tCG Torsion Pairs

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.