Noncommutative Tsen’s theorem in dimension one. (English) Zbl 1327.14017
The author considers noncommutative curves of genus zero over fields. These curves are defined as small \(k\)-linear abelian categories satisfying several restrictions. The author finds necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative \(\mathbb{P}^1\)-bundle. Then relations of this result with the one-dimensional version of Tsen’s theorem and Bondel-Orlov’s theorem are outlined. As a special case, it is proved that each arithmetic noncommutative projective line is a noncommutative curve. Then, noncommutative arithmetic curves of genus zero are exactly characterized. Finally applications to homogeneous noncommutative curves of genus zero are given in the paper.
Reviewer: Sergey Ludkovsky (Moskva)
MSC:
| 14A22 | Noncommutative algebraic geometry |
| 14H45 | Special algebraic curves and curves of low genus |
| 16S38 | Rings arising from noncommutative algebraic geometry |
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