\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Let $(R, \frak m)$ be a Noetherian local ring of depth $d$ and $C$
a semidualizing $R$-complex. Let $M$ be a finite $R$-module and
$t$ an integer between 0 and $d$. If $G_{C}$-dimension of $M/\frak
a M$ is finite for all ideals $\frak a$ generated by an
$R$-regular sequence of length at most $d-t$ then either
$G_{C}$-dimension of $M$ is at most $t$ or $C$ is a dualizing
complex. Analogous results for other homological dimensions are
also given.
\end{document}