a tool used to compute invariants of group actions using methods from homology theory, such as invariants, coinvariants, extensions... Use with (homology-cohomology).

Given a group $G$ and a $G$-module $M$, it is possible to define invariants:
$$H_n(G;M) \qquad H^n(G; M)$$
for all $n \ge 0$ called respectively the **homology** and the **cohomology** of the group $G$ with coefficients in $M$. These invariants are generalizations of two well-known constructions, the **invariants** and coinvariants:
$$\begin{align}
M^G & = \{ m \in M : g \cdot m = m \forall g \in G \} \\
M_G & = M / ( g \cdot m \sim m )
\end{align}$$
and they fit in long exact sequences, given a short exact sequence $0 \to L \to M \to N \to 0$ of $G$-modules:
$$0 \to \underbrace{L^G}_{= H^0(G; L)} \to M^G \to N^G \to H^1(G; L) \to H^1(G; M) \to H^1(G; M) \to \dots$$
$$0 \leftarrow \underbrace{L_G}_{= H_0(G; L)} \leftarrow M_G \leftarrow N_G \leftarrow H_1(G; L) \leftarrow H_1(G; M) \leftarrow H_1(G; M) \leftarrow \dots$$

This tag should be used in conjunction with homology-cohomology. More information about group cohomology can be found on Wikipedia.