---
tag: P9RN33VDFS
kind: definition
name: Group homology
msc: [20J06, 20J05]
defines: CFAG3VRXEA
uses: [G5S1M9, 0NZ8JA25WK, HK2AA1YSQ2, F0HQXQ0HCD, DDQRERAQCV]
sources:
  - url: https://en.wikipedia.org/wiki/Group_cohomology
---

## Definition

Let $G$ be a [[group|G5S1M9]] and let $\Z[G]$ be its group ring.
A $G$-module is a left [[module|0NZ8JA25WK]] over $\Z[G]$.
Let $\Z$ be the right $\Z[G]$-module with the trivial action $n \cdot g = n$.
Let $P_\bullet \to \Z$ be a [[projective resolution|HK2AA1YSQ2]] of $\Z$ by right $\Z[G]$-modules.
Let $M$ be a $G$-module and let $n \ge 0$.
The $n$-th *homology group* of $G$ with coefficients in $M$ is
$$H_n(G, M) = H_n(P_\bullet \otimes_{\Z[G]} M) = \mathrm{Tor}^{\Z[G]}_n(\Z, M),$$
the homology of the chain complex obtained by the [[tensor product|F0HQXQ0HCD]] with $M$.

## Remarks

The groups do not depend on the choice of projective resolution, up to canonical isomorphism.
In degree $0$, $H_0(G, M) = M_G$ is the module of coinvariants, the quotient of $M$ by the subgroup generated by all $gm - m$.
In degree $1$, $H_1(G, \Z) \cong G / [G, G]$.

Example: $G = \Z$ written multiplicatively with generator $t$, so that $\Z[G] = \Z[t, t^{-1}]$.
The sequence $0 \to \Z[G] \xrightarrow{\ t - 1\ } \Z[G] \to \Z \to 0$ is a free resolution of $\Z$.
Tensoring with the trivial module $\Z$ gives the complex $0 \to \Z \xrightarrow{0} \Z \to 0$.
So $H_0(\Z, \Z) = \Z$, $H_1(\Z, \Z) = \Z$, and $H_n(\Z, \Z) = 0$ for $n \ge 2$.

Non-example: the groups $\mathrm{Tor}^{\Z}_n(\Z, M)$, taken over $\Z$ instead of $\Z[G]$, forget the action.
They vanish for $n \ge 1$, while $H_1(\Z, \Z) = \Z$.