Topics in Cohomology of Groups_Serge Lang.pdf

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given by the homomorphism Aa/SGA --~ AG'/SG, A, with G' act-
ing on A in the manner prescribed above, via A.
By the uniqueness theorem, there exists a unique morphism of
cohomological functor (5-morphism)
A* : HG "-+ HG' o ~x or HG--~HG, o~x,
the second possibility arising when G and G' are finite. We shall
now make this map )~* explicit in various special cases.
Suppose A is surjective. Then we call A* the lifting morphism,
and we denote it by lif(gG,. In this case, G may be viewed as a factor
group of G ~ and the lifting goes from the factor group to the group.
On the other hand, when G ~ is a subgroup of G, then A* will be
called the restriction, and will be studied in detail below.
Let A E Mod(G) and B E Mod(G'). We may consider A as
a G'-module as above (via the given A). Let v : A --* B be a
G'-morphism. Then we say that the pair (A,v) is a morphism
of (G, A) to (G', B). One can define formally a category whose
objects axe pairs (G, A) for which the morphisms are precisely the
pairs (k, v). Every morphism (k, v) induces a homomorphism
(A,v). : H~(G,A) ~ H~(G',B),
and similarly replacing H by the special H if G and G' are finite,
by taking the composite
H"(G,A) x*, Hr(G',A) H~,(,)H~(G, B)"
Of course, we should write more correctly Hr(G ', ~5~(A)), but usu-
ally we delete the explicit reference to r when the reference is
clear from the context.
Proposition 1.1. Let (A,v) be a morphism of (G,A) to G',B),
is a morphism of (G,A) to (G",C), and one has