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
kQ U .
(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]).