On proper and exact relative homological dimensions

TL;DR

This paper explores different types of relative homological dimensions, examining their relationships, properties, and applications in derived functors, with a focus on exact sequences and self-orthogonal subcategories.

Contribution

It investigates the connection between proper and exact relative homological dimensions, establishing transfer results and characterizations of global dimensions and properties of subcategories.

Findings

Established relations between different relative homological dimensions.

Characterized properties of self-orthogonal subcategories.

Generalized balance results for relative derived functors.

Abstract

In Enochs' relative homological dimension theory occur the so called (co)resolvent and (co)proper dimensions which are defined using proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. We first study the relation between these two kinds of relative homological dimensions and establish some "transfer results" under adjoint pairs. Then, relative global dimensions are studied which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of the paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established.