Keller, Bernhard

On triangulated orbit categories

Doc. Math., J. DMV 10, 551-581 (2005)


Summary: 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 citeBuanMarshReinekeReitenTodorov04 with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras ($\cf $also citeCalderoChapotonSchiffler04) 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 citeHappelPreiserRingel80, whose triangulated structure goes back to Auslander-Reiten's work citeAuslanderReiten87, citeReiten87, citeAuslanderReiten96.

Mathematics Subject Classification

18E30, 16G20


derived category, triangulated category, orbit category, Calabi-Yau category