Torsion Pairs and Quasi-abelian Categories

Tattar, Aran

doi:10.1007/s10468-020-10004-y

Cited by 13 publications

(3 citation statements)

References 26 publications

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“…In a recent paper [34], Treffinger has introduced an axiomatic approach to Harder-Narasimhan filtrations for abelian categories, by showing that the existence of such a filtration for every object in an abelian category is equivalent to the existence of a chain of torsion classes in the category. Since this construction of Harder-Narasimhan filtrations does not depend on the existence of a stability condition, it allows the introduction of Harder-Narasimhan filtrations to nonabelian settings such as quasi-abelian categories [32]. In the second main result of this paper, we push this idea further by showing that chains of n-torsion classes induce Harder-Narasimhan filtrations in n-abelian categories.…”

Section: T : {N-torsion Classes In M } → {Torsion Classes In A}mentioning

confidence: 96%

On Higher Torsion Classes

et al. 2022

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Building on the embedding of an n-abelian category $\mathscr {M}$ into an abelian category $\mathcal {A}$ as an n-cluster-tilting subcategory of $\mathcal {A}$ , in this paper, we relate the n-torsion classes of $\mathscr {M}$ with the torsion classes of $\mathcal {A}$ . Indeed, we show that every n-torsion class in $\mathscr {M}$ is given by the intersection of a torsion class in $\mathcal {A}$ with $\mathscr {M}$ . Moreover, we show that every chain of n-torsion classes in the n-abelian category $\mathscr {M}$ induces a Harder–Narasimhan filtration for every object of $\mathscr {M}$ . We use the relation between $\mathscr {M}$ and $\mathcal {A}$ to show that every Harder–Narasimhan filtration induced by a chain of n-torsion classes in $\mathscr {M}$ can be induced by a chain of torsion classes in $\mathcal {A}$ . Furthermore, we show that n-torsion classes are preserved by Galois covering functors, thus we provide a way to systematically construct new (chains of) n-torsion classes.

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Section: T : {N-torsion Classes In M } → {Torsion Classes In A}mentioning

confidence: 96%

On Higher Torsion Classes

et al. 2022

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“…Given an E-stratifying system Φ in A, we will study the extriangulated subcategory F(Φ) of Φ-filtered objects of A. We can immediately prove that F(Φ) has a 'weak' Harder-Narasimhan filtration property [HN75,Rei03,Che10,Tre18,Tat21a]. This and similar phenomena will be explored more in the extriangulated setting in [TT22].…”

Section: Stratifying Systemsmentioning

confidence: 99%

Stratifying systems and Jordan-Hölder extriangulated categories

Brüstle¹,

Hassoun²,

Shah³

et al. 2022

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Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system Φ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory F(Φ) of objects admitting a composition series-like filtration with factors in Φ has the Jordan-Hölder property on these filtrations.This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series relative to the extriangulated structure in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise these categories in terms of the associated Grothendieck monoid and Grothendieck group.Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system Φ in an extriangulated category is part of a projective one (Φ, Q). We prove that F(Φ) is a Jordan-Hölder extriangulated category when (Φ, Q) satisfies some extra conditions.

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“…Proof. We do not write the proof of (a) since it is a straightforward generalisation of the proof of [49,Theorem 4.4]. Part (b) can be proved in a similar way.…”

Section: Corollary 47mentioning

confidence: 99%