Ulrich split rings

TL;DR

This paper introduces the concept of Ulrich split rings, characterizes their properties, and explores their structure, especially in low dimensions, with applications to Cohen-Macaulay modules and singularity theory.

Contribution

It initiates the study of Ulrich split rings, providing criteria, characterizations, constructions, and applications in algebraic geometry and commutative algebra.

Findings

Ulrich split rings are characterized by their syzygies and cohomology annihilators.

2-dimensional normal Ulrich split rings with minimal multiplicity are cyclic quotient singularities.

Various constructions and applications of Ulrich split rings are provided.

Abstract

A local Cohen--Macaulay ring is called Ulrich-split if any short exact sequence of Ulrich modules split. In this paper we initiate the study of Ulrich split rings. We prove several necessary or sufficient criteria for this property, linking it to syzygies of the residue field and cohomology annihilator. We characterize Ulrich split rings of small dimensions. Over complex numbers, $2$-dimensional Ulrich split rings, which are normal and have minimal multiplicity, are precisely cyclic quotient singularities with at most two indecomposable Ulrich modules up to isomorphism. We give several ways to construct Ulrich split rings, and give a range of applications, from test ideal of the family of maximal Cohen--Macaulay modules, to detecting projective/injective modules via vanishing of $\operatorname{Ext}$.