tCG Torsion Pairs

TL;DR

This paper characterizes when the t-structure associated with a torsion pair is compactly generated, introduces the concept of tCG torsion pairs, and describes their properties over specific rings.

Contribution

It introduces the concept of tCG torsion pairs, characterizes them in module categories, and describes their structure over Noetherian and von Neumann regular rings.

Findings

tCG torsion pairs are characterized by finitely presented modules.

Every tCG torsion pair is of finite type, but not vice versa.

Explicit descriptions of tCG torsion pairs over certain rings are provided.

Abstract

We investigate conditions for when the $t$-structure of Happel-Reiten-Smal{\o} associated to a torsion pair is a compactly generated $t$-structure. The concept of a $t$CG torsion pair is introduced and for any ring $R$, we prove that $\mathbf{t}=(\mathcal{T},\mathcal{F})$ is a $t$CG torsion pair in $R\text{-Mod}$ if, and only if, there exists, $\{T_{\lambda}\}$ a set of finitely presented $R$-modules in $\mathcal{T}$, such that $\mathcal{F}=\bigcap \text{Ker}({\text{Hom}}_{R}(T_{\lambda},?))$. We also show that every $t$CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the $t$CG torsion pairs over Noetherian rings and von Neumman regular rings.