Torsion and torsion-free classes from objects of finite type in Grothendieck categories

TL;DR

This paper characterizes when classes of certain objects in Grothendieck categories form torsion or torsion-free classes, introduces $n$-hereditary categories, and explores their applications across various algebraic contexts.

Contribution

It establishes necessary and sufficient conditions for $ ext{FP}_n$-injective objects to form torsion classes and introduces the concept of $n$-hereditary categories with broad applications.

Findings

Conditions for $ ext{FP}_n$-injective objects to be torsion classes

Characterization of $n$-hereditary categories

Applications to module categories, chain complexes, and functor categories

Abstract

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, we propose a notion of $n$-hereditary categories. We also define and study the class of $\text{FP}_n$-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. In the end, we present several applications and examples of $n$-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are $n$-hereditary in terms of the domain additive category.