McKay correspondence for canonical orders

Chan, Daniel

doi:10.1090/S0002-9947-09-05010-7

McKay correspondence for canonical orders
HTML articles powered by AMS MathViewer

by Daniel Chan PDF

Trans. Amer. Math. Soc. 362 (2010), 1765-1795 Request permission

Abstract:

Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities $k[[s,t]]^G$, $G < SL_2$ and their associated skew group rings $k[[s,t]]*G$. In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit the numerical version of the McKay correspondence for canonical orders noted in Chan, Hacking, and Ingalls, which relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order.

References

M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1986), no. 1, 195–222. MR 830045, DOI 10.1007/BF01388739
Michael Artin, Two-dimensional orders of finite representation type, Manuscripta Math. 58 (1987), no. 4, 445–471. MR 894864, DOI 10.1007/BF01277604
M. Artin. J. de Jong, “Stable Orders over Surfaces”, in preparation.
M. Artin and J.-L. Verdier, Reflexive modules over rational double points, Math. Ann. 270 (1985), no. 1, 79–82. MR 769609, DOI 10.1007/BF01455531
M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249–271. MR 1067406, DOI 10.1016/0021-8693(90)90269-T
Tom Bridgeland, Alastair King, and Miles Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554. MR 1824990, DOI 10.1090/S0894-0347-01-00368-X
Daniel Chan, Noncommutative cyclic covers and maximal orders on surfaces, Adv. Math. 198 (2005), no. 2, 654–683. MR 2183391, DOI 10.1016/j.aim.2005.06.012
Daniel Chan, Paul Hacking, and Colin Ingalls, Canonical singularities of orders over surfaces, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 83–115. MR 2472162, DOI 10.1112/plms/pdn025
Daniel Chan and Colin Ingalls, The minimal model program for orders over surfaces, Invent. Math. 161 (2005), no. 2, 427–452. MR 2180454, DOI 10.1007/s00222-005-0438-z
Daniel Chan and Rajesh S. Kulkarni, del Pezzo orders on projective surfaces, Adv. Math. 173 (2003), no. 1, 144–177. MR 1954458, DOI 10.1016/S0001-8708(02)00020-8
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Yujiro Kawamata, Equivalences of derived categories of sheaves on smooth stacks, Amer. J. Math. 126 (2004), no. 5, 1057–1083. MR 2089082, DOI 10.1353/ajm.2004.0036
M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), no. 3, 565–576. MR 1752785, DOI 10.1007/s002080050344
L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded orders, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1003605, DOI 10.1007/978-1-4612-3944-4
Michel Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. MR 2057015, DOI 10.1215/S0012-7094-04-12231-6