Quantization of minimal resolutions of Kleinian singularities

In this paper we prove an analogue of a recent result of Gordon and Stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the nth symmetric power of C2 with the geometry of the Hilbert scheme of n points in C2 through the formalism of Z-algebras. Our work produces, for every regular noncommutative deformation Oλ of a Kleinian singularity X = C2/Γ , as defined by Crawley-Boevey and Holland, a filtered Z-algebra which is Morita equivalent to Oλ, such that the associated graded Z-algebra is Morita equivalent to the minimal resolution of X. The construction uses the description of the algebras Oλ as quantum Hamiltonian reductions, due to Holland, and a GIT construction of minimal resolutions of X, due to Cassens and Slodowy. © 2006 Elsevier Inc. All rights reserved.