Year: 2025
Type: article
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 (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 nonempty 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.
Source: Israel Journal of Mathematics
Authors: Philip Dittmann, Erik Walsberg, Jinhe Ye
Institutions: University of Manchester, University of Oxford
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Award: G4880459266
OpenAlex ID: W4409239380