively prime, then
Res: Inv(G, G) - Inv(# , G)
is injective, and its image is the subgroup of Inv (if, G ) made up of the stable ele-
ments.
This applie s i n particula r whe n i f i s a p-Sylo w subgrou p o f G , an d G i s a
p-group.
Previous Page Next Page