Homological Dimensions Relative to Preresolving Subcategories

TL;DR

This paper introduces and studies relative homological dimensions in abelian categories, unifying known concepts and applying them to module categories, while proposing open questions related to Nakayama conjectures.

Contribution

It defines relative preresolving and precoresolving subcategories, establishing properties of their homological dimensions and applying these to module categories.

Findings

Unified properties of homological dimensions across subcategories

Application to module categories with new insights

Open questions related to Nakayama conjectures

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 generalized Nakayama conjecture and the strong Nakayama conjecture.