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McKay correspondence for canonical orders


Author: Daniel Chan
Journal: Trans. Amer. Math. Soc. 362 (2010), 1765-1795
MSC (2000): Primary 14B05, 16G30, 16H05, 16S35
DOI: https://doi.org/10.1090/S0002-9947-09-05010-7
Published electronically: November 12, 2009
MathSciNet review: 2574877
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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.


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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.

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