Relative Gorenstein objects in abelian categories

Abstract

Abstract Let be an abelian category. For a pair of classes of objects in we define the weak and the -Gorenstein relative projective objects in We point out that such objects generalize the usual Gorenstein projective objects and other generalizations appearing in the literature as Ding-projective, Ding-injective, -Gorenstein projective, Gorenstein AC-projective and GC -projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein projective modules remain true for the weak and the -Gorenstein relative objects. Furthermore, by using Auslander-Buchweitz approximation theory, a relative version of Gorenstein homological dimension is developed. Finally, we introduce the notion of -cotilting pair in the abelian category which is very strong connected with the cotorsion pairs related with relative Gorenstein objects in It is worth mentioning that the -cotilting pairs generalize the notion of cotilting objects in the sense of L. Angeleri Hügel and F. Coelho [3].