Testing for the Gorenstein property

TL;DR

This paper characterizes Gorenstein rings via the existence of a special integrally closed ideal with finite Gorenstein dimension, advancing understanding of ring properties through test complexes.

Contribution

It provides a new characterization of Gorenstein rings using integrally closed ideals and introduces test complexes that distinguish Gorenstein dimension from projective dimension.

Findings

Gorenstein rings are characterized by specific ideals of finite Gorenstein dimension.

Constructed test complexes that detect Gorenstein dimension finiteness.

Differentiated Gorenstein dimension from projective dimension using new complexes.

Abstract

We answer a question of Celikbas, Dao, and Takahashi by establishing the following characterization of Gorenstein rings: a commutative noetherian local ring $(R,\mathfrak m)$ is Gorenstein if and only if it admits an integrally closed $\mathfrak m$-primary ideal of finite Gorenstein dimension. This is accomplished through a detailed study of certain test complexes. Along the way we construct such a test complex that detect finiteness of Gorenstein dimension, but not that of projective dimension.