# Inflation functor on cohomology

From Groupprops

## Definition

Suppose is a group and is a normal subgroup of . Suppose is an abelian group and is a homomorphism of groups, making into a -module. Denote by the subgroup of fixed pointwise by all elements of .

Then, the **inflation homomorphism** is defined as the composite:

where the first map is the restriction homomorphism corresponding to the quotient map and the second map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the injection of -modules.