2006/04/02 by Farkhod Eshmatov, Eshmatov, Farkhod
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/0604011
27 pages
arxiv created 2006/04/02 · arxiv updated 2009/12/01
Associated to each finite group Γ in SL2(C) there is a family of noncommutative algebras which deforms the coordinate ring of the Kleinian singularity corresponding to that group. These algebras were defined by W. Crawley-Boevey and M. Holland, who also suggested a conjectural correspondence between the set of isomorphism classes of rank one projective modules over these algebras and associated Nakajima quiver varieties. In \citeBGK, V.Baranovski, V.Ginzburg and A.Kuznetsov proved the Crawley-Boevey-Holland conjecture using the methods of noncommutative projective geometry. In this paper we will state a refined (G-equivariant) version of this conjecture and, in the case of cyclic groups, give a new construction of this correspondence based on the notion of DG-model of a rank one projective module. This construction leads to a completely explicit description of ideals of the Crawley-Boevey-Holland algebras.