Definition 1

Let M be a left $\mathbf{Z}^{(G)}$-module (resp. right), n an integer $\geqslant 0$. The group $\mathrm{Ext}_{\mathbf{Z}^{(G)}}^n(\mathbf{Z}, M)$ (resp. $\mathrm{Tor}_n^{\mathbf{Z}^{(G)}}(M, \mathbf{Z})$) is denoted by $H^n(G, M)$ (resp. $H_n(G, M)$) and is called the nth cohomology group (resp. homology group) of G with coefficients in M.

The standard resolution (X, p. 58) $B(\mathbf{Z}^{(G)}, \mathbf{Z})$ is a free resolution of the $\mathbf{Z}^{(G)}$-module $\mathbf{Z}$; it follows that the groups $H^n(G, M)$ (resp. $H_n(G, M)$) identify with the homology groups of the complex:

Using the canonical isomorphism of $(\mathbf{Z}^{(G)})^{\otimes n}$ onto $\mathbf{Z}^{(G^n)}$ (III, p. 36) and the properties of extension of scalars (II, p. 82), one concludes that $H^n(G, M)$ is canonically isomorphic to the homology group of ascending degree $n$ of the complex $C(G, M)$ defined as follows: $C^n(G, M) = 0$ for $n < 0$; for $n \geqslant 0$, $C^n(G, M)$ is the $\mathbf{Z}$-module of mappings of $G^n$ into M; for $n \geqslant 0$, the differential $d^n : C^n(G, M) \to C^{n+1}(G, M)$ is given by

for any $f$ in $C^n(G, M)$ and $g_0, ..., g_n$ in G.

Analogously, $H_n(G, M)$ identifies with the homology group of degree $n$ of the complex $C'(G, M)$, where $C'_n(G, M) = M \otimes_{\mathbf{Z}} \mathbf{Z}^{(G^n)}$ for $n \geqslant 0$, $C'_n(G, M) = 0$ for $n < 0$, the differential $d_n : C'_n(G, M) \to C'_{n-1}(G, M)$ being defined by:

for any $n \geqslant 1$, $m$ in M and $g_1, ..., g_n$ in G.