On the theory of generalized Ulrich modules

Cleto B. Miranda-Neto, Douglas S. Queiroz and Thyago S. Souza

Vol. 323 (2023), No. 2, 307–335

DOI: 10.2140/pjm.2023.323.307

Abstract

We further develop the theory of generalized Ulrich modules introduced in 2014 by Goto et al. Our main goal is to address the problem as to 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.

Dedicated with gratitude to the memory of Professor Shiro Goto

Keywords

Ulrich module, maximal Cohen–Macaulay module, horizontal linkage, module of minimal multiplicity, blowup module

Mathematical Subject Classification

Primary: 13C05, 13C14, 13H10

Secondary: 13A30, 13C13, 13C40, 13D07

Milestones

Received: 18 August 2022

Revised: 15 March 2023

Accepted: 18 March 2023

Published: 2 June 2023