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Abelian subcategories of triangulated categories induced by simple minded systems
Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review
DOI
- https://doi.org/10.1007/s00209-021-02913-5
Forlagets udgivne version
If k is a field, A a finite dimensional k-algebra, then the simple A-modules form a simple minded collection in the derived category Db(modA). Their extension closure is modA; in particular, it is abelian. This situation is emulated by a general simple minded collection S in a suitable triangulated category C. In particular, the extension closure ⟨ S⟩ is abelian, and there is a tilting theory for such abelian subcategories of C. These statements follow from ⟨ S⟩ being the heart of a bounded t-structure. It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree. Relaxing this to vanishing in degrees { - w+ 1 , … , - 1 } where w is a positive integer leads to the rich, parallel notion of w-simple minded systems, which have recently been the subject of vigorous interest. If S is a w-simple minded system for some w⩾ 2 , then ⟨ S⟩ is typically not the heart of a t-structure. Nevertheless, using different methods, we will prove that ⟨ S⟩ is abelian and that there is a tilting theory for such abelian subcategories. Our theory is based on Quillen’s notion of exact categories, in particular a theorem by Dyer which provides exact subcategories of triangulated categories. The theory of simple minded systems can be viewed as “negative cluster tilting theory”. In particular, the result that ⟨ S⟩ is an abelian subcategory is a negative counterpart to the result from (higher) positive cluster tilting theory that if T is a cluster tilting subcategory, then (T∗ Σ T) / [T] is an abelian quotient category.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Mathematische Zeitschrift |
| Vol/bind | 301 |
| Nummer | 1 |
| Sider (fra-til) | 565-592 |
| Antal sider | 28 |
| ISSN | 0025-5874 |
| DOI | |
| Status | Udgivet - maj 2022 |
Bibliografisk note
Funding Information:
The main item of Sect. 3 is a theorem by Dyer on exact subcategories of triangulated categories, see [16 , p. 1] and Theorem 3.5. We are grateful to Professor Dyer for permitting the proof of Theorem 3.5 to appear in this paper. We thank the referee for several useful corrections and suggestions. We are grateful to Raquel Coelho Sim?es and David Pauksztello for patiently answering numerous questions and to Osamu Iyama and Haibo Jin for comments on an earlier version. This work was supported by a DNRF Chair from the Danish National Research Foundation (Grant DNRF156), by a Research Project 2 from the Independent Research Fund Denmark (Grant 1026-00050B), by the Aarhus University Research Foundation (Grant AUFF-F-2020-7-16), and by the Engineering and Physical Sciences Research Council (Grant EP/P016014/1). Thanks to Aleksandr and Sergei the Meerkats for years of inspiration.
Funding Information:
The main item of Sect. is a theorem by Dyer on exact subcategories of triangulated categories, see [, p. 1] and Theorem . We are grateful to Professor Dyer for permitting the proof of Theorem to appear in this paper. We thank the referee for several useful corrections and suggestions. We are grateful to Raquel Coelho Simões and David Pauksztello for patiently answering numerous questions and to Osamu Iyama and Haibo Jin for comments on an earlier version. This work was supported by a DNRF Chair from the Danish National Research Foundation (Grant DNRF156), by a Research Project 2 from the Independent Research Fund Denmark (Grant 1026-00050B), by the Aarhus University Research Foundation (Grant AUFF-F-2020-7-16), and by the Engineering and Physical Sciences Research Council (Grant EP/P016014/1). Thanks to Aleksandr and Sergei the Meerkats for years of inspiration.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
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