Partial Trace Ideals, Torsion and Canonical Module

TL;DR

This paper investigates the numerical invariant h(M) for modules over one-dimensional Noetherian local domains, establishing bounds and properties that relate to torsion, canonical modules, and classifications, extending previous work on trace ideals.

Contribution

The paper introduces new bounds on h(M) that connect torsion properties and canonical modules, generalizing prior results and addressing open questions in the field.

Findings

Established bounds on h(M) related to torsion submodules.

Analyzed properties of h(M) for canonical modules.

Provided classifications based on these bounds.

Abstract

For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's Conjecture". We establish a bound on it which helps capture information about the torsion submodule of $M$ when $M$ has rank one and it also generalizes the discussion in the mentioned previous article. We further study bounds and properties of $h(M)$ in the case when $M$ is the canonical module $\omega_R$. This in turn helps in answering a question of S. Greco and then provide some classifications. Most of the results in this article are based on the results presented in the author's doctoral dissertation "Partial Trace Ideals, The Conductor and Berger's Conjecture".