Abstract
We define torsion pairs for quasi-abelian categories and give several characterisations. We show that many of the torsion theoretic concepts translate from abelian categories to quasi-abelian categories. As an application, we generalise the recently defined algebraic Harder-Narasimhan filtrations to quasi-abelian categories.
Highlights
Torsion classes were introduced for abelian categories by Dickson [11] to generalise the notion of torsion and torsionfree groups
In [6] and [8] it was observed, for abelian categories, that each stability function induces a chain of torsion classes; and in [32] the above is generalised to show that every chain of torsion classes satisfying mild finiteness conditions in an abelian category induces Harder-Narasimhan filtrations
Theorem C (Corollary 6.9) Every chain of torsion classes satisfying mild finiteness conditions in a quasi-abelian category induces a Harder-Narasimhan filtration of each object that is unique up to isomorphism
Summary
Torsion classes were introduced for abelian categories by Dickson [11] to generalise the notion of torsion and torsionfree groups. In [6] and [8] it was observed, for abelian categories, that each stability function induces a chain of torsion classes; and in [32] the above is generalised to show that every chain of torsion classes satisfying mild finiteness conditions in an abelian category induces Harder-Narasimhan filtrations. Theorem C (Corollary 6.9) Every chain of torsion classes satisfying mild finiteness conditions (see Definition 6.1) in a quasi-abelian category induces a Harder-Narasimhan filtration of each object that is unique up to isomorphism. In the third Section, we prove that the heart of twin torsion pairs is quasi-abelian This provides us with a way to generate examples of quasi-abelian categories that are not naturally arising as torsion(free) classes. We explore topological properties of the set of chains of torsion classes in a quasi-abelian category
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