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