# Dualizing module

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In [abstract algebra](/source/Abstract_algebra), a **dualizing module**, also called a **canonical module**, is a [module](/source/Module_(mathematics)) over a [commutative ring](/source/Commutative_ring) that is analogous to the [canonical bundle](/source/Canonical_bundle) of a [smooth variety](/source/Smooth_variety). It is used in [Grothendieck local duality](/source/Grothendieck_local_duality).

## Definition

A dualizing module for a [Noetherian ring](/source/Noetherian_ring) *R* is a [finitely generated module](/source/Finitely_generated_module) *M* such that for any [maximal ideal](/source/Maximal_ideal) *m*, the *R*/*m* [vector space](/source/Vector_space) Ext*n* *R*(*R*/*m*,*M*) vanishes if *n* ≠ height(*m*) and is [1-dimensional](/source/Dimension_(vector_space)) if *n* = height(*m*).

A dualizing module need not be unique because the [tensor product](/source/Tensor_product) of any dualizing module with a rank 1 [projective module](/source/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](/source/Cohen%E2%80%93Macaulay_ring). Conversely if a Cohen–Macaulay ring is a quotient of a [Gorenstein ring](/source/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](/source/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](/source/Artinian_ring) [local ring](/source/Local_ring) then the [Matlis module](/source/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

- [dualizing sheaf](/source/Dualizing_sheaf)

## References

- [Bourbaki, N.](/source/Nicolas_Bourbaki) (2007), *Algèbre commutative. Chapitre 10*, Éléments de mathématique (in French), Springer-Verlag, Berlin, [ISBN](/source/ISBN_(identifier)) [978-3-540-34394-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-34394-3), [MR](/source/MR_(identifier)) [2333539](https://mathscinet.ams.org/mathscinet-getitem?mr=2333539)

- Bruns, Winfried; Herzog, Jürgen (1993), [*Cohen-Macaulay rings*](https://books.google.com/books?id=LF6CbQk9uScC), Cambridge Studies in Advanced Mathematics, vol. 39, [Cambridge University Press](/source/Cambridge_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-521-41068-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-41068-7), [MR](/source/MR_(identifier)) [1251956](https://mathscinet.ams.org/mathscinet-getitem?mr=1251956)

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