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  "body": "\n## Summary\n\nThe canonical module $\\omega_R$ of a [[Cohen–Macaulay|N1QJQEGBFC]] local ring $R$ is a finitely generated module that serves as a dualizing object for the Cohen–Macaulay $R$-modules.\nIt is the algebraic counterpart of the sheaf of top differential forms on a smooth variety.\nA Cohen–Macaulay local ring is [[Gorenstein|T6HCX959PJ]] exactly when $\\omega_R \\cong R$, and the canonical module appears in [[Grothendieck's local duality|VA3DY2YX83]].\n\n## Explanations\n\n### L1 human\n\nFor a Gorenstein ring the ring itself has perfect duality properties.\nFor a general Cohen–Macaulay ring this fails, but a substitute module $\\omega_R$ often exists that has those properties.\nIt is unique up to isomorphism when it exists.\nFor a finite-dimensional local algebra over a field, the canonical module is the dual vector space of the algebra, with the induced module structure.\n",
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