Homological dimensions relative to preresolving subcategories

Abstract

We introduce relative preresolving subcategories and precoresolving subcategories of an abelian category and define homological dimensions and codimensions relative to these subcategories, respectively. We study the properties of these homological dimensions and codimensions and unify some important properties possessed by some known homological dimensions. Then we apply the obtained properties to special subcategories and in particular to module categories. Finally we propose some open questions and conjectures which are closely related to the Auslander–Reiten conjecture and the strong Nakayama conjecture.