Finiteness conditions and cotorsion pairs

TL;DR

This paper explores the relationship between $n$-coherent rings and finitely $n$-presented modules, establishing equivalences and constructing cotorsion pairs to deepen understanding of their homological properties.

Contribution

It introduces new characterizations of $n$-coherent rings via the thickness of finitely $n$-presented modules and constructs cotorsion pairs from their orthogonal complements.

Findings

$n$-coherence is equivalent to the thickness of finitely $n$-presented modules

Cotorsion pairs can be constructed from orthogonal complements related to ${ m Ext}^1_R$ and ${ m Tor}_1^R$

Provides new characterizations of $n$-coherent rings

Abstract

We study the interplay between the notions of $n$-coherent rings and finitely $n$-presented modules, and also study the relative homological algebra associated to them. We show that the $n$-coherency of a ring is equivalent to the thickness of the class of finitely $n$-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to ${\rm Ext}^1_R(F,-)$ and ${\rm Tor}_1^R(F,-)$. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of $n$-coherent rings.