A change of rings result for Matlis reflexivity

TL;DR

This paper investigates how Matlis reflexivity of modules over a Noetherian ring behaves under localization, establishing conditions under which reflexivity is preserved or not.

Contribution

It proves that reflexivity is preserved under localization for any multiplicative set and characterizes when the converse holds, extending understanding of module duality.

Findings

Reflexivity is preserved under localization for any multiplicative set.

The converse holds when the multiplicative set is the complement of finitely many minimal primes.

Reflexivity may fail to be preserved in general localization scenarios.

Abstract

Let $R$ be a commutative Noetherian ring and $E$ the minimal injective cogenerator of the category of $R$-modules. An $R$-module $M$ is (Matlis) reflexive if the natural evaluation map $M \to \operatorname{Hom}_R(\operatorname{Hom}_R(M,E),E)$ is an isomorphism. We prove that if $S$ is a multiplicatively closed subset of $R$ and $M$ is a reflexive $R$-module, then $M$ is a reflexive $R_S$-module. The converse holds when $S$ is the complement of the union of finitely many minimal primes of $R$, but fails in general.