Bar resolution

Remarks

It is a free resolution of $\Z$ over $\Z[G]$: one checks $d \circ d = 0$, and a contracting homotopy over $\Z$ is given by $s(g_0 [g_1 | \cdots | g_n]) = [g_0 | g_1 | \cdots | g_n]$. Check of $d \circ d = 0$ in low degree: $d_1 [g] = g[\,] - [\,]$ and $d_2 [g | h] = g[h] - [gh] + [g]$, so $d_1 d_2 [g | h] = (gh - g)[\,] - (gh - 1)[\,] + (g - 1)[\,] = 0$. Applying $\Hom_{\Z[G]}(-, M)$ identifies $\Hom_{\Z[G]}(B_n, M)$ with the set of functions $G^n \to M$, the inhomogeneous cochains of group cohomology. The normalized bar resolution uses only symbols with all $g_i \ne 1$; it is also a free resolution of $\Z$. Non-example: for $G = \Z/2\Z = \set{1, t}$, the periodic complex $\cdots \to \Z[G] \xrightarrow{1 - t} \Z[G] \xrightarrow{1 + t} \Z[G] \xrightarrow{1 - t} \Z[G] \to \Z \to 0$ is a free resolution of $\Z$, but it is not the bar resolution, whose term $B_n$ has rank $2^n$.