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Remarks on almost Gorenstein rings

Naoki Endo, Naoyuki Matsuoka

Abstract

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once $R$ is an almost Gorenstein graded ring, the localization $R_M$ of $R$ at the graded maximal ideal $M$ is almost Gorenstein as a local ring. The converse does not hold true in general. However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well.

Remarks on almost Gorenstein rings

Abstract

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once is an almost Gorenstein graded ring, the localization of at the graded maximal ideal is almost Gorenstein as a local ring. The converse does not hold true in general. However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well.
Paper Structure (3 sections, 5 theorems, 26 equations)

This paper contains 3 sections, 5 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Examples of Theorem \ref{['main']}

Key Result

Theorem 1.1

There exists a graded canonical ideal $J$ of $R$ containing a parameter ideal as a reduction, and the following conditions are equivalent.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['main']}
  • Corollary 2.2: GKMT
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Corollary 2.7: cf. GTT
  • ...and 8 more