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  "body": "\n## Statement\n\nLet $(R, \\mathfrak{m})$ be a commutative [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].\nLet $M$ be a nonzero [[finitely generated|05QY1VEBYD]] $R$-module of finite [[projective dimension|MR9QCH1PNA]].\nThen\n$$\\operatorname{pd}_R M + \\operatorname{depth} M = \\operatorname{depth} R,$$\nwhere $\\operatorname{depth}$ is the [[depth|M2RADBFW99]] with respect to $\\mathfrak{m}$.\n\n## Remarks\n\nOne proof goes by induction on $\\operatorname{depth} R$, using a minimal free resolution of $M$.\nWhen the depths are positive, one divides by an element of $\\mathfrak{m}$ that is a non-zero-divisor on both $R$ and $M$.\nExample: for $R = K[\\![x, y]\\!]$, which has depth $2$, the residue field $K$ has projective dimension $2$ and depth $0$, and $R/(x)$ has projective dimension $1$ and depth $1$.\nThe hypothesis $\\operatorname{pd}_R M \u003c \\infty$ is needed: for $R = K[x]/(x^2)$ and $M = K$, the depths are $0$ and $0$, but $\\operatorname{pd}_R K = \\infty$.\nIn particular, a finitely generated module of finite projective dimension has projective dimension at most $\\operatorname{depth} R$.\n\n## Explanations\n\n### L1 human\n\nFor a finitely generated module of finite projective dimension over a local ring, two numbers add up to a constant.\nThe projective dimension counts how complicated the module is homologically.\nThe depth counts how much room the module has for regular sequences.\nThe more depth a module has, the shorter its resolution.\n\n### L2 human\n\nThe formula is the main link between resolutions and regular sequences.\nOver a [[regular local ring|63XXTF6B5M]], every finitely generated module has finite projective dimension, so the formula always applies there.\nIt shows that a module of finite projective dimension over a Cohen–Macaulay local ring is maximal Cohen–Macaulay exactly when it is free.\n",
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