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  "body": "\n## Definition\n\nLet $(R, \\mathfrak{m}, k)$ be a [[Cohen–Macaulay|N1QJQEGBFC]] [[local ring|MZ02BKPHJA]] of dimension $d$ with [[residue field|NJ3APQ8BHH]] $k$.\nA *canonical module* of $R$ is a [[finitely generated|05QY1VEBYD]] $R$-module $\\omega_R$ such that:\n\n1. $\\omega_R$ is a maximal Cohen–Macaulay module, that is, $\\operatorname{depth} \\omega_R = d$, where $\\operatorname{depth}$ is the [[depth|M2RADBFW99]];\n2. $\\omega_R$ has finite injective dimension;\n3. $\\dim_k \\operatorname{Ext}^d_R(k, \\omega_R) = 1$, where $\\operatorname{Ext}$ is the [[Ext functor|4P7GBWJ4RR]].\n\n## Remarks\n\nA canonical module is unique up to isomorphism when it exists.\nIt exists if and only if $R$ is a quotient of a Gorenstein local ring; in particular every complete Cohen–Macaulay local ring has one.\nExample: $R$ is [[Gorenstein|T6HCX959PJ]] if and only if $R$ itself is a canonical module; for instance $\\omega_R = R$ for $R = K[\\![x_1, \\ldots, x_n]\\!]$.\nExample: for an Artinian local ring $R$, the canonical module is the [[injective hull|QCJN05GN64]] $E_R(k)$ of the residue field.\nNon-example: for $R = K[x,y]/(x^2, xy, y^2)$ the module $R$ is not a canonical module, because $\\operatorname{Hom}_R(k, R)$ is the socle $Kx \\oplus Ky$, of dimension $2$, so clause 3 fails.\n",
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