Abelian subcategories of triangulated categories induced by simple minded systems

Abstract

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 \(\mathscr {D}^{ {\text {b}}}( {\text {mod}}\,A )\). Their extension closure is \({\text {mod}}\,A\); in particular, it is abelian. This situation is emulated by a general simple minded collection \(\mathscr {S}\) in a suitable triangulated category \(\mathscr {C}\). In particular, the extension closure \(\langle \mathscr {S}\rangle \) is abelian, and there is a tilting theory for such abelian subcategories of \(\mathscr {C}\). These statements follow from \(\langle \mathscr {S}\rangle \) 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 \(\{ -w+1, \ldots , -1 \}\) 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 \(\mathscr {S}\) is a w-simple minded system for some \(w \geqslant 2\), then \(\langle \mathscr {S}\rangle \) is typically not the heart of a t-structure. Nevertheless, using different methods, we will prove that \(\langle \mathscr {S}\rangle \) 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 \(\langle \mathscr {S}\rangle \) is an abelian subcategory is a negative counterpart to the result from (higher) positive cluster tilting theory that if \(\mathscr {T}\) is a cluster tilting subcategory, then \(( \mathscr {T}* \Sigma \mathscr {T})/[ \mathscr {T}]\) is an abelian quotient category.