A change of rings result for Matlis reflexivity

Dailey, Douglas; Marley, Thomas

doi:10.1090/proc/13287

A change of rings result for Matlis reflexivity
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by Douglas J. Dailey and Thomas Marley PDF

Proc. Amer. Math. Soc. 145 (2017), 1837-1841 Request permission

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 $M{\longrightarrow }\mathrm {Hom}_R(\mathrm {Hom}_R(M,E),E)$ is an isomorphism. We prove that if $S$ is a multiplicatively closed subset of $R$ and $M$ is an $R_S$-module which is reflexive as an $R$-module, then $M$ is a reflexive $R_S$-module. The converse holds when $S$ is the complement of the union of finitely many nonminimal primes of $R$, but fails in general.

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