Large implies henselian
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910044428173312 |
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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 | Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the étale-open topology over $K$: if $K$ is not separably closed and $V \to W$ is an étale morphism of $K$-varieties then $V(K) \to W(K)$ is a local homeomorphism in the étale-open topology. This, in turn, follows from results comparing the étale-open topology on $V(K)$ and the finite-closed topology on $V(K)$, newly introduced in this paper. We show that the étale-open topology refines the finite-closed topology when $K$ is perfect, and that the finite-closed topology refines the étale-open topology when $K$ is bounded. It follows that these two topologies agree in many natural examples. On the other hand, we construct several examples where these two differ, which allows us to answer a question of Lampe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large implies henselian Johnson, Will Tran, Chieu-Minh Walsberg, Erik Ye, Jinhe Logic Algebraic Geometry Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the étale-open topology over $K$: if $K$ is not separably closed and $V \to W$ is an étale morphism of $K$-varieties then $V(K) \to W(K)$ is a local homeomorphism in the étale-open topology. This, in turn, follows from results comparing the étale-open topology on $V(K)$ and the finite-closed topology on $V(K)$, newly introduced in this paper. We show that the étale-open topology refines the finite-closed topology when $K$ is perfect, and that the finite-closed topology refines the étale-open topology when $K$ is bounded. It follows that these two topologies agree in many natural examples. On the other hand, we construct several examples where these two differ, which allows us to answer a question of Lampe. |
| title | Large implies henselian |
| topic | Logic Algebraic Geometry |
| url | https://arxiv.org/abs/2508.10886 |