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.