## Abstract

We investigate complete exceptional sequences E = (E_{1},…, E_{n}) in the derived category D^{b} ∧ of finite-dimensional modules over a canonical algebra, equivalently in the derived category D^{b} (X double struck) of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E) = (〈[E_{i}], [E_{j}]〉). 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 (E_{1},…, E_{n})→ (E_{1} [i_{1}],…, E_{n} [i_{n}]) of D^{b} ∧. 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.

Original language | English |
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Pages (from-to) | 201-209 |

Number of pages | 9 |

Journal | Algebras and Representation Theory |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - May 2002 |

## Keywords

- Canonical algebra
- Derived equivalence
- Exceptional sequence
- Tilting complex
- Weighted projective line

## ASJC Scopus subject areas

- Mathematics (all)