ABSTRACT
In this chapter, we study Gorenstein homological dimensions in a general setting, i.e., in any Abelian category. We begin defining Gorensteinprojective and Gorenstein-injective objects. We show that these two classes of objects are Frobenius categories. Then, we present the Gorenstein-projective and Gorenstein-injective dimensions, in order to show that every object with finite Gorenstein-projective (respectively, Gorenstein-injective) dimension has a Gorenstein-projective cover (respectively, Gorenstein-injective envelope). In the next chapters, we will study cotorsion pairs associated to these homological dimensions. The corresponding model structures obtained from each homological dimension will be presented in Chapters 12 and 13. The study of the Gorenstein-flat dimension and the associated model structures will be left for Chapter 14.
