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.