Remote Access Journal of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14E15, 14J30, 18E30, 19L47

Retrieve articles in all journals with MSC (2000): 14E15, 14J30, 18E30, 19L47

Additional 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

Alastair King
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Miles Reid
Affiliation: Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

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