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  "body": "\n## Statement\n\nLet $(R, \\mathfrak{m}, k)$ be a [[Cohen–Macaulay|N1QJQEGBFC]] [[local ring|MZ02BKPHJA]] of dimension $d$ with [[residue field|NJ3APQ8BHH]] $k$, and assume that $R$ has a [[canonical module|W5ECXPM6FW]] $\\omega_R$.\nLet $E = E_R(k)$ be the [[injective hull|QCJN05GN64]] of $k$.\nLet $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module and let $i$ be an integer.\nThen there is an isomorphism, natural in $M$,\n$$H^i_{\\mathfrak{m}}(M) \\cong \\Hom_R\\left(\\operatorname{Ext}^{d-i}_R(M, \\omega_R), E\\right),$$\nwhere $H^i_{\\mathfrak{m}}$ is [[local cohomology|6JB30BXDA2]] and $\\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].\n\n## Remarks\n\nIf $R$ is Gorenstein, then $\\omega_R = R$ and the formula reads $H^i_{\\mathfrak{m}}(M) \\cong \\Hom_R(\\operatorname{Ext}^{d-i}_R(M, R), E)$.\nFor $M = R$ and $i = d$ it gives $H^d_{\\mathfrak{m}}(R) \\cong \\Hom_R(\\omega_R, E)$.\nExample: for $R = K[\\![x]\\!]$, one has $d = 1$, $\\omega_R = R$ and $E = K[\\![x]\\!]_x / K[\\![x]\\!]$, and indeed $H^1_{\\mathfrak{m}}(R) = R_x / R \\cong E = \\Hom_R(R, E)$.\nWhen $R$ is complete, [[Matlis duality|2V9BR5XX6A]] turns the formula around: $\\operatorname{Ext}^{d-i}_R(M, \\omega_R) \\cong \\Hom_R(H^i_{\\mathfrak{m}}(M), E)$.\n\n## Explanations\n\n### L1 human\n\nLocal cohomology modules are large and hard to see directly.\nLocal duality says that they are the duals of Ext modules, which are finitely generated and computable.\nIt is an algebraic analogue of Serre duality for projective varieties.\n\n### L2 human\n\nCombined with the vanishing pattern of local cohomology, local duality shows that $M$ is Cohen–Macaulay of dimension $t$ exactly when $\\operatorname{Ext}^{d-i}_R(M, \\omega_R) = 0$ for all $i \\ne t$.\nIt also identifies the canonical module as the Matlis dual of the top local cohomology $H^d_{\\mathfrak{m}}(R)$ when $R$ is complete.\n",
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