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

TL;DR

This paper explores when the étale open topology on a field can be derived from a field topology, revealing conditions under which they coincide and identifying cases where they do not, especially in relation to local domains and Pseudo-algebraically closed fields.

Contribution

It characterizes when the étale open topology matches a field topology, introduces generalized t-henselianity, and identifies pathological cases in non-quasi-excellent fields.

Findings

Étale open topology equals the $R$-adic topology for certain local domains.

Introduces generalized t-henselianity (gt-henselianity) for locally bounded topologies.

Shows that for Pseudo-algebraically closed fields, the étale open topology is never induced by a field topology.

Abstract

We investigate the following question: Given a field $K$, when is the \'etale 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 $\tau$, the \'etale open topology is induced by $\tau$ if and only if $\tau$ is gt-henselian and some non-empty \'etale image is $\tau$-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.