Homological dimensions under Foxby equivalence

Abstract

Let R and S be rings and _RC_S a semidualizing bimodule, and let be a subcategory of the Auslander class and . Then for any left R-module M, the -projective dimension of Hom_R (C,M) is at most the -projective dimension of M, and they are identical when M is in the Bass class . If _RC_S is faithful and is resolving, then in a short exact sequence of left R-modules, the -projective dimensions of any two terms can determine an upper bound of that of the third term. Furthermore, we apply these results to the cases of being the subcategories of (weak) flat modules, projective modules and respectively. Some known results are obtained as corollaries.