Matlis duality

Matlis duality is a concept in algebra that establishes a relationship between Artinian and Noetherian modules over a complete Noetherian local ring. It was introduced by Matlis (1958) and builds upon earlier work by Francis Sowerby Macaulay on polynomial rings, which is sometimes referred to as Macaulay duality. The duality is defined using an injective hull \( E \) of the residue field \( k \). For a given module \( M \), its dual \( D_R(M) \) is defined as \( \text{Hom}_R(M, E) \). Matlis duality states that this duality functor creates an anti-equivalence between the categories of Artinian and Noetherian \( R \)-modules. Specifically, it provides an anti-equivalence from the category of finite-length modules to itself. Examples of Matlis duality include cases where the ring \( R \) is a discrete valuation ring or the ring of \( p \)-adic numbers. In these instances, the dual of a finitely generated module corresponds to its Pontryagin dual as a locally compact abelian group. Additionally, for Cohen–Macaulay local rings of dimension \( d \), the Matlis module is given by the local cohomology group \( H^d_R(\Omega) \), where \( \Omega \) is the dualizing module. The theory can be explained using adjoint functors and derived categories. The right adjoint of a functor between derived categories of \( R \)- and \( k \)-modules sends the injective hull \( E(k) \) to \( k \), establishing a dualizing object in the category of \( k \)-modules. Matlis duality is closely related ...