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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\nThe *cohomological dimension* $\\mathrm{cd}(G)$ is the smallest integer $n \\ge 0$ such that $H^k(G, M) = 0$ for every $G$-module $M$ and every $k \u003e n$.\nHere $H^k(G, M)$ is the cohomology group.\nIf there is no such integer, then $\\mathrm{cd}(G) = \\infty$.\n\n## Remarks\n\nEquivalently, $\\mathrm{cd}(G)$ is the smallest $n$ such that $\\Z$, with trivial action, has a [[projective resolution|HK2AA1YSQ2]] of length $n$ over the group ring $\\Z[G]$.\nFor a subgroup $H$ of $G$, $\\mathrm{cd}(H) \\le \\mathrm{cd}(G)$.\nThe only group with $\\mathrm{cd}(G) = 0$ is the trivial group.\n\nExample: $\\mathrm{cd}(\\Z) = 1$.\nThe resolution $0 \\to \\Z[t, t^{-1}] \\xrightarrow{\\ t - 1\\ } \\Z[t, t^{-1}] \\to \\Z \\to 0$ has length $1$, and $H^1(\\Z, \\Z) \\cong \\Z$ is not zero.\nMore generally, $\\mathrm{cd}(\\Z^n) = n$, and a nontrivial [[free group|0KTW26C80J]] has cohomological dimension $1$.\n\nNon-example: a nontrivial finite [[cyclic group|0KVGH0J5B3]] $C_m$ has $H^{2k}(C_m, \\Z) \\cong \\Z/m\\Z \\ne 0$ for every $k \\ge 1$, so no integer $n$ works and $\\mathrm{cd}(C_m) = \\infty$.\nSince a group containing $C_m$ has dimension at least that of $C_m$, every group of finite cohomological dimension is [[torsion-free|PEMNN32A43]].\n",
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