Restriction functor on cohomology
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 .
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.