Separable pairs, Gorenstein categories and induced triangle equivalences

We prove that any separable pair of functors between abelian categories preserves relative Gorenstein categories. As a consequence, it preserves and reflects Gorenstein projective and Gorenstein injective objects in abelian categories. Furthermore, we discuss when a separable pair induces triangle equivalences on the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. By applying these results to the category of modules, we obtain new properties for separably equivalent rings.