Quantization of minimal resolutions of Kleinian singularities

Abstract

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 n-th symmetric power of C^2 with the geometry of the Hilbert scheme of n points in C^2 through the formalism of Z-algebras. Our work produces, for every regular noncommutative deformation O^\lambda of a Kleinian singularity X=C^2/\Gamma, as defined by Crawley-Boevey and Holland, a filtered Z-algebra which is Morita equivalent to O^\lambda, such that the associated graded Z-algebra is Morita equivalent to the minimal resolution of X. The construction uses the description of the algebras O^\lambda as quantum Hamiltonian reductions, due to Holland, and a GIT construction of minimal resolutions of X, due to Cassens and Slodowy.