When is the étale open topology a field topology?
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911521200668672 |
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| author | Dittmann, Philip Walsberg, Erik Ye, Jinhe |
| author_facet | Dittmann, Philip Walsberg, Erik Ye, Jinhe |
| contents | We investigate the following question: Given a field $K$, when is the étale open topology $\mathcal{E}_K$ induced by a field topology? On the positive side, when $K$ is the fraction field of a local domain $R\neq K$, using a weak form of resolution of singularities due to Gabber, we show that $\mathcal{E}_K$ agrees with the $R$-adic topology when $R$ is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gt-henselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology $τ$, the étale open topology is induced by $τ$ if and only if $τ$ is gt-henselian and some non-empty étale image is $τ$-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field $K$, $\mathcal{E}_K$ is never induced by a field topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02398 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | When is the étale open topology a field topology? Dittmann, Philip Walsberg, Erik Ye, Jinhe Logic Commutative Algebra We investigate the following question: Given a field $K$, when is the étale open topology $\mathcal{E}_K$ induced by a field topology? On the positive side, when $K$ is the fraction field of a local domain $R\neq K$, using a weak form of resolution of singularities due to Gabber, we show that $\mathcal{E}_K$ agrees with the $R$-adic topology when $R$ is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gt-henselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology $τ$, the étale open topology is induced by $τ$ if and only if $τ$ is gt-henselian and some non-empty étale image is $τ$-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field $K$, $\mathcal{E}_K$ is never induced by a field topology. |
| title | When is the étale open topology a field topology? |
| topic | Logic Commutative Algebra |
| url | https://arxiv.org/abs/2208.02398 |