On Triangulated Orbit Categories

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 Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study \cite{BuanMarshReinekeReitenTodorov04} with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (\cf also \cite{CalderoChapotonSchiffler04}) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many 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 \cite{HappelPreiserRingel80}, whose triangulated structure goes back to Auslander-Reiten's work \cite{AuslanderReiten87}, \cite{Reiten87}, \cite{AuslanderReiten96}.

2000 Mathematics Subject Classification: Primary 18E30; Secondary 16G20.

Keywords and Phrases: Derived category, Triangulated category, Orbit category, Calabi-Yau category

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