Noncommutative curves and noncommutative surfaces.(English)Zbl 1042.16016
Summary: 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.
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.
MSC:
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 14A22 | Noncommutative algebraic geometry |
| 16P90 | Growth rate, Gelfand-Kirillov dimension |
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 18E15 | Grothendieck categories (MSC2010) |
Keywords:
Noetherian graded rings;noncommutative projective geometry;deformations;twisted homogeneous coordinate ringsCite
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