Testing for the Gorenstein property

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.