Reflexive modules and Auslander-type conditions

TL;DR

This paper investigates the structure of reflexive modules over Noetherian rings, establishing conditions for their category to be quasi-abelian and providing a Morita theorem characterizing these categories.

Contribution

It introduces a new characterization of the category of reflexive modules as quasi-abelian under Auslander-type conditions and proves a Morita theorem for these categories.

Findings

Reflexive module category is quasi-abelian iff the ring satisfies an Auslander-type condition.

A Morita theorem characterizes reflexive modules among quasi-abelian categories.

Conditions on injective resolutions relate to the categorical properties of reflexive modules.

Abstract

We study the category $\mathop{\mathrm{ref}}\Lambda$ of reflexive modules over a two-sided Noetherian ring $\Lambda$. We show that the category $\mathop{\mathrm{ref}}\Lambda$ is quasi-abelian if and only if $\Lambda$ satisfies certain Auslander-type condition on the minimal injective resolution of the ring itself. Furthermore, we establish a Morita theorem which characterizes the category of reflexive modules among quasi-abelian categories in terms of generator-cogenerators.