When is the étale open topology a field topology?
We investigate the following question: Given a field K, when is the ´etale open topology EK induced by a field topology? On the positive side, when K is the fraction field of a local domain R 6= K, using a weak form of resolution of singularities due to Gabber, we show that EK agrees with the R-adic...
| Principais autores: | , , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado em: |
Springer
2023
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| _version_ | 1826309827618406400 |
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| author | Dittmann, P Walsberg, E Ye, J |
| author_facet | Dittmann, P Walsberg, E Ye, J |
| author_sort | Dittmann, P |
| collection | OXFORD |
| description | We investigate the following question: Given a field K, when is the ´etale open
topology EK induced by a field topology? On the positive side, when K is the fraction
field of a local domain R 6= K, using a weak form of resolution of singularities due to
Gabber, we show that EK 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 (gthenselianity) following Prestel and Ziegler. We establish the following: For a locally bounded
field topology τ, the ´etale open topology is induced by τ if and only if τ is gt-henselian and
some non-empty ´etale image is τ-bounded open. On the negative side, we obtain that for a
pseudo-algebraically closed field K, EK is never induced by a field topology. |
| first_indexed | 2024-03-07T07:41:32Z |
| format | Journal article |
| id | oxford-uuid:d41f9690-1299-4754-a3f3-c872dd4b4019 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:41:32Z |
| publishDate | 2023 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:d41f9690-1299-4754-a3f3-c872dd4b40192023-04-25T11:55:37ZWhen is the étale open topology a field topology?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d41f9690-1299-4754-a3f3-c872dd4b4019EnglishSymplectic ElementsSpringer2023Dittmann, PWalsberg, EYe, JWe investigate the following question: Given a field K, when is the ´etale open topology EK induced by a field topology? On the positive side, when K is the fraction field of a local domain R 6= K, using a weak form of resolution of singularities due to Gabber, we show that EK 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 (gthenselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology τ, the ´etale open topology is induced by τ if and only if τ is gt-henselian and some non-empty ´etale image is τ-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field K, EK is never induced by a field topology. |
| spellingShingle | Dittmann, P Walsberg, E Ye, J When is the étale open topology a field topology? |
| title | When is the étale open topology a field topology? |
| title_full | When is the étale open topology a field topology? |
| title_fullStr | When is the étale open topology a field topology? |
| title_full_unstemmed | When is the étale open topology a field topology? |
| title_short | When is the étale open topology a field topology? |
| title_sort | when is the etale open topology a field topology |
| work_keys_str_mv | AT dittmannp whenistheetaleopentopologyafieldtopology AT walsberge whenistheetaleopentopologyafieldtopology AT yej whenistheetaleopentopologyafieldtopology |