# Tag Info

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: $$0 \to \underbrace{L^G}_{= H^0(G; L)} \to M^G \to N^G \to H^1(G; L) \to H^1(G; M) \to H^1(G; M) \to \dots$$ $$0 \leftarrow \underbrace{L_G}_{= H_0(G; L)} \leftarrow M_G \leftarrow N_G \leftarrow H_1(G; L) \leftarrow H_1(G; M) \leftarrow H_1(G; M) \leftarrow \dots$$