We apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

For any ring $R$, we show that, in the bounded derived category ${{D}^{b}}(\text{Mod}\,R)$ of left $R$-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\text{GP}}(\text{Mod}\,R)\,(resp.\overline{GI}(Mod\,R))$ of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\text{GP}}(\text{Mod}\,R)$ and $\overline{\text{GI}}(\text{Mod}\,R)$ are equivalent.

Let $\Gamma $ be a group and ${\Gamma }^{\prime } $ be a subgroup of $\Gamma $ of finite index. Let $M$ be a $\Gamma $-module. It is shown that $M$ is (strongly) Gorenstein flat if and only if it is (strongly) Gorenstein flat as a ${\Gamma }^{\prime } $-module. We also provide some criteria in which the classes of Gorenstein projective and strongly Gorenstein flat $\Gamma $-modules are the same.

In basic homological algebra, projective, injective and flat modules play an important and fundamental role. In this paper, we discuss some properties of Gorenstein projective, injective and flat modules and study some connections between Gorenstein injective and Gorenstein flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under change of rings.