Canonical module

Summary

The canonical module $\omega_R$ of a Cohen–Macaulay local ring $R$ is a finitely generated module that serves as a dualizing object for the Cohen–Macaulay $R$-modules. It is the algebraic counterpart of the sheaf of top differential forms on a smooth variety. A Cohen–Macaulay local ring is Gorenstein exactly when $\omega_R \cong R$, and the canonical module appears in Grothendieck's local duality.