34
V. RESTRICTIO N AN D CORESTRICTIO N O F INVARIANT S
REMARK
13.3. On e ma y as k th e followin g questions :
(a) Whe n i s Res: Inv(G, H) -• Inv(# , H) injective ?
(b) Wha t i s th e imag e o f Res ? I n particular , whe n i s i t equa l t o th e se t o f
elements fixed b y N(ko)/H(ko)?
When G i s finite, w e give in §15 some partia l answer s t o thes e questions .
14. Corestrictio n o f invariant s
(The materia l i n §14 and §15 will only b e use d i n §34. )
14.1. I n this section, we assume that G is a finite group (viewe d as a "constant "
group schem e o f dimension 0 ) an d H i s a subgroup o f G. W e also assum e tha t th e
functor H is the functo r k i- » H(k,C) associate d wit h som e Galoi s modul e C ove r
k0.
14.2. Befor e definin g th e corestriction , i t i s convenien t t o mak e som e defini -
tions. I f k is an extension o f ko and K i s an etale k-algebra wit h K = YYi=i ^ (wit h
the ki finite extension s o f /c) , we pu t
n n
H(K,C) = Y[H(ki,C) an d H
X(K,G)
= ]jH
1G).u{k
i=l i=l
We writ e Cor : H(K,C) » H(k,C) fo r th e su m o f th e corestriction s H(ki,C)
H(k,C). I f Y Spec(K) i s th e etal e /c-schem e associate d wit h K, w e als o writ e
H(Y, C) fo r H(K, G) , an d Cor
y
o r Cor f fo r Cor : H(K, C) - H(k, G) .
(Note that H(Y, C) i s the etale cohomology of Y = ] J Spec(A^), with coefficient s
in th e shea f define d b y C.)
If a i s a cohomological invarian t i n Inv(G, G), and i f x = (xi,... , xn) i s an ele-
ment o f H l(K, G) , then a(x) G H(K, C) i s defined a s the tupl e (a(xi),... , a(x
n
)).
DEFINITION
14.3. Le t a b e i n Inv(i7 , G), an d le t T b e a G-torso r ove r k. Th e
action of H o n T make s T int o an iJ-torsor T # ove r T/H, an d T/H i s a finite etal e
/c-scheme, henc e o f th e for m Spec(if) , wher e K i s a n etal e k- algebra, cf . 2.1. B y
the above ,
CL(TH)
is a well-defined elemen t o f H{K, G) . Pu t
b(T) = Cor(a(T
H
)) i n H(k, G) ,
where Cor : H(K, C) - » H(k, C) i s as in 14.2 above. I t i s clear tha t T \-+ b(T) i s an
element o f Inv(G, G), which w e call th e corestriction of a an d denot e b y Cor^(a )
or simpl y b y Co r (a). Thi s define s a homomorphis m
Cor:Inv(tf,G)-Inv(G,G).
We have th e usua l formula :
PROPOSITION 14.4. Co r o Res = n, where n = ( G : H).
PROOF.
Le t a be a n elemen t o f Inv(G , G). Le t a\ b e it s restrictio n t o H an d
put b = Cor(ai) . W e have to prov e tha t b = n a, i.e. , tha t
b(T) = n a{T)
for ever y G-torso r T ove r ever y extensio n k of ko. A s i n Def . 14.3, let TH b e th e
i7-torsor ove r T/H define d b y th e actio n o f H o n T , an d le t u s writ e T/H a s
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