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.