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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nA *$G$-module* is an [[abelian group|082GHNPJ53]] $M$ with an [[action|0WCY42EV9V]] of $G$ such that $g(m + m') = gm + gm'$ for all $g \\in G$ and $m, m' \\in M$.\nEquivalently, it is a module over the [[group ring|MNP2HBWB5Q]] $\\Z[G]$.\nLet every element of $G$ act on $\\Z$ as the identity.\nFor $n \\ge 0$, the $n$-th *cohomology group* of $G$ with coefficients in $M$ is\n$$H^n(G, M) = \\operatorname{Ext}^n_{\\Z[G]}(\\Z, M),$$\nwhere $\\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].\n\n## Remarks\n\nWith the [[bar resolution|8BGJ6J9PYF]], $H^n(G, M)$ is the cohomology of the complex of functions $f \\colon G^n \\to M$ with\n$$(df)(g_1, \\dots, g_{n+1}) = g_1 f(g_2, \\dots, g_{n+1}) + \\sum_{i=1}^{n} (-1)^i f(g_1, \\dots, g_i g_{i+1}, \\dots, g_{n+1}) + (-1)^{n+1} f(g_1, \\dots, g_n).$$\nSo $H^0(G, M) = M^G$, the elements fixed by $G$.\nAnd $H^1(G, M)$ is the group of crossed homomorphisms, the maps with $f(gh) = f(g) + g f(h)$, modulo those of the form $f(g) = gm - m$.\nIf every element of $G$ acts on $M$ as the identity, then $H^1(G, M) = \\Hom(G, M)$.\nExample: 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.\nThen $H^1(G, M) \\cong \\Z/2\\Z$.\nIndeed, a crossed homomorphism is determined by $f(t) \\in \\Z$, with $f(1) = 0$ and no further condition.\nThe principal ones have $f(t) = tm - m = -2m$.\nNon-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.\n",
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