Exceptional Cycles in the Bounded Derived Categories of Quivers

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.