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arXiv · 2502.15175

Complements of the point schemes of noncommutative projective lines

Jackson Ryder
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Abstract

Recently, Chan and Nyman constructed noncommutative projective lines via a noncommutative symmetric algebra for a bimodule $V$ over a pair of fields. These noncommutative projective lines of contain a canonical closed subscheme (the point scheme) determined by a normal family of elements in the noncommutative symmetric algebra. We study the complement of this subscheme when $V$ is simple, the coordinate ring of which is obtained by inverting said normal family. We show that this localised ring is a noncommutative Dedekind domain of Gelfand-Kirillov dimension 1. Furthermore, the question of simplicity of these Dedekind domains is answered by a similar dichotomy to an analogous open subscheme of the noncommutative quadrics of Artin, Tate and Van den Bergh.

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Jackson Ryder. 2025-02-21. Complements of the point schemes of noncommutative projective lines. https://arxiv.org/abs/2502.15175

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