On the theory of generalized Ulrich modules

TL;DR

This paper advances the theory of generalized Ulrich modules, examining how Hom and linkage operations affect their properties, and provides new characterizations and connections to other algebraic concepts.

Contribution

It investigates the preservation of Ulrich properties under Hom and linkage, offers a new characterization of quadratic hypersurface rings, and links Ulrich modules to modules with minimal multiplicity.

Findings

Hom and linkage operations can preserve Ulrich properties under certain conditions

A new characterization of quadratic hypersurface rings is provided

Ulrich modules' Chern number and Rees module regularity are determined

Abstract

In this paper we further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem of when the operations of taking the Hom functor and horizontal linkage preserve the Ulrich property. One of the applications is a new characterization of quadratic hypersurface rings. Moreover, in the Gorenstein case, we deduce that applying linkage to sufficiently high syzygy modules of Ulrich ideals yields Ulrich modules. Finally, we explore connections to the theory of modules with minimal multiplicity, and as a byproduct we determine the Chern number of an Ulrich module as well as the Castelnuovo-Mumford regularity of its Rees module.