Group cohomology

Remarks

With the bar resolution, $H^n(G, M)$ is the cohomology of the complex of functions $f \colon G^n \to M$ with $$(df)(g_1, \dots, g_{n+1}) = g_1 f(g_2, \dots, g_{n+1}) + \sum_{i=1}^{n} (-1)^i f(g_1, \dots, g_i g_{i+1}, \dots, g_{n+1}) + (-1)^{n+1} f(g_1, \dots, g_n).$$ So $H^0(G, M) = M^G$, the elements fixed by $G$. And $H^1(G, M)$ is the group of crossed homomorphisms, the maps with $f(gh) = f(g) + g f(h)$, modulo those of the form $f(g) = gm - m$. If every element of $G$ acts on $M$ as the identity, then $H^1(G, M) = \Hom(G, M)$. Example: let $G = \Z/2\Z = \set{1, t}$ act on $M = \Z$ by $t m = -m$; this is a $G$-module, since $m \mapsto -m$ is additive. Then $H^1(G, M) \cong \Z/2\Z$. Indeed, a crossed homomorphism is determined by $f(t) \in \Z$, with $f(1) = 0$ and no further condition. The principal ones have $f(t) = tm - m = -2m$. Non-example: $\Z/2\Z$ acting on $\Z$ by $m \mapsto 1 - m$ is an action, but not by additive maps, since $0$ is not fixed, so $\Z$ is not a $G$-module for it.