Stratifying systems and Jordan-Hölder extriangulated categories
Thomas Brüstle, Souheila Hassoun, Amit Shah, Aran Tattar
TL;DR
The paper extends stratifying systems to artin extriangulated categories by introducing $(\mathbb{E},\mathfrak{s})$-composition series and a Jordan-Hölder framework, enabling a length notion for $\mathcal{F}(\Phi)$. It shows every $\mathbb{E}$-stratifying system $\Phi$ sits inside a minimal projective $\mathbb{E}$-stratifying system $(\Phi,Q)$ and, under a left-exactness condition, that $\mathcal{F}(\Phi)$ is length and Jordan-Hölder. A central contribution is the Grothendieck-monoid/group characterization: length and Jordan-Hölder are equivalent to the monoid (and the group) being free, with basis given by the $(\mathbb{E},\mathfrak{s})$-simple objects; this yields a practical invariant to detect JH-filtrations. The authors provide numerous examples, including cases where a projective $\mathbb{E}$-stratifying system in an extriangulated category yields a Jordan-Hölder subcategory that is not exact, and they give counterexamples to conjectures about reduced Grothendieck monoids implying exactness, along with negative answers to questions of Enomoto–Saito.
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 $\mathcal{F}(Φ)$ 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 $\mathcal{F}(Φ)$ 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.