by Lenzing, Helmut; Meltzer, Hagen

We investigate complete exceptional sequences E=(E
₁,¨,E

_n
) in the derived category D
^b
Λ of finite-dimensional modules over a canonical algebra, equivalently in the derived category D
^b

X 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
Λ. 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.

DOI: 10.1023/A:1015646412663
Print publication date: 5/1/2002
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