When is the étale open topology a field topology?

2022/08/04 by Philip Dittmann, Erik Walsberg, Dittmann, Philip +3 · 4 citations
Mathematics · #Advanced Topology and Set Theory #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2208.02398

openalex publication_date 2022/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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.

Cited by

• Derivations and gt-henselian field topologies
• Largeness and generalized t-henselianity
• Large implies henselian
• Nash maps over large fields

Related

• Large implies henselian
• Topological proofs of results on large fields
• Largeness and generalized t-henselianity
• Derivations and gt-henselian field topologies
• Topological fields with a generic derivation