Using semidualizing complexes to detect Gorenstein rings

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.