Noncommutative curves and noncommutative surfaces
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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.References
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Additional Information
- 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
- MR Author ID: 176980
- Email: vdbergh@luc.ac.be
- Received by editor(s): October 18, 1999
- Received by editor(s) in revised form: May 20, 2000
- Published electronically: 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 2001 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 38 (2001), 171-216
- MSC (2000): Primary 14A22, 14F05, 16D90, 16P40, 16S80, 16W50, 18E15
- DOI: https://doi.org/10.1090/S0273-0979-01-00894-1
- MathSciNet review: 1816070