McKay correspondence for canonical orders

Author:
Daniel Chan

Journal:
Trans. Amer. Math. Soc. **362** (2010), 1765-1795

MSC (2000):
Primary 14B05, 16G30, 16H05, 16S35

Published electronically:
November 12, 2009

MathSciNet review:
2574877

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Abstract | References | Similar Articles | Additional Information

Abstract: Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities , and their associated skew group rings . 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.

**1.**M. Artin,*Maximal orders of global dimension and Krull dimension two*, Invent. Math.**84**(1986), no. 1, 195–222. MR**830045**, 10.1007/BF01388739**2.**Michael Artin,*Two-dimensional orders of finite representation type*, Manuscripta Math.**58**(1987), no. 4, 445–471. MR**894864**, 10.1007/BF01277604**3.**M. Artin. J. de Jong, ``Stable Orders over Surfaces'', in preparation.**4.**M. Artin and J.-L. Verdier,*Reflexive modules over rational double points*, Math. Ann.**270**(1985), no. 1, 79–82. MR**769609**, 10.1007/BF01455531**5.**M. Artin and M. Van den Bergh,*Twisted homogeneous coordinate rings*, J. Algebra**133**(1990), no. 2, 249–271. MR**1067406**, 10.1016/0021-8693(90)90269-T**6.**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 (electronic). MR**1824990**, 10.1090/S0894-0347-01-00368-X**7.**Daniel Chan,*Noncommutative cyclic covers and maximal orders on surfaces*, Adv. Math.**198**(2005), no. 2, 654–683. MR**2183391**, 10.1016/j.aim.2005.06.012**8.**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**, 10.1112/plms/pdn025**9.**Daniel Chan and Colin Ingalls,*The minimal model program for orders over surfaces*, Invent. Math.**161**(2005), no. 2, 427–452. MR**2180454**, 10.1007/s00222-005-0438-z**10.**Daniel Chan and Rajesh S. Kulkarni,*del Pezzo orders on projective surfaces*, Adv. Math.**173**(2003), no. 1, 144–177. MR**1954458**, 10.1016/S0001-8708(02)00020-8**11.**Charles W. Curtis and Irving Reiner,*Methods of representation theory. Vol. I*, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR**632548****12.**Yujiro Kawamata,*Equivalences of derived categories of sheaves on smooth stacks*, Amer. J. Math.**126**(2004), no. 5, 1057–1083. MR**2089082****13.**M. Kapranov and E. Vasserot,*Kleinian singularities, derived categories and Hall algebras*, Math. Ann.**316**(2000), no. 3, 565–576. MR**1752785**, 10.1007/s002080050344**14.**L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen,*Graded orders*, Birkhäuser Boston, Inc., Boston, MA, 1988. MR**1003605****15.**Michel Van den Bergh,*Three-dimensional flops and noncommutative rings*, Duke Math. J.**122**(2004), no. 3, 423–455. MR**2057015**, 10.1215/S0012-7094-04-12231-6

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Additional Information

**Daniel Chan**

Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

Email:
danielc@unsw.edu.au

DOI:
https://doi.org/10.1090/S0002-9947-09-05010-7

Received by editor(s):
July 24, 2007

Published electronically:
November 12, 2009

Additional Notes:
This project was supported by ARC Discovery project grant DP0557228.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.