CHAPTE R I I
Cohomological preliminaries : th e local cas e
The result s o f this chapte r an d of the next on e are essentially standard s Th e
basic ide a of Section 9 originates wit h [F a 51]. Muc h o f §§6, 7, and 9 can be found i n
[Ar 75] when C = Z/2 Z an d k has characteristic ^ 2 and in [El 82] when C = Z/nZ
and k contain s a primitiv e n-t h root o f unity . A resum e o f the general cas e was
given i n [S e 92a] an d reproduced i n the fifth editio n o f [S e 65].
6. A n exac t sequenc e
Let G be a profinit e grou p an d N a (closed ) norma l subgrou p o f G. Se t F =
G/N, an d let C be a discrete T-module , whic h w e also vie w a s a discrete G-modul e
with trivia l actio n b y N.
T H E O R E M 6.1. Suppose that
(1) H i(N,C) = 0 ifi 1.
(2) The exact sequence 1—iV—G^T—* 1 splits.
Then for all i 0, we have an exact sequence
o - #*(r, c) -^ H\G, c) A ir-^r,Hom(7v, c)) -+ o,
where TT is the natural map defined by G T and r is as described below.
[Here, Hom(iV , C) denote s th e group o f continuous homomorphism s N C
(i.e., thos e wit h ope n kerne l an d hence finite image). ]
P R O O F . Th e spectral sequenc e o f group extensions , a s constructed i n [HS53] ,
gives
Hp(T,Hq(N,C)) = H(G,C).
We hav e H°(N,C) = C an d H^N.C) = Hom(AT,C) . B y hypothesis (1), we also
have H l(N, C) = 0 for i 1, hence th e spectral sequenc e reduce s t o a lon g exac t
sequence
iT(T, C) - ^ H\G, C) ^ H*- 1 ^, Hom(AT , C)) -
^ H l+1 (T , C) - Hi+1 (G , C) - .
Here d is the "^-differential" o f the spectra l sequence . Hypothesi s (2 ) implies tha t
7 T is injective, henc e d = 0 , so that th e exact sequenc e abov e split s int o shor t exac t
sequences
0 - i T ( I \ C) ^ H\G, C) ^ IP' 1 ^, Hom(AT , C)) - 0.
a
It woul d b e a good exercis e fo r the reader t o deduce th e content o f these tw o chapters fro m
the genera l machiner y o f etal e cohomology . Th e main difficult y i s the compariso n betwee n th e
residues give n b y the etale theor y an d the residues define d i n the present text ; the y ma y differ b y
a sign . Whic h one ?
15
http://dx.doi.org/10.1090/ulect/028/04
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