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Proceedings of the London Mathematical Society Advance Access published online on June 3, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn025
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© 2008 London Mathematical Society

Canonical singularities of orders over surfaces

Daniel Chan

School of Mathematics
University of New South Wales
Sydney
2052 NSW
Australia

Paul Hacking

Department of Mathematics
University of Washington
Box 354350 Seattle
WA 98195
USA
hacking@math.washington.edu

Colin Ingalls

Department of Mathematics and Statistics
University of New Brunswick
Fredericton, NB
Canada E3B 5A3
colin@math.unb.ca

Received 3 January 2006. Revision received 16 August 2007.

We define and study canonical singularities of orders over surfaces. These are non-commutative analogues of Kleinian singularities that arise naturally in the minimal model program for orders over surfaces D. Chan and C. Ingalls, Invent. Math. 161 (2005) 427–452. We classify canonical singularities of orders using their minimal resolutions (which we define). We describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring. We also prove that they are Gorenstein, describe their Auslander–Reiten quivers, and note a simple version of the McKay correspondence.

2000 Mathematics Subject Classification 14B05, 16H05.


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