Geodesic
The McKay correspondence as an equivalence of derived categories
Bridgeland, Tom1 ; King, Alastair2 ; Reid, Miles3
1 Department of Mathematics and Statistics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
3 Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Journal of the American Mathematical Society, Tome 14 (2001) no. 3, pp. 535-554

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-01-00368-X

Bridgeland, Tom 1 ; King, Alastair 2 ; Reid, Miles 3

1 Department of Mathematics and Statistics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
3 Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom 
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Bridgeland, Tom; King, Alastair; Reid, Miles. The McKay correspondence as an equivalence of derived categories. Journal of the American Mathematical Society, Tome 14 (2001) no. 3, pp. 535-554. doi: 10.1090/S0894-0347-01-00368-X

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