---
tag: 2PAS78F34T
kind: definition
name: Group cohomology
msc: [20J06, 18G15]
defines: YXJXA0Q52R
uses: [G5S1M9, 082GHNPJ53, MNP2HBWB5Q, 0WCY42EV9V, 4P7GBWJ4RR, 8BGJ6J9PYF]
sources:
  - url: https://en.wikipedia.org/wiki/Group_cohomology
---

## Definition

Let $G$ be a [[group|G5S1M9]].
A *$G$-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 $m, m' \in M$.
Equivalently, it is a module over the [[group ring|MNP2HBWB5Q]] $\Z[G]$.
Let every element of $G$ act on $\Z$ as the identity.
For $n \ge 0$, the $n$-th *cohomology group* of $G$ with coefficients in $M$ is
$$H^n(G, M) = \operatorname{Ext}^n_{\Z[G]}(\Z, M),$$
where $\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].

## Remarks

With the [[bar resolution|8BGJ6J9PYF]], $H^n(G, M)$ 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.