When is the étale open topology a field topology?

Abstract

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.