[Paper Review] On triangulated orbit categories

Bernhard Keller|ArXiv.org|Mar 13, 2005

Algebraic structures and combinatorial models28 references494 citations

TL;DR

This paper establishes that the orbit category of a bounded derived category of a hereditary category under a well-behaved autoequivalence admits a canonical triangulated structure, resolving a question by Buan, Marsh, and Reiten on cluster categories and extending results on Calabi-Yau categories. The key contribution is a general criterion ensuring triangulated orbit categories exist, with applications to cluster algebras and preprojective algebras of generalized Dynkin type.

ABSTRACT

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.

Motivation & Objective

• To resolve the open question of whether orbit categories of triangulated categories inherit a triangulated structure under certain autoequivalences.
• To provide a canonical triangulated structure on orbit categories of bounded derived categories of hereditary categories.
• To establish that cluster categories and preprojective algebras of generalized Dynkin type admit triangulated structures via this construction.
• To demonstrate that such orbit categories naturally yield examples of Calabi-Yau categories.
• To generalize the construction beyond module categories to arbitrary hereditary abelian categories with the Krull-Schmidt property.

Proposed method

• Constructs a 'triangulated hull' of the orbit category using dg categories and the formalism of derived categories.
• Uses the natural t-structure on the derived category of a hereditary category to prove the main theorem.
• Applies a second, 'Koszul-dual' construction based on the exterior algebra on the dual of a dg bimodule over a dg algebra.
• Employs dg lifts of standard functors to define the orbit category in the dg setting and prove equivalence to the pretriangulated hull.
• Characterizes the construction via universal properties in the 2-category of enhanced triangulated categories.
• Applies the results to specific cases, such as the derived category of a hereditary algebra and preprojective algebras of generalized Dynkin type.

Experimental results

Research questions

• RQ1 Under what conditions does the orbit category of a triangulated category under an autoequivalence inherit a triangulated structure?
• RQ2 Can the cluster category construction be rigorously shown to be triangulated via this framework?
• RQ3 How do triangulated orbit categories relate to Calabi-Yau categories, and can they be used to construct new examples?
• RQ4 What is the role of the dg category formalism in constructing the triangulated hull of an orbit category?
• RQ5 Can the construction be generalized beyond module categories to arbitrary hereditary abelian categories?

Key findings

• The orbit category of the bounded derived category of a hereditary category under a well-behaved autoequivalence admits a canonical triangulated structure.
• The construction proves that cluster categories, as defined by Buan, Marsh, and Reiten, are triangulated.
• The category of projective modules over a preprojective algebra of generalized Dynkin type is triangulated, confirming earlier results from Auslander and Reiten.
• The orbit category construction yields numerous examples of triangulated categories with the Calabi-Yau property.
• The triangulated hull of the orbit category is equivalent to the pretriangulated hull of the corresponding dg orbit category under suitable conditions.
• The main result holds for derived categories of hereditary abelian categories with the Krull-Schmidt property and finite-dimensional morphism and extension spaces.