Hilbert schemes and noncommutative deformations of type A Kleinian singularities

Abstract

Let H k be a symplectic reflection algebra corresponding to a cyclic subgroup Γ ⊆ SL 2 C of order n and U k = e H k e the spherical subalgebra of H k . We show that for suitable k there is a filtered Z n − 1 -algebra R such that (1) there is an equivalence of categories R - qgr ≃ U k - mod ; (2) there is an equivalence of categories gr R - qgr ≃ Coh ( Hilb Γ C 2 ) . Here Coh ( Hilb Γ C 2 ) is the category of coherent sheaves on the Γ-Hilbert scheme and, for a graded algebra R , we write R - qgr for the quotient category of finitely generated graded R -modules modulo torsion.