On triangulated orbit categories

  • Bernhard Keller

    UFR de Mathématiques Institut de Mathématiques UMR 7586 du CNRS Case 7012 Université Paris 7 2, place Jussieu 75251 Paris Cedex 05 France
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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\cfalso 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.

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Bernhard Keller, On triangulated orbit categories. Doc. Math. 10 (2005), pp. 551–581

DOI 10.4171/DM/199