  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.
