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On generalized Gorenstein local rings

Shiro Goto, Shinya Kumashiro

TL;DR

The paper introduces generalized Gorenstein local rings (GGL) as a broad framework that extends almost Gorenstein rings and unifies nearly Gorenstein phenomena. It develops Ulrich theory in this context, linking Ulrich modules and Ulrich ideals to the trace of the canonical module, and analyzes endomorphism algebras of the maximal ideal to reveal structural constraints. The core results establish strong one-dimensional characterizations, including when the defining m-primary ideal equals the trace of the canonical module, and extend these insights via flat base change and higher-dimensional constructions. The authors provide extensive constructions and examples across numerical semigroup rings, idealizations, and determinantal rings, demonstrating the versatility of GGL rings and illuminating connections to Rees algebras and minimal free resolutions. Collectively, these results yield a cohesive framework for non-Gorenstein Cohen–Macaulay rings with Ulrich-related invariants and broad applicability to explicit ring-constructs.

Abstract

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a Cohen-Macaulay local ring $R$, we explore the endomorphism algebra of the maximal ideal, the trace ideal of the canonical module, Ulrich ideals, and Rees algebras of parameter ideals in connection with the GGL property. We also give numerous examples of numerical semigroup rings, idealizations, and determinantal rings of certain matrices.

On generalized Gorenstein local rings

TL;DR

The paper introduces generalized Gorenstein local rings (GGL) as a broad framework that extends almost Gorenstein rings and unifies nearly Gorenstein phenomena. It develops Ulrich theory in this context, linking Ulrich modules and Ulrich ideals to the trace of the canonical module, and analyzes endomorphism algebras of the maximal ideal to reveal structural constraints. The core results establish strong one-dimensional characterizations, including when the defining m-primary ideal equals the trace of the canonical module, and extend these insights via flat base change and higher-dimensional constructions. The authors provide extensive constructions and examples across numerical semigroup rings, idealizations, and determinantal rings, demonstrating the versatility of GGL rings and illuminating connections to Rees algebras and minimal free resolutions. Collectively, these results yield a cohesive framework for non-Gorenstein Cohen–Macaulay rings with Ulrich-related invariants and broad applicability to explicit ring-constructs.

Abstract

In this paper, we introduce generalized Gorenstein local (GGL) rings. The notion of GGL rings is a natural generalization of the notion of almost Gorenstein rings, which can thus be treated as part of the theory of GGL rings. For a Cohen-Macaulay local ring , we explore the endomorphism algebra of the maximal ideal, the trace ideal of the canonical module, Ulrich ideals, and Rees algebras of parameter ideals in connection with the GGL property. We also give numerous examples of numerical semigroup rings, idealizations, and determinantal rings of certain matrices.
Paper Structure (14 sections, 67 theorems, 127 equations)

This paper contains 14 sections, 67 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ulrich modules with respect to an $\mathfrak{m}$-primary ideal
  4. Ulrich ideals and the trace ideal of the canonical module
  5. $\operatorname{GGL}$ rings
  6. One-dimensional $\operatorname{GGL}$ rings
  7. Defining $\mathfrak{m}$-primary ideals of one-dimensional $\operatorname{GGL}$ rings
  8. Endomorphism algebra of the maximal ideal
  9. The Ulrich property for $\mathfrak{c}$ and $\mathrm{tr}_R(\mathrm{K}_R)$
  10. $\operatorname{GGL}$ rings arising from idealizations
  11. Numerical semigroup rings
  12. Minimal free resolutions of $\operatorname{GGL}$ rings
  13. $\operatorname{GGL}$ rings of higher dimension
  14. Determinantal $\operatorname{GGL}$ rings of certain specific matrices

Key Result

Theorem 1.3

The following assertions hold true.

Theorems & Definitions (152)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Theorem \ref{['a3.5']}
  • Theorem 1.4: Theorem \ref{['aaaa72']}
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 142 more