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 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.
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 HdR(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)
This right adjoint sends the injective hull E ( k ) {\displaystyle E(k)} mentioned above to k, which is a dualizing object in D ( k ) {\displaystyle D(k)} . This abstract fact then gives rise to the above-mentioned equivalence.
Paul Balmer, Ivo Dell'Ambrogio, and Beren Sanders. Grothendieck-Neeman duality and the Wirthmüller isomorphism, 2015. Example 7.2. /wiki/Paul_Balmer ↩