Arf rings, simple singularities and reflexive modules

TL;DR

This paper explores the properties of Arf rings, focusing on their reflexive modules and self-duality, and extends the investigation to higher-dimensional commutative Noetherian local rings.

Contribution

It generalizes the self-dual property of Arf rings to higher Krull dimension rings, expanding understanding of reflexive modules in this broader context.

Findings

Arf rings have finitely many indecomposable reflexive modules.

Reflexive modules over Arf rings are self-dual.

The self-dual property is characterized for Arf rings.

Abstract

In a pandemic era preprint, Dao showed showed two remarkable properties of Arf rings: under some mild conditions, they admit finitely many indecomposable reflexive modules up to isomorphism and every reflexive module is actually isomorphic to its own dual. In fact, the latter property characterises Arf rings. Arf rings are one dimensional rings and it is natural to wonder what happens in higher Krull dimension. In this paper, we investigate the self-dual property for commutative Noetherian local rings.