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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $\\Z[G]$ be its [[group ring|MNP2HBWB5Q]].\nFor $n \\ge 0$, let $B_n$ be the [[free|0DZ9DKAKZR]] left $\\Z[G]$-module with basis the symbols $[g_1 | \\cdots | g_n]$ with $g_1, \\dots, g_n \\in G$; the basis of $B_0$ is the single symbol $[\\,]$.\nFor $n \\ge 1$, the $\\Z[G]$-linear map $d_n \\colon B_n \\to B_{n-1}$ is given on the basis by\n$$d_n [g_1 | \\cdots | g_n] = g_1 [g_2 | \\cdots | g_n] + \\sum_{i=1}^{n-1} (-1)^i [g_1 | \\cdots | g_i g_{i+1} | \\cdots | g_n] + (-1)^n [g_1 | \\cdots | g_{n-1}].$$\nThe augmentation $\\varepsilon \\colon B_0 \\to \\Z$ sends $[\\,]$ to $1$, where every element of $G$ acts on $\\Z$ as the identity.\nThe *bar resolution* of $G$ is the complex $\\cdots \\to B_2 \\xrightarrow{d_2} B_1 \\xrightarrow{d_1} B_0 \\xrightarrow{\\varepsilon} \\Z \\to 0$.\n\n## Remarks\n\nIt is a free [[resolution|HK2AA1YSQ2]] of $\\Z$ over $\\Z[G]$: one checks $d \\circ d = 0$, and a contracting homotopy over $\\Z$ is given by $s(g_0 [g_1 | \\cdots | g_n]) = [g_0 | g_1 | \\cdots | g_n]$.\nCheck of $d \\circ d = 0$ in low degree: $d_1 [g] = g[\\,] - [\\,]$ and $d_2 [g | h] = g[h] - [gh] + [g]$, so $d_1 d_2 [g | h] = (gh - g)[\\,] - (gh - 1)[\\,] + (g - 1)[\\,] = 0$.\nApplying $\\Hom_{\\Z[G]}(-, M)$ identifies $\\Hom_{\\Z[G]}(B_n, M)$ with the set of functions $G^n \\to M$, the inhomogeneous cochains of group cohomology.\nThe *normalized* bar resolution uses only symbols with all $g_i \\ne 1$; it is also a free resolution of $\\Z$.\nNon-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 $2^n$.\n",
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