Abelian subcategories of triangulated categories induced by simple minded systems
Peter Jorgensen
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
$\mathscr{D}^{\rm b}(\operatorname{mod}\,A)$. Their extension closure is
$\operatorname{mod}\,A$; in particular, it is abelian. This situation is
emulated by a general simple minded collection $\mathscr{S}$ 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 $\{ -w+1, \ldots, -1 \}$ only, 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.