A new approach to projectivity in the categories of complexes

TL;DR

This paper introduces a novel perspective on projectivity in the category of complexes using subprojectivity, offering new tools to analyze exactness and null-homotopic morphisms with applications to ring characterization.

Contribution

It extends the concept of subprojectivity to complexes, providing new insights into null-homotopic morphisms and classical ring characterizations.

Findings

Subprojectivity offers a new way to measure exactness in complexes.

Characterization of classical rings using subprojectivity.

Examples illustrating the scope and limits of the approach.

Abstract

Recently, several authors have adopted new alternative approaches in the study of some classical notions of modules. Among them, we find the notion of subprojectivity which was introduced to measure in a way the degree of projectivity of modules. The study of subprojectivity has recently been extended to the context of abelian categories, which has brought to light some interesting new aspects. For instance, in the category of complexes, it gives a new way to measure, among other things, the exactness of complexes. In this paper, we prove that the subprojectivity notion provides a new sight of null-homotopic morphisms in the category of complexes. This will be proven through two main results. Moreover, various results which emphasize the importance of subprojectivity in the category of complexes are also given. Namely, we give some applications by characterizing some classical rings and establish various examples that allow us to reflect the scope and limits of our results.