The McKay correspondence as an equivalence of derived categories

Let G G be a finite group of automorphisms of a nonsingular three-dimensional complex variety M M, whose canonical bundle ω M M is locally trivial as a G G -sheaf. We prove that the Hilbert scheme Y = G Y = G - Hilb ⁡ M HilbM parametrising G G -clusters in M M is a crepant resolution of X = M / G X=M/G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on Y Y and coherent 𝐺-sheaves on M M. This identifies the K theory of Y Y with the equivariant K theory of M M, and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.