Restriction functor on cohomology

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Definition

General definition

Suppose G_1,G_2 are groups and A is an abelian group. Suppose \varphi_1:G_1 \to \operatorname{Aut}(A), \varphi_2: G_2 \to \operatorname{Aut}(A), and \alpha: G_1 \to G_2 are homomorphisms such that \varphi_2 \circ \alpha = \varphi_1, i.e., the G_1-action and G_2-action on A are compatible. Then, we get an induced homomorphism between the cohomology groups:

\operatorname{res}^{G_2}_{G_1}: H^*_{\varphi_2}(G_2,A) \to H^*_{\varphi_1}(G_1,A)

This homomorphism is termed the restriction homomorphism and the functor that sends the map \alpha:G_1 \to G_2 to this homomorphism is the restriction functor on cohomology.

Note that the direction of this homomorphism is reverse to the direction of the original homomorphism. The association gives a contravariant functor.

Note that it is sufficient to specify \varphi_2 and \alpha since \varphi_1 is defined in terms of them. In other words, we can define the restriction functor in terms of \varphi_2:G_2 \to \operatorname{Aut}(A) and \alpha: G_1 \to G_2.

Typical case

The typical case where we talk of the restriction functor is where the map \alpha: G_1 \to G_2 is injective, and we naturally identify G_1 with its image subgroup of G_2. We thus talk of restricting cohomology to a subgroup.

Related notions