# Restriction functor on cohomology

## Definition

### General definition

Suppose are groups and is an abelian group. Suppose , , and are homomorphisms such that , i.e., the -action and -action on are compatible. Then, we get an induced homomorphism between the cohomology groups:

This homomorphism is termed the *restriction homomorphism* and the functor that sends the map 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 and since is defined in terms of them. In other words, we can define the restriction functor in terms of and .

### Typical case

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