Altmann, Klaus; Hille, Lutz
A Vanishing Result for the Universal Bundle on a Toric Quiver Variety
Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976)

MSC:

16G20 Representations of quivers and partially ordered sets

14M25 Toric varieties, Newton polyhedra

16D90 Module categories, See also {16Exx, 16Gxx, 16S90}; module theory in a category-theoretic context; Morita equivalence and duality

52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry), See also {06A08, 13F20, 13Hxx}

Abstract: Let $Q$ be a finite quiver without oriented cycles. Denote by $\UB \ra \cwtM$ the fine moduli space of stable thin sincere representations of $Q$ with respect to the canonical stability notion. We prove $\mExt^l_{\cwtM}(\UB,\UB) = 0$ for all $l>0$ and compute the endomorphism algebra of the universal bundle $\UB$. 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 $\ck Q$-modules and that of coherent sheaves on $\cwtM$ are related via the full and faithful functor $- \otimes^{\Ll}_{\ck Q}\UB$.
Keywords: quivers, reflexive polytopes, repsesentations, cohomology