16 II . COHOMOLOGICA L PRELIMINARIES : TH E LOCA L CAS E
This prove s th e theorem .
As for the explicit definitio n o f r, i t i s as follows: B y [H S 53, p . 121], an elemen t
a G Hl(G, C) ca n be represented by a cocycle f{gi^...1gi) whic h is normalized (i.e. ,
is equal to 0 when one of the gj i s equal to 1) and which only depends on the imag e
of gj i n r fo r j 2. Fo r 71, ..., 7^_ i i n T , le t r(/)(7i,... , 7i-i) b e th e elemen t o f
Hom(N, C) define d b y
r(/)(7i, •,7i-i)(™) = f(n,g
u
... , g %-i)
where th e gj ar e representative s o f the 7 ^ in G and n run s throug h th e element s of
N. Th e cochai n r(f) i s an (i l)-cocycle of T with values in Hom(iV, C). Th e clas s
of r(f) i n i7 z _ 1(r, Hom(A^, C)) i s independent o f the choic e o f / ; i t i s r(a).
REMARK
6.2 . Whe n i = 0 , ir
_ 1
(r,Hom(N,C) ) i s to b e interprete d a s 0.
When i = 1, the ma p r: H^G.C) -* tf°(r, Hom(N , C)) = Hom(N,C)
r
i s th e
restriction ma p H^G, C) - H
l(N,
C) = Rom(N, C).
REMARK
6.3 . Bewar e that differen t construction s o f "the " spectra l sequence of
group extensions (se e [B e 81]) could lead to an r ma p different fro m our s by the sign
(—1)2_1,
an d the n Propositio n 6. 6 woul d als o involv e a sign . Suc h sig n problem s
are on e o f th e reason s wh y Galoi s cohomolog y i s bein g use d her e instea d o f th e
(more powerful ) etal e cohomology .
REMARK
6.4 . Th e r-ma p i s obviously functorial : A commutative diagra m
1 N' G' T 1
1
N G
r 1
gives rise t o a diagra m
Hl(G',C)
^-
1(r /
,Hom(A^
/
,C))
i 1
H^G.C) ^- 1(r,Hom(A^,C))
which i s commutative . (Assuming , o f course , tha t H l{N,C) an d H l(N'\C) ar e 0
for al l i 1.)
6.5.
CU P PRODUC T FORMULAS .
Le t Ci , C
2
, C
3
b e T-modules , wit h a
pairing C\ x C2 —• C3. Assum e th e hypothese s o f the theore m fo r C\ an d C3 . Le t
ai G H^iCCi) an d a
2
G #2 2 (r,C
2
) C ^ 2 ( G , C
2
) . Le t a
x
-a
2
G H l^(G,C3)
be th e cu p produc t o f a\ an d a
2
relativ e t o th e give n pairing .
PROPOSITION 6.6 . We haver(ai -a
2
) = r(a{) -a
2
in iT1+i2_1(r,Hom(Af, C
3
)).
(Wehaver(ai) G i T ^ Q ^ H o m ^, d)) an d a
2
G
iP2
(r,C
2
); th e cup produc t
r{a\) -a
2
i s relative t o th e natura l pairin g Hom(A/' , C±) x C
2
Hom(iV, C3).)
PROOF.
Choos e a normalized cocycl e j \ representing
OL\
o f the typ e describe d
in th e proo f o f Th . 6.1. Le t f
2
b e a normalize d r-cocycl e representin g a 2, whic h
we vie w als o a s a G-cocycle . The n -a2 ca n b e represente d b y th e cu p produc t
A 'A; recal l tha t
(A 'AX^i, -,^1+22) = A(#i , •••,&!)•
gi '"9ilf2(9i1
+i, ,0ii+i2)-
A direct computatio n show s that r(/ i -^2 ) = r (A) 'A - Th e propositio n follows .
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