Inflation functor on cohomology

Definition

Suppose $G$ is a group and $N$ is a normal subgroup of $G$. Suppose $A$ is an abelian group and $\varphi :G\to \operatorname {Aut} (A)$ is a homomorphism of groups, making $A$ into a $G$-module. Denote by $A^{N}$ the subgroup of $A$ fixed pointwise by all elements of $N$.

Then, the inflation homomorphism $\operatorname {inf} :H^{*}(G/N;A^{N})\to H^{*}(G;A)$ is defined as the composite:

$H^{*}(G/N;A^{N}){\stackrel {\operatorname {res} }{\to }}H^{*}(G;A^{N})\to H^{*}(G;A)$

where the first map is the restriction homomorphism $H^{*}(G/N;A^{N})\to H^{*}(G;A^{N})$ corresponding to the quotient map $G\to G/N$ and the second map is the natural map obtained by viewing cohomology as a covariant functor in its second coordinate, applied to the injection $A^{N}\to A$ of $G$-modules.

Related notions

• Restriction functor on cohomology
• Corestriction functor on homology
• Coinflation functor on homology