# Matlis duality

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Theorem in algebra

In [algebra](/source/Algebra), **Matlis duality** is a [duality](/source/Duality_(mathematics)) between [Artinian](/source/Artinian_module) and [Noetherian](/source/Noetherian_module) [modules](/source/Module_(mathematics)) over a complete Noetherian [local ring](/source/Local_ring). In the special case when the local ring contains a field mapping to the [residue field](/source/Residue_field) it is closely related to earlier work by [Francis Sowerby Macaulay](/source/Francis_Sowerby_Macaulay) on [polynomial rings](/source/Polynomial_ring) and is sometimes called **Macaulay duality**, and the general case was introduced by [Matlis](/source/Eben_Matlis) ([1958](#CITEREFMatlis1958)).

## Statement

Suppose that *R* is a Noetherian complete local ring with residue field *k*, and choose *E* to be an [injective hull](/source/Injective_hull) of *k* (sometimes called a **Matlis module**). The dual *D**R*(*M*) of a module *M* is defined to be Hom*R*(*M*,*E*). Then Matlis duality states that the duality functor *D**R* gives an anti-equivalence between the categories of Artinian and Noetherian *R*-modules. In particular the duality functor gives an anti-equivalence from the category of finite-length modules to itself.

## Examples

Suppose that the Noetherian complete local ring *R* has a subfield *k* that maps onto a subfield of finite index of its residue field *R*/*m*. Then the Matlis dual of any *R*-module is just its dual as a [topological vector space](/source/Topological_vector_space) over *k*, if the module is given its *m*-adic topology. In particular the dual of *R* as a topological vector space over *k* is a Matlis module. This case is closely related to work of Macaulay on graded polynomial rings and is sometimes called Macaulay duality.

If *R* is a [discrete valuation ring](/source/Discrete_valuation_ring) with [quotient field](/source/Quotient_field) *K* then the Matlis module is *K*/*R*. In the special case when *R* is the ring of [*p*-adic numbers](/source/P-adic_number), the Matlis dual of a [finitely-generated module](/source/Finitely-generated_module) is the [Pontryagin dual](/source/Pontryagin_dual) of it considered as a [locally compact](/source/Locally_compact_group) [abelian group](/source/Abelian_group).

If *R* is a Cohen–Macaulay local ring of dimension *d* with [dualizing module](/source/Dualizing_module) Ω, then the Matlis module is given by the [local cohomology](/source/Local_cohomology) group H*d* *R*(Ω). In particular if *R* is an Artinian local ring then the Matlis module is the same as the dualizing module.

## Explanation using adjoint functors

Matlis duality can be conceptually explained using the language of [adjoint functors](/source/Adjoint_functor) and [derived categories](/source/Derived_category):[1] the functor between the derived categories of *R*- and *k*-modules induced by regarding a *k*-module as an *R*-module, admits a right adjoint (derived [internal Hom](/source/Internal_Hom))

- D ( k ) ← D ( R ) : R Hom R ⁡ ( k , − ) . {\displaystyle D(k)\gets D(R):R\operatorname {Hom} _{R}(k,-).}

This right adjoint sends the injective hull E ( k ) {\displaystyle E(k)} mentioned above to *k*, which is a [dualizing object](/source/Dualizing_object) in D ( k ) {\displaystyle D(k)} . This abstract fact then gives rise to the above-mentioned equivalence.

## See also

- [Grothendieck local duality](/source/Grothendieck_local_duality)

## References

1. **[^](#cite_ref-1)** [Paul Balmer](/source/Paul_Balmer), Ivo Dell'Ambrogio, and Beren Sanders. [*Grothendieck-Neeman duality and the Wirthmüller isomorphism*](https://arxiv.org/abs/1501.01999v1), 2015. Example 7.2.

- 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)

- [Matlis, Eben](/source/Eben_Matlis) (1958), ["Injective modules over Noetherian rings"](https://web.archive.org/web/20140503194835/http://projecteuclid.org/euclid.pjm/1103039896), *[Pacific Journal of Mathematics](/source/Pacific_Journal_of_Mathematics)*, **8**: 511–528, [doi](/source/Doi_(identifier)):[10.2140/pjm.1958.8.511](https://doi.org/10.2140%2Fpjm.1958.8.511), [ISSN](/source/ISSN_(identifier)) [0030-8730](https://search.worldcat.org/issn/0030-8730), [MR](/source/MR_(identifier)) [0099360](https://mathscinet.ams.org/mathscinet-getitem?mr=0099360), archived from [the original](https://projecteuclid.org/euclid.pjm/1103039896) on 2014-05-03

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