Generalized Matlis duality

Abstract

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