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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].\nThen $R$ is a *Cohen–Macaulay local ring* if $\\operatorname{depth} R = \\dim R$, where $\\operatorname{depth}$ is the [[depth|M2RADBFW99]] with respect to $\\mathfrak{m}$ and $\\dim$ is the [[Krull dimension|MVGBS3J1P7]].\nA commutative Noetherian ring $R$ is a *Cohen–Macaulay ring* if the [[localization|X6JD4AG973]] $R_{\\mathfrak{p}}$ is a Cohen–Macaulay local ring for every [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$ of $R$.\n\n## Remarks\n\nFor a Noetherian local ring one always has $\\operatorname{depth} R \\le \\dim R$, so the condition is that the depth is as large as possible.\nIt suffices to test the localizations at maximal ideals, since localizations of Cohen–Macaulay local rings at primes are Cohen–Macaulay.\nExample: every Noetherian local ring of dimension $0$ is Cohen–Macaulay, since then $\\operatorname{depth} R = \\dim R = 0$.\nExample: $K[x_1, \\ldots, x_n]$ localized at $(x_1, \\ldots, x_n)$ is Cohen–Macaulay, since $x_1, \\ldots, x_n$ is a regular sequence of length $n = \\dim R$.\nNon-example: $R = K[x,y]/(x^2, xy)$ localized at $(x, y)$ has dimension $1$, because its primes are $(x)$ and $(x, y)$, but depth $0$, because $x$ kills the maximal ideal.\nFor a local ring, Cohen–Macaulay is equivalent to: some (equivalently every) system of parameters is a regular sequence.\nA finitely generated module $M \\ne 0$ over a Noetherian local ring is a *Cohen–Macaulay module* if $\\operatorname{depth} M = \\dim M$.\n",
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