Exceptional Sequences Determined by their Cartan Matrix

Abstract

We investigate complete exceptional sequences E=(E1,¨,En) in the derived category DbΛ of finite-dimensional modules over a canonical algebra, equivalently in the derived category DbX of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=(〈 [Ei],[Ej]〉). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E1,¨,En)↦(E1[i1],¨,En[in]) of DbΛ. Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.