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Abelian subcategories of triangulated categories induced by simple minded systems

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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.

Original languageEnglish
JournalMathematische Zeitschrift
Pages (from-to)565-592
Number of pages28
Publication statusPublished - May 2022

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

    Research areas

  • Cluster category, Derived category, Heart, Orbit category, Orthogonal collection, Simple minded collection, t-structure, tilting

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