Matlis duality

Suppose that R is a Noetherian complete local ring with residue field k, and choose E to be an 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.

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 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 with quotient field K then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis dual of a finitely-generated module is the Pontryagin dual of it considered as a locally compact abelian group.

If R is a Cohen–Macaulay local ring of dimension d with dualizing module Ω, then the Matlis module is given by the 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.

Matlis duality can be conceptually explained using the language of adjoint functors and derived categories:[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)

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

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

• Grothendieck local duality

• Bruns, Winfried; Herzog, Jürgen (1993), Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956
• Matlis, Eben (1958), "Injective modules over Noetherian rings", Pacific Journal of Mathematics, 8: 511–528, doi:10.2140/pjm.1958.8.511, ISSN 0030-8730, MR 0099360, archived from the original on 2014-05-03