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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The McKay correspondence as an equivalence of derived categories


Authors: Tom Bridgeland, Alastair King and Miles Reid
Journal: J. Amer. Math. Soc. 14 (2001), 535-554
MSC (2000): Primary 14E15, 14J30; Secondary 18E30, 19L47
Published electronically: March 22, 2001
MathSciNet review: 1824990
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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 $Y={\operatorname{\hbox{$G$ }-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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Additional Information

Tom Bridgeland
Affiliation: Department of Mathematics and Statistics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
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

DOI: http://dx.doi.org/10.1090/S0894-0347-01-00368-X
PII: S 0894-0347(01)00368-X
Keywords: Quotient singularities, McKay correspondence, derived categories
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”
Dedicated: To Andrei Tyurin on his 60th birthday
Article copyright: © Copyright 2001 American Mathematical Society