---
tag: W5ECXPM6FW
kind: concept
name: Canonical module
msc: [13H10, 13C14, 13D45]
sources:
  - url: https://en.wikipedia.org/wiki/Dualizing_module
---

## Summary

The 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.
It is the algebraic counterpart of the sheaf of top differential forms on a smooth variety.
A Cohen–Macaulay local ring is [[Gorenstein|T6HCX959PJ]] exactly when $\omega_R \cong R$, and the canonical module appears in [[Grothendieck's local duality|VA3DY2YX83]].

## Explanations

### L1 human

For a Gorenstein ring the ring itself has perfect duality properties.
For a general Cohen–Macaulay ring this fails, but a substitute module $\omega_R$ often exists that has those properties.
It is unique up to isomorphism when it exists.
For a finite-dimensional local algebra over a field, the canonical module is the dual vector space of the algebra, with the induced module structure.