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

Enrique Duarte, Juan Ramón García Rozas, Hanane Ouberka and Luis Oyonarte ()
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Enrique Duarte: Department of Mathematics, University of Almeria, 04071 Almería, Spain
Juan Ramón García Rozas: Department of Mathematics, University of Almeria, 04071 Almería, Spain
Hanane Ouberka: Department of Mathematics, University of Almeria, 04071 Almería, Spain
Luis Oyonarte: Department of Mathematics, University of Almeria, 04071 Almería, Spain

Mathematics, 2024, vol. 12, issue 8, 1-30

Abstract: 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 .

Keywords: Gorenstein categories; Gorenstein injective module; Gorenstein projective module; Foxby classes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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