# Dualizing module

In abstract algebra, a **dualizing module**, also called a **canonical module**, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.

## Definition

A dualizing module for a Noetherian ring *R* is a finitely generated module *M* such that for any maximal ideal *m*, the *R*/*m* vector space Ext^{n}_{R}(*R*/*m*,*M*) vanishes if *n* ≠ height(*m*) and is 1-dimensional if *n* = height(*m*).

A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module. In particular if the ring is local the dualizing module is unique up to isomorphism.

A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute.

## Examples

If *R* is a Gorenstein ring, then *R* considered as a module over itself is a dualizing module.

If *R* is an Artinian local ring then the Matlis module of *R* (the injective hull of the residue field) is the dualizing module.

The Artinian local ring *R* = *k*[*x*,*y*]/(*x*^{2},*y*^{2},*xy*) has a unique dualizing module, but it is not isomorphic to *R*.

The ring **Z**[√–5] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.

The local ring *k*[*x*,*y*]/(*y*^{2},*xy*) is not Cohen–Macaulay so does not have a dualizing module.

## See also

## References

- Bourbaki, N. (2007),
*Algèbre commutative. Chapitre 10*, Éléments de mathématique (in French), Springer-Verlag, Berlin, ISBN 978-3-540-34394-3, MR 2333539 - Bruns, Winfried; Herzog, Jürgen (1993),
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics,**39**, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956