Full-Trace Modules

TL;DR

This paper introduces the concept of full-trace modules over commutative Noetherian local rings, explores their existence, and characterizes Cohen-Macaulay rings with minimal multiplicity via full-trace Ulrich modules.

Contribution

It defines full-trace modules, proves their existence in certain rings, and characterizes minimal multiplicity Cohen-Macaulay rings through full-trace Ulrich modules.

Findings

Every positive syzygy of the residue field is full-trace in non-regular, non-principal rings.

A Cohen-Macaulay local ring has minimal multiplicity iff it admits a full-trace Ulrich module.

Full-trace Ulrich modules decompose into the maximal ideal plus a zero or Ulrich module in numerical semigroup rings.

Abstract

Motivated by the definition of nearly Gorenstein rings, we introduce the notion of full-trace modules over commutative Noetherian local rings--namely, finitely generated modules whose trace equals the maximal ideal. We investigate the existence of such modules and prove that, over rings that are neither regular nor principal ideal rings, every positive syzygy module of the residue field is full-trace. Moreover, over Cohen-Macaulay rings, we study full-trace Ulrich modules--that is, maximally generated maximal Cohen-Macaulay modules that are full-trace. We establish the following characterization: a non-regular Cohen-Macaulay local ring has minimal multiplicity if and only if it admits a full-trace Ulrich module. Finally, for numerical semigroup rings with minimal multiplicity, we show that each full-trace Ulrich module decomposes as the direct sum of the maximal ideal and a module that is either zero or Ulrich.