An algebraic approach to Harder-Narasimhan filtrations

In this article we study chains of torsion classes in an abelian category $A$. We prove that chains of torsion classes satisfying mild technical conditions induce a Harder-Narasimhan filtration for every non-zero object M in $A$, generalising a well-known property of stability conditions. We also characterise the slicings of $A$ in terms of chains 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.