The inf-res Sequence

Suppose $G$ is a group and $H$ is a normal subgroup of $G$, and $A$ is a $G$-module. Then for any $n\geq 0$, there are natural homomorphisms

$$\res : \H^n(G,A) \to \H^n(H,A)$$

and

$$\inf : \H^n(G/H,A^H) \to \H^n(G,A)$$

Require that we view $n$-cocycles as certain maps on the $n$-fold product of the group. On cocycles, the map $\res$ is obtained by simply restricting a cocycle, which is a map $G^i\to A$, to a map $H^i\to A$. The second map $\inf$ is obtained by precomposing a cocycle $(G/H)^i\to A^H$ with the natural map $G^i \to (G/H)^i$.

Proposition 2.4   The inf-res sequence

$$0 \to \H^1(G/H,A^H) \xrightarrow{\inf} \H^1(G,A) \xrightarrow{\res } \H^1(H,A)$$

is exact.