Large implies henselian

TL;DR

The paper characterizes large fields via their elementary extensions and henselian local domains, introducing new topologies on $K$-varieties and exploring their properties and differences.

Contribution

It establishes a new equivalence for large fields involving henselian local domains and introduces the 'finite-closed' topology, comparing it with the 'étale-open' topology.

Findings

A new characterization of large fields using henselian local domains.

Introduction of the finite-closed topology on $K$-varieties.

Demonstration of the conditions under which the two topologies agree or differ.

Abstract

Fix a field $K$. We show that $K$ is large if and only if some elementary extension of $K$ is the fraction field of a henselian local domain which is not a field. The proof uses a new result about the \'etale-open topology over $K$: if $K$ is not separably closed and $V \to W$ is an \'etale morphism of $K$-varieties then $V(K) \to W(K)$ is a local homeomorphism in the \'etale-open topology. This, in turn, follows from results comparing the \'etale-open topology on $V(K)$ and the finite-closed topology on $V(K)$, newly introduced in this paper. We show that the \'etale-open topology refines the finite-closed topology when $K$ is perfect, and that the finite-closed topology refines the \'etale-open topology when $K$ is bounded. It follows that these two topologies agree in many natural examples. On the other hand, we construct several examples where these two differ, which allows us to answer a question of Lampe.