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Stratifying systems and Jordan-Hölder extriangulated categories

Part of: Abelian categories Homological algebra

Published online by Cambridge University Press:  27 August 2025

Thomas Brüstle ,
Souheila Hassoun ,
Amit Shah Open the ORCID record for Amit Shah [Opens in a new window]  and
Aran Tattar
Thomas Brüstle
Affiliation:
Départment de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada
Souheila Hassoun
Affiliation:
Départment de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada
Amit Shah*
Affiliation:
Department of Mathematics, Aarhus University, Aarhus, Denmark
Aran Tattar
Affiliation:
Mathematical Institut, University of Cologne, Köln, Germany
*
Corresponding author: Amit Shah; Email: a.shah1728@gmail.com
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Abstract

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 $\Phi$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(\Phi )$ of objects admitting a composition series-like filtration with factors in $\Phi$ 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 for an extriangulated category, in order to define a Jordan-Hölder extriangulated category. Moreover, we characterise Jordan-Hölder, length, weakly idempotent complete extriangulated 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 $\Phi$ in an extriangulated category is part of a minimal projective one $(\Phi ,Q)$. We prove that $\mathcal{F}(\Phi )$ is a length, Jordan-Hölder extriangulated category when $(\Phi ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto–Saito in the negative.

Keywords

Extriangulated categoryfiltrationGrothendieck monoidlengthstratifying systemprojective stratifying systemJordan-Hölder

MSC classification

Primary: 18E05: Preadditive, additive categories
Secondary: 18E10: Exact categories, abelian categories 18G25: Relative homological algebra, projective classes

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

Dedicated to the late Brian Parshall in honour of his contributions to representation theory.

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