In the special case when the local ring has a field[clarification needed] mapping to the residue field it is closely related to earlier work by Francis Sowerby Macaulay on polynomial rings and is sometimes called Macaulay duality, and the general case was introduced by Matlis (1958).
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).
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.
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