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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]] and let $\\hat{R}$ be its $\\mathfrak{m}$-adic [[completion|KBN13NPYJQ]].\nThen $R$ is a *complete intersection ring* if the following holds.\nThere are a [[complete|G9Q61B8A7C]] [[regular local ring|63XXTF6B5M]] $S$ and an [[$S$-regular sequence|69R5R2FCS5]] $f_1, \\ldots, f_r$ such that\n$$\\hat{R} \\cong S/(f_1, \\ldots, f_r).$$\n\n## Remarks\n\nThe completion is used because a Noetherian local ring need not be a quotient of a regular local ring, while its completion always is, by the [[Cohen structure theorem|N3XH42KVV3]].\nIf $R = S/(f_1, \\ldots, f_r)$ with $S$ a regular local ring and $f_1, \\ldots, f_r$ an $S$-regular sequence, then $R$ is a complete intersection.\nExample: $K[x,y]$ localized at $(x, y)$ and divided by $(xy)$ is a complete intersection, since $xy \\ne 0$ is a non-zero-divisor in the domain $K[x,y]_{(x,y)}$.\nExample: $K[x]/(x^n)$ is a complete intersection, with $S = K[\\![x]\\!]$ and $f_1 = x^n$.\nNon-example: $K[x,y]/(x^2, xy, y^2)$ is not a complete intersection, because it is not [[Gorenstein|T6HCX959PJ]], while every complete intersection is Gorenstein.\n",
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