14. CORESTRICTIO N O F INVARIANT S
35
Spec (if), wher e K i s a n etal e k- algebra. I f T^.G denote s th e G-torso r deduce d
from TH b y th e embeddin g H G, w e have
ai
(TH) = a(T
H
.G) i n H(K,C) an d b(T) = COT% (ai(TH)) i n H(k,C).
There i s a commutativ e diagra m o f G-torsors :
TH.G ^T
Y Y
Spec(X) = T/H - T/G = Spec(fc),
where th e to p ma p i s give n b y (£,# ) i— tg. Thi s diagra m show s tha t th e G-torso r
TH.G i s isomorphic to the pullback o f T b y T/H - T/ G = Spec(/c) ; hence a(T
H
.G)
is th e imag e o f a(T) b y th e natura l ma p H(k
1
C) H(K,C). Sinc e [i f : fc] n ,
this show s that Cor (a(TH.G)) = n-a(T), i.e. , tha t 6(T ) = n-a{T). D
14.5. TH E DOUBL E COSE T FORMUL A ( A L A MACKEY) . Le t H an d H' b e tw o
subgroups o f G . W e wish t o comput e th e ma p
Resg' o Corg : Inv(iJ, G) - Inv(#' , G).
To do so, choose a set S o f representatives o f H\G/H
r.
Fo r each s G S , on e define s
Hs = sH's- 1C\H an d H'
s
= s^HsC] H'.
One ha s s~ 1Hss = H'
s
. Defin e
H g + n
H s
(14.6) /
s
: Inv(iJ, G) ^ ^ Inv(ff
a
, G ) ^ Inv(i^ , G) ^ ^ InvCff' , G),
where i* i s th e isomorphis m associate d wit h th e conjugatio n i
s
: H'
s
iis , cf .
Prop. 13.1. (Th e righ t arro w make s sens e becaus e H f
s
C H f.)
REMARK
14.7. Not e that, whe n s is replaced b y fish', wit h h e H an d ft' G iiT ,
then ii"
s
i s replaced b y hHsh -1 an d if £ b y (h ,)~1H's ^ This doe s no t chang e /
s
,
because o f Prop. 13.1.
Note als o that z * o Res#s i s none othe r tha n th e restrictio n ma p
# : I n v ( f f , C ) ^ I n v ( J C C )
corresponding t o th e embeddin g (j) s: H's H give n b y x i—» sxs
- 1
.
PROPOSITION
14.8. Res^ f oCor ^ = J ] /
s
/o r f
s
as in (14.6).
PROOF. Le t a b e i n Inv(i7 , G). Writ e a
s
fo r th e imag e o f a i n Irw(H
s
,C) b y
restriction; writ e a ^ fo r th e imag e o f a
s
i n Inv(iJ
7,G)
b y conjugation , an d pu t
cs = Cor^(a^ ) = /
s
(a), whic h belong s t o Inv(/f
/,G).
Le t 6 = Cor G(a) an d
c = Res ^ (b). W e have t o sho w tha t c = J2c s, i.e. , that , i f T' i s an if'-torso r ove r
an extensio n fc/fco, we have c(T" ) = X^sCO -
Let T'.G b e the G-torso r deduce d fro m T ' b y i/'— G ; the natura l actio n o f H
on T'. G make s T'. G int o a n iJ-torso r T ove r T/H] w e denot e th e etal e fc-scheme
T/H b y y . W e have a(T) G i/(F, G), an d th e definitio n o f c shows tha t
c(T') = Cor y(a(T)).
If s G 5, le t G
s
H's~lH b e th e doubl e cose t o f G containin g s _ 1. W e hav e
G = IJ
s(E
s Gs (disjoin t union) . Thi s decompositio n o f G give s a correspondin g
Previous Page Next Page