Using semidualizing complexes to detect Gorenstein rings

TL;DR

This paper extends Foxby's criterion for Gorenstein rings by using semidualizing complexes and homological dimensions, providing a broader characterization of Gorenstein rings in commutative algebra.

Contribution

It improves Foxby's result by showing that the existence of certain complexes with finite homological dimensions involving a semidualizing complex implies the ring is Gorenstein.

Findings

Established that specific homological conditions involving semidualizing complexes characterize Gorenstein rings.

Extended Foxby's criterion to include $ ext{F}_C$-projective and $ ext{I}_C$-injective dimensions.

Provided an affirmative answer to a question of Takahashi and White.

Abstract

A result of Foxby states that if there exists a complex with finite depth, finite flat dimension, and finite injective dimension over a local ring $R$, then $R$ is Gorenstein. In this paper we investigate some homological dimensions involving a semidualizing complex and improve on Foxby's result by answering a question of Takahashi and White. In particular, we prove for a semidualizing complex $C$, if there exists a complex with finite depth, finite $\mathcal{F}_C$-projective dimension, and finite $\mathcal{I}_C$-injective dimension over a local ring $R$, then $R$ is Gorenstein.