Gorenstein Projective Dimension Relative to a Semidualizing Bimodule

摘要

Let S and R be rings and SCR 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 G:...→ G_1 → G_0 → G~0 → G~1 →... is an exact sequence of S-modules with all G _i, G~iG_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 _SC, 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 _SC_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.