Topics in Cohomology of Groups_Serge Lang.pdf

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Theorem 3.6. Let a be a finite group. Let X = {Xr}(r E Z)
be the standard complex. Then X is Z[G]-free, acyclic, and such
that the association
A ~ Horns(X, A)
is an exact functor of Mod(G) into the category of complexes of
abeIian groups. The corresponding cohomological functor H is
such that H~ = Aa/SGA.
Examples. In the standard complex, the group of 1-cocycles
consists of maps f : G ---* A such that
f(a) + crf(v) = f(ar) for all o, r E G.
The 1-coboundaries consist of maps f of the form f(~) = aa - a
for some a E A. Observe that if G has trivial action on A, then by
the above formulas,
HI(A)=Hom(G,A).
In particular, H 1 (Q/Z) = G is the character group of G.
The 2-cocycles have also been known as factor sets, and are
maps f(a, r) of two variables in G satisfying
f(o', T) + f(o'r,p) = o'f(r,p) + f(cr, vp).
In Theorem 3.5, we showed that for finite groups, H is erased by
Ms. The analogous statement in Theorem 3.1 has been left open.
We can now settle it by using the standard complex.
Theorem 3.7. Let G be any group. Let B E Mod(Z). Then
for all subgroups G' of G we have
Hr(G',Ma(B)) = 0 for r > O.
Proof. By Proposition 2.6 it suffices to prove the theorem when
G' = G. Define a map h on the chains of the standard complex by
h: Cr(G, MG(B)) ---. Cr-'(G, Ma(B))