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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2502.15175 |
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| _version_ | 1866915619506487296 |
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| author | Ryder, Jackson |
| author_facet | Ryder, Jackson |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15175 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complements of the point schemes of noncommutative projective lines Ryder, Jackson Rings and Algebras Algebraic Geometry 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. |
| title | Complements of the point schemes of noncommutative projective lines |
| topic | Rings and Algebras Algebraic Geometry |
| url | https://arxiv.org/abs/2502.15175 |