Combinatorial Restrictions on the Tree Class of the Auslander-Reiten Quiver of a Triangulated Category

TL;DR

This paper investigates the structure of Auslander-Reiten quivers in triangulated categories, establishing conditions under which the entire quiver has a Dynkin or extended Dynkin tree class, extending classical theorems to a broader context.

Contribution

It generalizes Auslander’s theorem to triangulated categories and characterizes the possible tree classes of Auslander-Reiten quiver components under certain homomorphism conditions.

Findings

If a component has Dynkin tree class, the entire quiver has that class and has Auslander-Reiten triangles.

Components with extended Dynkin class imply other components are also extended Dynkin or small infinite trees.

The proofs utilize the theory of additive functions to analyze the quiver structure.

Abstract

We show that if a connected, Hom-finite, Krull-Schmidt triangulated category has an Auslander-Reiten quiver component with Dynkin tree class then the category has Auslander-Reiten triangles and that component is the entire quiver. This is an analogue for triangulated categories of a theorem of Auslander, and extends a previous result of Scherotzke. We also show that if there is a quiver component with extended Dynkin tree class, then other components must also have extended Dynkin class or one of a small set of infinite trees, provided there is a non-zero homomorphism between the components. The proofs use the theory of additive functions.