Bulletin of the American Mathematical Society

Author(s): J. T. Stafford; M. Van den Bergh
Journal: Bull. Amer. Math. Soc. 38 (2001), 171-216.
MSC (2000): Primary 14A22, 14F05, 16D90, 16P40, 16S80, 16W50, 18E15
Posted: January 9, 2001
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Abstract: In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories.

Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic, growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has led to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

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J. T. Stafford
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
Email: jts@math.lsa.umich.edu

M. Van den Bergh
Affiliation: Departement WNI, Limburgs Universitair Centrum, 3590 Diepenbeek, Belgium
Email: vdbergh@luc.ac.be

DOI: 10.1090/S0273-0979-01-00894-1
PII: S 0273-0979(01)00894-1
Keywords: Noetherian graded rings, noncommutative projective geometry, deformations, twisted homogeneous coordinate rings
Received by editor(s): October 18, 1999, and in revised form May 20, 2000
Posted: January 9, 2001
Additional Notes: The first author was supported in part by an NSF grant
The second author is a senior researcher at the FWO and was partially supported by the Clay Research Institute during the preparation of this article.
Copyright of article: Copyright 2001, American Mathematical Society

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