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  "body": "\n## Summary\n\nGroup cohomology attaches to a [[group|G5S1M9]] $G$ and a $G$-module $M$ a sequence of abelian groups $H^n(G, M)$, for $n \\ge 0$.\nThey are the right [[derived functors|NPMC3K6J2F]] of the functor of invariants $M \\mapsto M^G$, or equivalently the groups $\\operatorname{Ext}^n_{\\Z[G]}(\\Z, M)$.\nIn low degrees they have classical meanings.\n$H^0$ is the invariants, $H^1$ classifies crossed homomorphisms up to principal ones, and $H^2$ classifies extensions of $G$ by $M$.\nIt is dual to [[group homology|CFAG3VRXEA]], and [[Galois cohomology|9MPCZ3EAJM]] is the case of Galois groups.\n\n## Explanations\n\n### L1 human\n\nA group acting on an abelian group $M$ leaves some elements fixed; these form $H^0(G, M)$.\nTaking fixed points does not respect quotients well, and the higher cohomology groups record this failure.\nThey can be computed with cochains, which are functions of several group elements with values in $M$.\nWhen every element of $G$ acts on $M$ as the identity, $H^1(G, M)$ is just the group of homomorphisms from $G$ to $M$.\nFor a cyclic group of order $m$ acting on $\\Z$ as the identity, the cohomology is $\\Z/m\\Z$ in every positive even degree and $0$ in every odd degree.\n",
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