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