An algebraic approach to Harder-Narasimhan filtrations

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.

This is the final and published version of the present article. Minor typos corrected, some figures added

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• A survey on maximal green sequences
• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• On higher torsion classes
• On maximal green sequences in abelian length categories
• Stratifying systems and Jordan-Hölder extriangulated categories