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.

14 pages; v.2 includes minor revisions; v.3 includes some clarifications, minor corrections, a few new examples, and a question