Matlis duality

<h2 id="statement">Statement</h2>
Suppose that R is a Noetherian complete local ring with residue field k, and choose E to be an <a href="/facts/Injective_hull/VoENIwTw">injective hull</a> of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then Matlis duality states that the duality functor DR 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.

<h2 id="examples">Examples</h2>
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 <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> 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 <a href="/facts/Discrete_valuation_ring/DaHoTBSH">discrete valuation ring</a> with <a href="/facts/Quotient_field/cmlJrDSS">quotient field</a> K then the Matlis module is K/R. In the special case when R is the ring of <a href="/facts/P-adic_number/SDgDbkLF">p-adic numbers</a>, the Matlis dual of a <a href="/facts/Finitely-generated_module/6Njcpgzr">finitely-generated module</a> is the <a href="/facts/Pontryagin_dual/fbD3d8kz">Pontryagin dual</a> of it considered as a <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact</a> <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>.
If R is a Cohen–Macaulay local ring of dimension d with <a href="/facts/Dualizing_module/W6XrJr5R">dualizing module</a> Ω, then the Matlis module is given by the <a href="/facts/Local_cohomology/NjY1n2Vl">local cohomology</a> group HdR(Ω). In particular if R is an Artinian local ring then the Matlis module is the same as the dualizing module.

<h2 id="explanation-using-adjoint-functors">Explanation using adjoint functors</h2>
Matlis duality can be conceptually explained using the language of <a href="/facts/Adjoint_functor/bJ1P7AN2">adjoint functors</a> and <a href="/facts/Derived_category/pBQHnFYo">derived categories</a>:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 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 <a href="/facts/Internal_Hom/H76BLKCD">internal Hom</a>)

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 <a href="/facts/Dualizing_object/kF0mrSrI">dualizing object</a> in 
 
 
 
 D
 (
 k
 )
 
 
 {\displaystyle D(k)}
 
. This abstract fact then gives rise to the above-mentioned equivalence.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Grothendieck_local_duality/EczNNxQ0">Grothendieck local duality</a></li></ul>

<ul><li>Bruns, Winfried; Herzog, Jürgen (1993), <a href="https://books.google.com/books?id=LF6CbQk9uScC">Cohen-Macaulay rings</a>, Cambridge Studies in Advanced Mathematics, vol. 39, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-41068-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1251956">1251956</a></li>
<li><a href="/facts/Eben_Matlis/vYqJkMxC">Matlis, Eben</a> (1958), <a href="https://web.archive.org/web/20140503194835/http://projecteuclid.org/euclid.pjm/1103039896">"Injective modules over Noetherian rings"</a>, <a href="/facts/Pacific_Journal_of_Mathematics/fBmfJLky">Pacific Journal of Mathematics</a>, 8: 511–528, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1958.8.511">10.2140/pjm.1958.8.511</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0030-8730">0030-8730</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0099360">0099360</a>, archived from <a href="https://projecteuclid.org/euclid.pjm/1103039896">the original</a> on 2014-05-03</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Paul Balmer, Ivo Dell'Ambrogio, and Beren Sanders.
Grothendieck-Neeman duality and the Wirthmüller isomorphism, 2015. Example 7.2. <a href="/wiki/Paul_Balmer" target="_blank">/wiki/Paul_Balmer</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Matlis duality open-in-new

Matlis duality