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  "body": "\n## Summary\n\nThe global dimension of a ring is the supremum of the [[projective dimensions|MR9QCH1PNA]] of all its modules.\nIt is $0$ for fields and $1$ for principal ideal domains that are not fields.\nIt is $n$ for a polynomial ring in $n$ variables over a field.\nBy [[Serre's theorem|93RQRG9R1P]], a Noetherian local ring has finite global dimension exactly when it is a [[regular local ring|63XXTF6B5M]].\n\n## Explanations\n\n### L1 human\n\nThe global dimension says how complicated the modules over a ring can be, in the worst case.\nA ring of global dimension $0$ is one where every module is projective, as for vector spaces over a field.\nOver the integers, every module has a resolution of length at most $1$, so the global dimension of $\\Z$ is $1$.\nOver $\\Z/4\\Z$, the module $\\Z/2\\Z$ needs an infinite resolution, so the global dimension is infinite.\n",
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