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 $\phi :G\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$ 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 $\mathrm{i}\mathrm{n}\mathrm{f}:H^{*}(G/N;A^N)\to H^{*}(G;A)$ is defined as the composite:

$H^{*}(G/N;A^N)\stackrel{\mathrm{r}\mathrm{e}\mathrm{s}}{\to }{H}^{*}(G;A^N)\to H^{*}(G;A)$

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