Abelian subcategories of triangulated categories induced by simple minded systems
Peter Jorgensen

TL;DR
This paper explores abelian subcategories induced by simple minded systems in triangulated categories, extending classical results to a broader setting involving $w$-simple minded systems and their tilting theory.
Contribution
It generalizes the concept of simple minded collections to $w$-simple minded systems and proves these induce abelian subcategories with tilting theory, even when not hearts of $t$-structures.
Findings
Extension closure of simple minded systems is abelian.
$w$-simple minded systems induce abelian subcategories.
Develops tilting theory for these abelian subcategories.
Abstract
If is a field, a finite dimensional -algebra, then the simple -modules form a simple minded collection in the derived category . Their extension closure is ; in particular, it is abelian. This situation is emulated by a general simple minded collection in a suitable triangulated category . In particular, the extension closure is abelian, and there is a tilting theory for such abelian subcategories of . These statements follow from being the heart of a bounded -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 where is a positive integer leads to the rich,…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Rings, Modules, and Algebras · Homotopy and Cohomology in Algebraic Topology
