An algebraic approach to Harder-Narasimhan filtrations

TL;DR

This paper develops an algebraic framework linking chains of torsion classes in abelian categories to Harder-Narasimhan filtrations, characterising slicings and deriving wall-crossing formulas in Hall algebras.

Contribution

It generalizes the concept of Harder-Narasimhan filtrations via chains of torsion classes and connects these to slicings and wall-crossing phenomena.

Findings

Chains of torsion classes induce Harder-Narasimhan filtrations.

Characterization of slicings in terms of torsion class chains.

Wall-crossing formulas derived in the Hall algebra context.

Abstract

In this article we study chains of torsion classes in an abelian category $\mathcal{A}$. We prove that each chain of torsion classes induce a Harder-Narasimhan filtration for every nonzero object $M$ in $\mathcal{A}$, generalising a well-known property of stability conditions. We also characterise the slicings of $\mathcal{A}$ in terms of chain of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.