Differential Geometry of Synthetic Schemes
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909822104895488 |
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| author | Cherubini, Felix Hutzler, Matthias Moeneclaey, Hugo Wärn, David |
| author_facet | Cherubini, Felix Hutzler, Matthias Moeneclaey, Hugo Wärn, David |
| contents | Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use homotopy type theory only for convenience. We define étale, smooth and unramified maps between schemes in synthetic algebraic geometry using a new synthetic definition. We give the usual characterizations of these classes of maps in terms of injectivity, surjectivity and bijectivity of differentials. We also show that the tangent spaces of smooth schemes are finite free modules. Finally, we show that unramified, étale and smooth schemes can be understood very concretely via the expected local algebraic description. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Differential Geometry of Synthetic Schemes Cherubini, Felix Hutzler, Matthias Moeneclaey, Hugo Wärn, David Algebraic Geometry Logic 14A99 (Primary), 03B38, 18N99 (Secondary) Synthetic algebraic geometry uses homotopy type theory extended with three axioms to develop algebraic geometry internal to a higher version of the Zariski topos. In this article we make no essential use of the higher structure and use homotopy type theory only for convenience. We define étale, smooth and unramified maps between schemes in synthetic algebraic geometry using a new synthetic definition. We give the usual characterizations of these classes of maps in terms of injectivity, surjectivity and bijectivity of differentials. We also show that the tangent spaces of smooth schemes are finite free modules. Finally, we show that unramified, étale and smooth schemes can be understood very concretely via the expected local algebraic description. |
| title | Differential Geometry of Synthetic Schemes |
| topic | Algebraic Geometry Logic 14A99 (Primary), 03B38, 18N99 (Secondary) |
| url | https://arxiv.org/abs/2504.08495 |