City Research Online

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 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 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 Φ in an extriangulated category is part of a minimal projective one (Φ,Q). We prove that F(Φ) is a length, Jordan-Hölder extriangulated category when (Φ,Q) satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto–Saito in the negative.

Publication Type:	Article
Additional Information:	This article has been published in its final form in Glasgow Mathematical Journal and it's available online at: https://doi.org/10.1017/S0017089525100621
Publisher Keywords:	Extriangulated category, filtration, Grothendieck monoid, length, stratifying system, projective stratifying system, Jordan-Hölder
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology School of Science & Technology > Department of Mathematics
SWORD Depositor:	Symplectic Administrator

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