Weakly stable torsion classes

TL;DR

This paper explores weakly stable torsion classes, their relation to localization and completion, and their connection to dg rings and homological epimorphisms, advancing the understanding of torsion theoretic characterizations in algebra.

Contribution

It characterizes when torsion classes from denominator sets and ideals are weakly stable and links weakly stable torsion classes to dg rings and homological epimorphisms.

Findings

Characterization of weakly stable torsion classes from denominator sets and ideals

Association of weakly stable torsion classes with dg rings

Existence of homological epimorphisms in well-behaved cases

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.