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  "body": "\n## Statement\n\nLet $R$ be a [[regular local ring|63XXTF6B5M]].\nThen $R$ is a [[unique factorization domain|0W4X9G6263]].\n\n## Remarks\n\nThis is the Auslander–Buchsbaum theorem.\nProof idea: $R$ is a domain by [[regular local rings are domains|V3ZBMRKC08]]; one argues by induction on $\\dim R$.\nPick $x \\in \\mathfrak m \\setminus \\mathfrak m^2$; then $x$ is prime, since $R/xR$ is regular, hence a domain.\nBy [[Nagata's criterion|GGEMJF449D]], it is enough to show that $R[1/x]$ is a UFD.\nIts localizations at primes are regular of smaller dimension, hence UFDs by induction.\nSo each height one prime of $R[1/x]$ is locally principal.\nFinite free resolutions, which exist by [[Serre's theorem|93RQRG9R1P]], then show that it is principal.\n\nExample: the power series ring $k[\\![x_1, \\ldots, x_n]\\!]$ over a field $k$ and the localizations $k[x_1, \\ldots, x_n]_{\\mathfrak p}$ of a polynomial ring at prime ideals are UFDs.\n\nThe regularity hypothesis matters.\nFor a field $k$, the local ring $R = k[x, y, z]_{(x, y, z)}/(xy - z^2)$ is a domain of dimension $2$ and embedding dimension $3$.\nIt is not a UFD, since $xy = z \\cdot z$ are two factorizations of the same element into irreducible elements that are not associate.\n\n## Explanations\n\n### L1 human\n\nAt a smooth point of a variety, every function factors uniquely into irreducible pieces, just as integers factor uniquely into primes.\nAt a singular point this can fail: on the cone $xy = z^2$ the function $z^2$ factors in two different ways.\n\n### L2 human\n\nThe proof sketched above relies on homological algebra, through Serre's theorem and finite free resolutions.\nIn geometry it means that on a smooth variety every hypersurface through a point is defined locally by one equation, so Weil divisors and Cartier divisors agree.\nIt is used with the [[divisor class group|WN8PK6RQZR]]: the class group of a regular local ring is trivial.\n",
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