Abelian subcategories of triangulated categories induced by simple minded systems

TY - JOUR

T1 - Abelian subcategories of triangulated categories induced by simple minded systems

AU - Jørgensen, Peter

N1 - Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Cluster category

KW - Derived category

KW - Heart

KW - Orbit category

KW - Orthogonal collection

KW - Simple minded collection

KW - t-structure

KW - tilting

UR - http://www.scopus.com/inward/record.url?scp=85123265285&partnerID=8YFLogxK

U2 - 10.1007/s00209-021-02913-5

DO - 10.1007/s00209-021-02913-5

M3 - Journal article

AN - SCOPUS:85123265285

SN - 0025-5874

VL - 301

SP - 565

EP - 592

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1

ER -