Arf rings, simple singularities and reflexive modules

2025/05/22 by Özgür Esentepe, Esentepe, Özgür
Mathematics · #13C05 #13C14 #13C60 #13D07 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2505.16568

openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Stable trace ideals and applications
• Reflexive modules, self-dual modules and Arf rings
• Reflexive modules over Arf local rings
• On the Subcategories of n-Torsionfree Modules and Related Modules
• On the ubiquity of Arf rings
• Finite Gorenstein representation type implies simple singularity
• Two theorems about maximal Cohen-Macaulay modules

Related

• Reflexive modules, self-dual modules and Arf rings
• Reflexive modules over Arf local rings
• When are syzygies of the residue field self-dual?
• Some classes of one-dimensional rings characterized by their reflexive ideals
• On the ubiquity of Arf rings