The McKay correspondence as an equivalence of derived categories
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- by Tom Bridgeland, Alastair King and Miles Reid
- J. Amer. Math. Soc. 14 (2001), 535-554
- DOI: https://doi.org/10.1090/S0894-0347-01-00368-X
- Published electronically: March 22, 2001
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Abstract:
Let $G$ be a finite group of automorphisms of a nonsingular three-dimensional complex variety $M$, whose canonical bundle $\omega _M$ is locally trivial as a $G$-sheaf. We prove that the Hilbert scheme $Y = G$-$\operatorname {Hilb}M$ 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 correspondence. Some higher-dimensional extensions are possible.
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Bibliographic Information
- Tom Bridgeland
- Affiliation: Department of Mathematics and Statistics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
- MR Author ID: 635821
- ORCID: 0000-0001-5120-006X
- Email: tab@maths.ed.ac.uk
- Alastair King
- Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
- Email: a.d.king@maths.bath.ac.uk
- Miles Reid
- Affiliation: Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
- Email: miles@maths.warwick.ac.uk
- Received by editor(s): May 1, 2000
- Received by editor(s) in revised form: November 1, 2000
- Published electronically: March 22, 2001
- Additional Notes: Earlier versions of this paper carried the additional title “Mukai implies McKay”
- © Copyright 2001 American Mathematical Society
- Journal: J. Amer. Math. Soc. 14 (2001), 535-554
- MSC (2000): Primary 14E15, 14J30; Secondary 18E30, 19L47
- DOI: https://doi.org/10.1090/S0894-0347-01-00368-X
- MathSciNet review: 1824990
Dedicated: To Andrei Tyurin on his 60th birthday