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 $$R\subset S$$ 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.