Remarks on almost Gorenstein rings

TL;DR

This paper explores the relationship between almost Gorenstein properties in graded and local rings, clarifying conditions under which these properties are equivalent, especially in one-dimensional graded domains and numerical semigroup rings.

Contribution

It establishes when the almost Gorenstein property in graded rings implies the same in localizations and analyzes defining ideals of numerical semigroup rings.

Findings

Localization of an almost Gorenstein graded ring is almost Gorenstein.

The converse implication holds under mild conditions in one-dimensional graded domains.

Characterization of defining ideals of almost Gorenstein numerical semigroup rings.

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.