The braid group action for exceptional curves

Abstract

We show that the operation of the braid group on the set of complete exceptional sequences in the category of coherent sheaves on an exceptional curve \( \mathbb{X} \) over a field k is transitive. As a consequence the list of endomorphism skew-fields of the indecomposable direct summands of a tilting complex is a derived invariant. Furthermore, we apply the result in order to establish a bijection (which is compatible with the K-theory) between the sets of translation classes of exceptional objects in the derived categories of two derived-canonical algebras with the same Cartan matrix, but which are defined over possibly distinct fields.