2020/10/22 by Peter Jørgensen, Jorgensen, Peter
Mathematics · #16G10 #16S90 #18E10 #18E40 #18G80 #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2010.11799
openalex publication_date 2020/10/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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( mod A ). Their extension closure is mod A; 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 \geqslant 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 * ΣT )/[ T ] is an abelian quotient category.