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  "body": "\n## Summary\n\nThe cohomological dimension of a [[group|G5S1M9]] $G$ is the largest degree in which group cohomology of $G$ can be nonzero.\nIt equals the shortest length of a projective resolution of $\\Z$ over the group ring $\\Z[G]$.\nFree groups are exactly the groups of dimension at most $1$, by the [[Stallings–Swan theorem|B41GE4M134]], and $\\Z^n$ has dimension $n$.\nA group of finite cohomological dimension is [[torsion-free|PEMNN32A43]].\n\n## Explanations\n\n### L1 human\n\nThe cohomological dimension says how many dimensions a group needs to be modelled as a shape.\nThe integers $\\Z$ need one dimension: they are the loops on a circle.\nThe group $\\Z^2$ needs two, like a torus.\nA nontrivial finite cyclic group cannot be modelled in any finite dimension, and its dimension is infinite.\n",
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