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 E K induced by a field topology? On the positive side, when K is the fraction field of a local domain R = K, using a weak form of resolution of singularities due to Gabber, we show that 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 (gthenselianity) 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, E K is never induced by a field topology.