When is the \'etale open topology a field topology?
Abstract
We investigate the following question: Given a field , when is the \'etale open topology induced by a field topology? On the positive side, when is the fraction field of a local domain , using a weak form of resolution of singularities due to Gabber, we show that agrees with the -adic topology when 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 , the \'etale open topology is induced by if and only if is gt-henselian and some non-empty \'etale image is -bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field , is never induced by a field topology.
Keywords
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}
}
Comments
24 pages