Inflation functor on cohomology

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Definition

Suppose G is a group and N is a normal subgroup of G. Suppose A is an abelian group and \varphi:G \to \operatorname{Aut}(A) is a homomorphism of groups, making A into a G-module. Denote by A^N the subgroup of A fixed pointwise by all elements of N.

Then, the inflation homomorphism \operatorname{inf}: H^*(G/N;A^N) \to H^*(G;A) is defined as the composite:

H^*(G/N;A^N) \stackrel{\operatorname{res}}{\to} H^*(G;A^N) \to H^*(G;A)

where the first map is the restriction homomorphism H^*(G/N;A^N) \to H^*(G;A^N) corresponding to the quotient map G \to G/N and the second map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the injection A^N \to A of G-modules.

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