On a generalization of Ulrich modules and its applications

TL;DR

This paper introduces a modified class of Ulrich modules called c-Ulrich modules, which always exist and help analyze Cohen-Macaulay properties and homological dimensions in algebraic structures.

Contribution

It defines c-Ulrich modules, demonstrating their existence and utility in studying Cohen-Macaulay obstructions and homological dimension finiteness.

Findings

c-Ulrich modules always exist unlike classical Ulrich modules

They serve as obstructions to Cohen-Macaulayness in tensor products

They are useful for testing finiteness of homological dimensions

Abstract

We study a modified version of the classical Ulrich modules, which we call $c$-Ulrich. Unlike the traditional setting, $c$-Ulrich modules always exist. We prove that these modules retain many of the essential properties and applications observed in the literature. Additionally, we reveal their significance as obstructions to Cohen-Macaulay properties of tensor products. Leveraging this insight, we show the utility of these modules in testing the finiteness of homological dimensions across various scenarios.