( X , Y )-Gorenstein Categories, Associated (Global) Homological Dimensions and Applications to Relative Foxby Classes

Recently, Gorenstein dimensions relative to a semidualizing module have been the subject of numerous studies with interesting extensions of the classical homological dimensions. Although all these studies share the same direction, a common basis, and similar final goals, there is no common framework encompassing them as parts of a whole, progressing, on different fronts, towards the same end. We provide this general and global framework in the context of abelian categories, standardizing terminology and notation: we establish a general context by defining Gorenstein categories relative to two classes of objects ( ( X , Y ) -Gorenstein categories, denoted G ( X , Y ) ), and carry out a study of the homological dimensions associated with them. We prove, under some mild standard conditions, the corresponding version of the Comparison Lemma that ensures the consistency of a homological-dimension theory. We show that Ext functors can be used as tools to compute these G ( X , Y ) -dimensions, and we compare the dimensions obtained using the classes G ( X ) with those computed using G ( X , Y ) . We also initiate a research of the global dimensions obtained with these classes G ( X , Y ) and find conditions for them to be finite. Finally, we show that these classes of Gorenstein objects are closely and interestingly related to the Foxby classes induced by a pair of functors. Namely, we prove that the Auslander and Bass classes are indeed G ( X , Y ) categories for some specific classes X and Y .