# Coinflation functor on homology

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## 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 quotient of by the subgroup of elements of the form , with .

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

where the first map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the surjection of -modules, and the second map is obtained by applying the corestriction functor on homology to the quotient map .