Injective envelopes and flat covers of Matlis reflexive modules
All rings in this paper are commutative with identity and noetherian. For a local ring \((R,{\mathfrak m})\), the Matlis dual \(\text{Hom}_ R(M,E)\), where \(E\) is the injective envelope \(E_ R(R/{\mathfrak m})\), is denoted by \(M^ \lor\). If the canonical map \(M\to M^{\lor\lor}\) is an isomorphism, then \(M\) is said to be (Matlis) reflexive. The relation between the reflexivity of a module \(M\) and its injective hull \(E_ R(M)\) is studied for different types of rings and modules. Amongst these results, the most important one states that if \((R,{\mathfrak m})\) is a complete local ring with \(\dim R\leq 1\), then an \(R\)-module \(M\) is reflexive if and only if \(E(M)\) is reflexive. The paper culminates in the main result in which the authors show that for a commutative noetherian local ring \(R\), every reflexive \(R\)-module has a reflexive injective envelope if and only if every reflexive \(R\)-module has a reflexive flat cover. This occurs if and only if \(R\) is complete and has Krull dimension less than or equal to 1.
scientific article; zbMATH DE number 61223
scientific article