Reflexive modules over the endomorphism algebras of reflexive trace ideals

TL;DR

This paper studies reflexive modules over endomorphism algebras of reflexive trace ideals in one-dimensional Cohen-Macaulay local rings, generalizing previous results and exploring conditions for finite representation type.

Contribution

It generalizes existing results on endomorphism algebras of maximal ideals and characterizes when the category of reflexive modules is of finite type.

Findings

Finite type category implies the ring is analytically unramified.

Finitely many Ulrich ideals in the ring under finite type conditions.

In Arf local rings with local normalization, only finitely many Ulrich ideals exist.

Abstract

In the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong and T. Kobayashi concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified and has only finitely many Ulrich ideals. As a consequence, there are only finitely many Ulrich ideals are contained in Arf local rings once the normalization is a local ring.