Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings

TL;DR

This paper characterizes modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings, linking their properties to trace and reflexive ideals, and explores their relation to Burch and Ulrich modules.

Contribution

It provides a characterization of minimal multiplicity modules via trace and reflexive ideals, extending classical concepts to modules over Cohen-Macaulay rings.

Findings

Trace and reflexive ideals characterize minimal multiplicity modules.

Minimal multiplicity of the canonical module implies the ring's minimal multiplicity.

Examples show the relationship between minimal multiplicity, Burch, and Ulrich modules.

Abstract

We study finitely generated modules of minimal multiplicity, a notion introduced by Puthenpurakal that extends the classical concept of minimal multiplicity from rings to modules. Our main result characterizes when trace ideals or reflexive ideals yield modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings. As a consequence, we show that a one-dimensional non-Gorenstein reduced local ring with a canonical module has minimal multiplicity if and only if its canonical module has minimal multiplicity as a module. We also construct several examples and compare them with Burch and Ulrich modules, highlighting cases where minimal multiplicity coincides with the Burch or Ulrich property.