Stratifying systems and Jordan-Hölder extriangulated categories
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866918459742355456 |
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| author | Brüstle, Thomas Hassoun, Souheila Shah, Amit Tattar, Aran |
| author_facet | Brüstle, Thomas Hassoun, Souheila Shah, Amit Tattar, Aran |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_07808 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stratifying systems and Jordan-Hölder extriangulated categories Brüstle, Thomas Hassoun, Souheila Shah, Amit Tattar, Aran Representation Theory Category Theory 18E05 (Primary) 18E10, 18G25, 18G80 (Secondary) 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. |
| title | Stratifying systems and Jordan-Hölder extriangulated categories |
| topic | Representation Theory Category Theory 18E05 (Primary) 18E10, 18G25, 18G80 (Secondary) |
| url | https://arxiv.org/abs/2208.07808 |