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

Ela Celikbas, Olgur Celikbas, Naoki Endo, Shinya Kumashiro

TL;DR

The paper advances the theory of minimal multiplicity by extending it to finitely generated modules over 1-dimensional CM local rings and by studying it with respect to trace or reflexive $\mathfrak{m}$-primary ideals. It introduces a central equivalence: for generically Gorenstein non-Gorenstein rings with canonical module $\omega_R$ and an $\mathfrak{m}$-primary ideal $I$ that is trace or reflexive, $\omega_R$ having minimal multiplicity with respect to $I$ is equivalent to $I$ being stable and to every torsion-free $R$-module having minimal multiplicity with respect to $I$, with further equivalences to $I$ being $\omega_R$-Ulrich and to ring-level minimal multiplicity properties of $R$ and related algebras. The results connect module-theoretic minimal multiplicity to the Ulrich property, the conductor, idealizations like $R \ltimes \mathfrak{m}$, and endomorphism algebras $\mathrm{Hom}_R(\mathfrak{m},\mathfrak{m})$, offering a framework to identify and construct new examples and to understand almost Gorenstein and Burch-type phenomena within this context. Overall, the work clarifies when canonical and reflexive/trace ideals enforce minimal multiplicity on a broad class of modules, and it links these conditions to practical constructions such as idealizations and conductor-based Ulrichs. $

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.

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

TL;DR

The paper advances the theory of minimal multiplicity by extending it to finitely generated modules over 1-dimensional CM local rings and by studying it with respect to trace or reflexive -primary ideals. It introduces a central equivalence: for generically Gorenstein non-Gorenstein rings with canonical module and an -primary ideal that is trace or reflexive, having minimal multiplicity with respect to is equivalent to being stable and to every torsion-free -module having minimal multiplicity with respect to , with further equivalences to being -Ulrich and to ring-level minimal multiplicity properties of and related algebras. The results connect module-theoretic minimal multiplicity to the Ulrich property, the conductor, idealizations like , and endomorphism algebras , offering a framework to identify and construct new examples and to understand almost Gorenstein and Burch-type phenomena within this context. Overall, the work clarifies when canonical and reflexive/trace ideals enforce minimal multiplicity on a broad class of modules, and it links these conditions to practical constructions such as idealizations and conductor-based Ulrichs. $

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.
Paper Structure (5 sections, 18 theorems, 24 equations)

This paper contains 5 sections, 18 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Definitions, Examples, and Remarks
  3. Modules of minimal multiplicity versus Ulrich modules
  4. Proof of Theorem \ref{['thm-intro']}
  5. An exact sequence involving modules of minimal multiplicity

Key Result

Theorem 1.1

Let $R$ be a one-dimensional generically Gorenstein non-Gorenstein ring with canonical ideal $\omega_R$, and let $I$ be an $\mathfrak{m}$-primary ideal of $R$. Assume $I$ is either a trace ideal of $R$, or reflexive as an $R$-module. Then the following conditions are equivalent:

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Example 2.7
  • Remark 2.8
  • Example 2.10
  • Remark 2.12
  • Remark 2.13
  • Remark 2.14
  • Example 2.15
  • Remark 2.16
  • ...and 38 more