Group cohomology

Summary

Group cohomology attaches to a group $G$ and a $G$-module $M$ a sequence of abelian groups $H^n(G, M)$, for $n \ge 0$. They are the right derived functors of the functor of invariants $M \mapsto M^G$, or equivalently the groups $\operatorname{Ext}^n_{\Z[G]}(\Z, M)$. In low degrees they have classical meanings. $H^0$ is the invariants, $H^1$ classifies crossed homomorphisms up to principal ones, and $H^2$ classifies extensions of $G$ by $M$. It is dual to group homology, and Galois cohomology is the case of Galois groups.