Reflexive modules, self-dual modules and Arf rings

TL;DR

This paper explores the relationship between reflexive modules and self-dual modules over one-dimensional rings, characterizing Arf rings and their modules, with implications for non-commutative desingularizations and similarities to rational singularities.

Contribution

It establishes a connection between reflexive and self-dual modules over Arf rings, characterizes Arf rings via reflexive modules, and explores applications in non-commutative desingularizations.

Findings

Complete local reduced Arf rings have finitely many indecomposable reflexive modules.

Arf rings are characterized by all reflexive modules being self-dual.

Results reveal similarities between Arf rings and rational singularities in higher dimensions.

Abstract

We prove a tight connection between reflexive modules over a one-dimensional ring $R$ and its birational extensions that are self-dual as $R$-modules. Consequently, we show that a complete local reduced Arf ring has finitely many indecomposable reflexive modules up to isomorphism, which can be represented precisely by the local rings infinitely near it. We also characterize Arf rings by the property that any reflexive module is self-dual. We give applications on dimension of subcategories and existence of endomorphism rings of small global dimension (non-commutative desingularizations). Our results indicate striking similarities between Arf rings in dimension one and rational singularities in dimension two from representation-theoretic and categorical perspectives.