Finiteness conditions and cotorsion pairs

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 ${\mathrm{Ext}}_R^1(F,-)$ and ${\mathrm{Tor}}_1^R(F,-)$. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of $n$-coherent rings.

18 pages

References in corpus (1)

• The stable module category of a general ring

Cited by in corpus (13)

• Relative FP-injective and FP-flat complexes and their model structures
• Torsion and torsion-free classes from objects of finite type in Grothendieck categories
• Complete cohomology for extriangulated categories
• Cut cotorsion pairs
• Locally type ${\text{FP}}_n$ and $n$-coherent categories
• Model structures and relative Gorenstein flat modules and chain complexes
• ${\mathrm{FP}}_{\mathrm{n}}$-injective and ${\mathrm{FP}}_{\mathrm{n}}$-flat covers and preenvelopes, and Gorenstein AC-flat covers
• Relative coherent modules
• Intersection of complete cotorsion pairs
• When the kernel of a complete hereditary cotorsion pair is the additive closure of a tilting module
• Gorenstein flat modules with respect to duality pairs
• Completeness of the induced cotorsion pairs in functor categories
• K-theory of n-coherent rings