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  "body": "\n## Statement\n\nLet $(R, \\mathfrak{m}, k)$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] with [[residue field|NJ3APQ8BHH]] $k$.\nThe following are equivalent:\n\n1. $R$ is a [[regular local ring|63XXTF6B5M]].\n2. $R$ has finite [[global dimension|GWA0T1WKCN]].\n3. The [[projective dimension|MR9QCH1PNA]] $\\operatorname{pd}_R k$ is finite.\n\nWhen they hold, $\\operatorname{gl.dim} R = \\operatorname{pd}_R k = \\dim R$, where $\\dim R$ is the [[Krull dimension|MVGBS3J1P7]].\n\n## Remarks\n\nIf $R$ is regular, then $\\mathfrak{m}$ is generated by a regular sequence of length $\\dim R$, and its [[Koszul complex|YZY11W08SD]] is a free resolution of $k$ of that length.\nFor a Noetherian local ring, the global dimension equals $\\operatorname{pd}_R k$, which gives the implication from 1 to 2.\nThe converse is the deep part: a finite resolution of $k$ forces the minimal number of generators of $\\mathfrak{m}$ to equal $\\dim R$.\nExample: $R = K[x]/(x^2)$ is not regular, and indeed $\\cdots \\to R \\xrightarrow{x} R \\xrightarrow{x} R \\to K \\to 0$ is a minimal resolution of $K$ that never stops.\nA major consequence is that every localization of a regular local ring at a prime ideal is again regular.\n\n## Explanations\n\n### L1 human\n\nRegular local rings are the algebraic version of smooth points of a space.\nThe theorem says that smoothness can be detected by homological algebra alone: a local ring is regular exactly when every module has a finite resolution.\nIt is enough to test one module, the residue field.\n\n### L2 human\n\nThe theorem was proved by Serre and, independently, by Auslander and Buchsbaum.\nThe homological criterion gives a short proof that a localization of a regular local ring is regular, since finite global dimension passes to localizations.\nIt also gives [[Hilbert's syzygy theorem|R0DSZPRG6T]] in the local setting, and, combined with the [[Auslander–Buchsbaum formula|SQB74J5E8Q]], it controls the resolutions of all finitely generated modules over a regular local ring.\n",
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