Canonical module

Builds on Cohen–Macaulay ring Local ring Residue field Finitely generated module Depth Ext functor Gorenstein ring Injective hull

Definition

Let $(R, \mathfrak{m}, k)$ be a Cohen–Macaulay local ring of dimension $d$ with residue field $k$ . A canonical module of $R$ is a finitely generated $R$ -module $\omega_R$ such that:

$\omega_R$ is a maximal Cohen–Macaulay module, that is, $\operatorname{depth} \omega_R = d$ , where $\operatorname{depth}$ is the depth;
$\omega_R$ has finite injective dimension;
$\dim_k \operatorname{Ext}^d_R(k, \omega_R) = 1$ , where $\operatorname{Ext}$ is the Ext functor.

Remarks

A canonical module is unique up to isomorphism when it exists. It exists if and only if $R$ is a quotient of a Gorenstein local ring; in particular every complete Cohen–Macaulay local ring has one. Example: $R$ is Gorenstein if and only if $R$ itself is a canonical module; for instance $\omega_R = R$ for $R = K[\![x_1, \ldots, x_n]\!]$ . Example: for an Artinian local ring $R$ , the canonical module is the injective hull $E_R(k)$ of the residue field. Non-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.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Cohen–Macaulay ring concept
Depth concept
Ext functor concept
Finitely generated module concept
Gorenstein ring concept
Injective hull concept
Local ring concept
Residue field concept
Canonical module definition

Tag	`ETSXVDDCYK`
URI	`m2://ETSXVDDCYK`
Kind	definition
Writing	W0 stub
MSC	13H10, 13C14, 13D45
Defines	Canonical module
Uses	Cohen–Macaulay ring, Local ring, Residue field, Finitely generated module, Depth, Ext functor, Gorenstein ring, Injective hull
Changed	2 November 2026

Canonical module

Definition

Remarks

Reading path

Further reading

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