The étale-open topology and the stable fields conjecture
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866909359109308416 |
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| author | Johnson, Will Tran, Chieu-Minh Walsberg, Erik Ye, Jinhe |
| author_facet | Johnson, Will Tran, Chieu-Minh Walsberg, Erik Ye, Jinhe |
| contents | For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_02319 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The étale-open topology and the stable fields conjecture Johnson, Will Tran, Chieu-Minh Walsberg, Erik Ye, Jinhe Logic Algebraic Geometry 03C60, 14G27, 12E30 For an arbitrary field $K$ and $K$-variety $V$, we introduce the étale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably closed, real closed, or $p$-adically closed, respectively. Topological properties of the étale-open topology corresponds to algebraic properties of $K$. For example, the étale-open topology on $\mathbb{A}^1(K)$ is not discrete if and only if $K$ is large. As an application, we show that a large stable field is separably closed. |
| title | The étale-open topology and the stable fields conjecture |
| topic | Logic Algebraic Geometry 03C60, 14G27, 12E30 |
| url | https://arxiv.org/abs/2009.02319 |