The authors introduce and investigate the T_C-Gorenstein projective,L_CGorenstein injective and H_C-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective,injective and flat modules,respectively.The authors prove that the classes of all the T_C-Gorenstein projective or the H_C-Gorenstein flat modules axe exactly those Gorenstein projective or fiat modules which are in the Auslander class with respect to C,respectively,and the classes of all the L_C-Gorenstein injective modules axe exactly those Gorenstein injective modules which are in the Bass class,so the authors get the relations between the Gorenstein projective,injective or flat modules and the C-Gorenstein projective,injective or flat modules.Moreover,the authors consider the T_C(R)-projective and L_C(R)-injective dimensions and T_C(R)-precovers and L_C(R)-preenvelopes.Finally,the authors study the H_C-Gorenstein flat modules and extend the Foxby equivalences.
