Proceedings of the London Mathematical Society Advance Access originally published online on January 7, 2009
Proceedings of the London Mathematical Society 2009 99(1):100-144; doi:10.1112/plms/pdn054
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© 2009 London Mathematical Society
A class of noncommutative projective surfaces
Department of Mathematics
UCSD
La Jolla, CA 92093-0112
USA
drogalsk@math.ucsd.edu
Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
USA
Current address:
Department of Mathematics
Alan Turing Building
University of Manchester
Manchester M13 9PL,
United Kingdom
Toby.Stafford@manchester.ac.uk
Received 23 November 2007. Revision received 7 October 2008.
Let A = 
i
0Ai be a connected graded, noetherian k-algebra that is generated in degree one over an algebraically closed field k. Suppose that the graded quotient ring Q(A) has the form Q(A) = k(X)[t, t–1; 
], where is an automorphism of the integral projective surface X. Then we prove that A can be written as a naïve blowup algebra of a projective surface 
birational to X. This enables one to obtain a deep understanding of the structure of these algebras; for example, generically they are not strongly noetherian and their point modules are not parametrized by a projective scheme. This is despite the fact that the simple objects in qgr-A will always be in (1-1) correspondence with the closed points of the scheme .
2000 Mathematics Subject Classification 14A22, 16P40, 16P90, 16S38, 16W50, 18E15 (primary).
The first author was partially supported by NSF grants DMS-0202479 and DMS-0600834 while the second author was partially supported by NSF grants DMS-0245320 and DMS-0555750 and also by the Leverhulme Research Interchange Grant F/00158/X. Part of this work was written up while the second author was visiting and supported by the Newton Institute, Cambridge. We would like to thank all three institutions for their financial support.