Bridgeland, Tom; King, Alastair; Reid, Miles The McKay correspondence as an equivalence of derived categories. (English) Zbl 0966.14028 J. Am. Math. Soc. 14, No. 3, 535-554 (2001). Summary: Let \(G\) be a finite group of automorphisms of a non-singular three-dimensional complex variety \(M\), whose canonical bundle \(\omega_M\) is locally trivial as a \(G\)-sheaf. We prove that the Hilbert scheme \(Y=G \text{-Hilb }{M}\) 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. Cited in 37 ReviewsCited in 226 Documents MSC: 14J50 Automorphisms of surfaces and higher-dimensional varieties 18E30 Derived categories, triangulated categories (MSC2010) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 19L47 Equivariant \(K\)-theory 14L30 Group actions on varieties or schemes (quotients) 14J30 \(3\)-folds Keywords:quotient singularities; McKay correspondence; derived categories; group of automorphisms; three-dimensional complex variety; Hilbert scheme; crepant resolution; Fourier-Mukai transform; equivariant K-theory × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151 – 166. · Zbl 0108.36402 · doi:10.1016/0040-9383(65)90023-6 [2] Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671 – 677. · Zbl 0708.19004 · doi:10.1016/0393-0440(89)90032-6 [3] W. Barth, C. 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