Mukai implies McKay: the McKay correspondence as an equivalence of derived categories

@article{Bridgeland1999MukaiIM,
  title={Mukai implies McKay: the McKay correspondence as an equivalence of derived categories},
  author={Tom Bridgeland and Alastair King and Miles Anthony Reid},
  journal={arXiv: Algebraic Geometry},
  year={1999}
}
Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay… 

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