Generalized Matlis duality

Abstract:

Let $R$ be a commutative noetherian ring and let $E$ be the minimal injective cogenerator of the category of $R$-modules. A module $M$ is said to be reflexive with respect to $E$ if the natural evaluation map from $M$ to $\operatorname {Hom}_R( \operatorname {Hom}_R(M,E), E)$ is an isomorphism. We give a classification of modules which are reflexive with respect to $E$. A module $M$ is reflexive with respect to $E$ if and only if $M$ has a finitely generated submodule $S$ such that $M/S$ is artinian and $R/\operatorname {ann}(M)$ is a complete semi-local ring.