English

When is the \'etale open topology a field topology?

Logic 2026-03-18 v5 Commutative Algebra

Abstract

We investigate the following question: Given a field KK, when is the \'etale open topology EK\mathcal{E}_K induced by a field topology? On the positive side, when KK is the fraction field of a local domain RKR\neq K, using a weak form of resolution of singularities due to Gabber, we show that EK\mathcal{E}_K agrees with the RR-adic topology when RR 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\tau, the \'etale open topology is induced by τ\tau if and only if τ\tau is gt-henselian and some non-empty \'etale image is τ\tau-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field KK, EK\mathcal{E}_K is never induced by a field topology.

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Cite

@article{arxiv.2208.02398,
  title  = {When is the \'etale open topology a field topology?},
  author = {Philip Dittmann and Erik Walsberg and Jinhe Ye},
  journal= {arXiv preprint arXiv:2208.02398},
  year   = {2026}
}

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24 pages