Ultrafilter and Constructible topologies on spaces of valuation domains

TL;DR

This paper explores the ultrafilter topology on valuation domain spaces, demonstrating its equivalence with the constructible topology and extending known spectral topology results in algebraic geometry.

Contribution

It establishes the equivalence of ultrafilter and constructible topologies on valuation spaces and extends spectral topology results.

Findings

Ultrafilter topology is compact and Hausdorff.

Ultrafilter topology coincides with the constructible topology.

Extended results on spectral topologies for valuation spaces.

Abstract

Let $K$ be a field and let $A$ be a subring of $K$. We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on the space Zar$(K|A)$ of all valuation domains having $K$ as quotient field and containing $A$. We show that the ultrafilter topology coincides with the constructible topology on the abstract Riemann-Zariski surface Zar$(K|A)$. We extend results regarding distinguished spectral topologies on spaces of valuation domains.