Gorenstein Projective Dimension Relative to a Semidualizing Bimodule

Let S and R be rings and S C R a semidualizing bimodule. We investigate the relation between the G C -syzygy with the C-syzygy of a module as well as the relation between the G C -projective resolution and the projective resolution of a module. As a consequence, we get that if is an exact sequence of S-modules with all G i , G i G C -projective, such that Hom S (𝔾, T) is still exact for any module T which is isomorphic to a direct summand of direct sums of copies of S C, then Im(G 0 → G 0) is also G C -projective. We obtain a criterion for computing the G C -projective dimension of modules. When S C R is a faithfully semidualizing bimodule, we study the Foxby equivalence between the subclasses of the Auslander class and that of the Bass class with respect to C.