Restriction functor on cohomology

Definition

General definition

Suppose $G_1,G_2$ are groups and $A$ is an abelian group. Suppose ${\phi }_1:G_1\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$, ${\phi }_2:G_2\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$, and $\alpha :G_1\to G_2$ are homomorphisms such that ${\phi }_2\circ \alpha ={\phi }_1$, i.e., the $G_1$-action and $G_2$-action on $A$ are compatible. Then, we get an induced homomorphism between the cohomology groups:

${\mathrm{r}}_{\mathrm{e}}^{\mathrm{s}}:H_{{\phi }_2}^{*}(G_2,A)\to H_{{\phi }_1}^{*}(G_1,A)$

This homomorphism is termed the restriction homomorphism and the functor that sends the map $\alpha :G_1\to G_2$ 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 ${\phi }_2$ and $\alpha$ since ${\phi }_1$ is defined in terms of them. In other words, we can define the restriction functor in terms of ${\phi }_2:G_2\to \mathrm{A}\mathrm{u}\mathrm{t}(A)$ and $\alpha :G_1\to G_2$.

Typical case

The typical case where we talk of the restriction functor is where the map $\alpha :G_1\to G_2$ is injective, and we naturally identify $G_1$ with its image subgroup of $G_2$. We thus talk of restricting cohomology to a subgroup.

Related notions

• Corestriction functor on homology
• Inflation functor on cohomology
• Coinflation functor on homology