Weakly Stable Torsion Classes

Abstract

Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class $\operatorname{\mathsf{T}}$ can be associated with a dg ring $A_{\operatorname{\mathsf{T}}}$; in well behaved situations there is a homological epimorphism $A\to A_{\operatorname{\mathsf{T}}}$. We end by studying torsion and completion with respect to a single regular and normal element.