Geodesic
On triangulated orbit categories
Documenta mathematica, Tome 10 (2005), pp. 551-581
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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. 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.
DOI : 10.4171/dm/199
Classification : 16G20
Mots-clés : derived category, triangulated category, orbit category, Calabi-Yau category 
@article{10_4171_dm_199,
     author = {Bernhard Keller},
     title = {On triangulated orbit categories},
     journal = {Documenta mathematica},
     pages = {551--581},
     year = {2005},
     volume = {10},
     doi = {10.4171/dm/199},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/199/}
}
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Bernhard Keller. On triangulated orbit categories. Documenta mathematica, Tome 10 (2005), pp. 551-581. doi: 10.4171/dm/199

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