On classes defining a homological dimension

TL;DR

This paper investigates classes in abelian categories that define a homological dimension, meaning the length of their resolutions is uniquely determined for all objects, which is fundamental for understanding homological properties.

Contribution

It characterizes and studies classes that define a homological dimension, providing insights into their properties and significance in abelian categories.

Findings

Identification of conditions for classes to define a homological dimension

Analysis of properties ensuring resolution length uniqueness

Contribution to the theory of homological dimensions in abelian categories

Abstract

A class $\mathcal F$ of objects of an abelian category $\mathcal A$ is said to define a \emph{homological dimension} if for any object in $\mathcal A$ the length of any $\mathcal F$-resolution is uniquely determined. In the present paper we investigate classes satisfying this property.