Exceptional sequences determined by their Cartan matrix

Michael A. Brehm, Amelia K. Pinto, Keith A. Daniels, Jonathan P. Schneck, Raymond M. Welsh, Liisa K. Selin

Research output: Contribution to journalArticlepeer-review

Abstract

We investigate complete exceptional sequences E = (E1,…, En) in the derived category Db ∧ of finite-dimensional modules over a canonical algebra, equivalently in the derived category Db (X double struck) 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 Db ∧. 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 languageEnglish (US)
Pages (from-to)201-209
Number of pages9
JournalAlgebras and Representation Theory
Volume5
Issue number2
DOIs
StatePublished - May 2002

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Fingerprint Dive into the research topics of 'Exceptional sequences determined by their Cartan matrix'. Together they form a unique fingerprint.

Cite this