Abstract
An exceptional n-cycle in a Horn-finite triangulated category with Serre functor has been recently introduced by Broomhead, Pauksztello and Ploog. When n = 1, it is a spherical object. We explicitly determine all the exceptional cycles in the bounded derived category Db (kQ) of a finite quiver Q without oriented cycles. In particular, if Q is an Euclidean quiver, then the length type of exceptional cycles in Db (kQ) is exactly the tubular type of Q; if Q is a Dynkin quiver of type Em (m = 6, 7, 8), or Q is a wild quiver, then there are no exceptional cycles in Db (kQ); and if Q is a Dynkin quiver of type An or Dn, then the length of an exceptional cycle in Db (kQ) is either h or \(\frac{h}{2}\), where h is the Coxeter number of Q.
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Supported by the NSFC (Grant No. 11971304)
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Guo, P., Zhang, P. Exceptional Cycles in the Bounded Derived Categories of Quivers. Acta. Math. Sin.-English Ser. 36, 207–223 (2020). https://doi.org/10.1007/s10114-020-9094-x
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DOI: https://doi.org/10.1007/s10114-020-9094-x