Gorenstein Homological Properties and Quasi-Frobenius Bimodules

We establish relations of Gorenstein homological properties of modules and rings linked by a fixed quasi-Frobenius bimodule. Particularly, let R⊂S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\subset S$$\end{document} be a strongly separable quasi-Frobenius extension. The left Gorenstein global dimensions and the left finitistic Gorenstein projective dimensions of rings S and R are equal. Moreover, R is left-Gorenstein (Cohen–Macaulay finite, Cohen–Macaulay free) if and only if so is S.