Shapes of Auslander-Reiten Triangles

TL;DR

This paper classifies Auslander-Reiten triangles based on morphism properties, enabling computation of mapping cones and providing a method to construct the derived category of tilted algebras, especially those of finite representation type.

Contribution

It introduces a classification of Auslander-Reiten triangles and offers a new technique for constructing derived categories of tilted algebras.

Findings

Classification of Auslander-Reiten triangles based on morphism properties

Method to compute the mapping cone of irreducible morphisms

Technique for constructing the derived category of tilted algebras

Abstract

Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for constructing the connecting component of the derived category of any tilted algebra. In particular we obtain a technique for constructing the derived category of any tilted algebra of finite representation type.