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Strong exceptional sequences provided by quivers Klaus Altmann Lutz Hille

Summary: Strong exceptional sequences provided by quivers
Klaus Altmann Lutz Hille
Let Q be a finite quiver without oriented cycles. Denote by U ! M(Q) the fine moduli
space of stable thin sincere representations of Q with respect to the canonical stability notion.
We prove Ext l
M(Q) (U ; U) = 0 for all l ? 0 and compute the endomorphism algebra of the
universal bundle U . Moreover, we obtain a necessary and sufficient condition for when this
algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived
categories of finitely generated right kQ­modules and that of coherent sheaves on M(Q) are
related via the full and faithful functor
\Gamma\Omega IL
kQ U .
1 Introduction
(1.1) Let Q be a quiver (i.e. an oriented graph) without oriented cycles; denote by Q 0 the
vertices and by Q 1 the arrows of Q. For a fixed dimension vector d, that is a map d : Q 0 ! ZZ –0 ,
we denote by IH(d) := f` : Q 0 ! IR j P
q2Q0 ` q d q = 0g the vector space of the so­called weights
with respect to d. We fix an algebraically closed field k. To each ` 2 IH(d) there exists the
moduli space M ` (Q; d) of `­semistable k­representations of Q with dimension vector d (cf. [Ki]).


Source: Altmann, Klaus - Fachbereich Mathematik und Informatik & Institut für Mathematik, Freie Universität Berlin


Collections: Mathematics