The constructible topology on spaces of valuation domains

TL;DR

This paper explores the ultrafilter (constructible) topology on spectral spaces of valuation domains, establishing its properties, and applying it to understand intersections of valuation overrings and representations of integrally closed domains.

Contribution

It demonstrates that the ultrafilter topology coincides with the constructible topology on spectral spaces and constructs explicit homeomorphisms for valuation domain spaces, extending spectral topology results.

Findings

Ultrafilter topology equals the constructible topology on spectral spaces.

Constructs explicit homeomorphisms between valuation domain spaces and Zariski spectra.

Shows valuation domain collections with the same ultrafilter closure define the same integrally closed domain.

Abstract

We consider properties and applications of a compact, Hausdorff topology called the "ultrafilter topology" defined on an {\sl arbitrary spectral space} and we observe that this topology coincides with the constructible topology. If $K$ is a field and $A$ a subring of $K$, we show that the space Zar$(K|A)$ of all valuation domains, having $K$ as quotient field and containing $A$, (endowed with the Zariski topology) is a spectral space by giving in this general setting the explicit construction of a ring whose Zariski spectrum is homeomorphic to Zar$(K|A)$. We extend results regarding spectral topologies on the spaces of all valuation domains and apply the theory developed to study representations of integrally closed domains as intersections of valuation overrings. As a very particular case, we prove that two collections of valuation domains of $K$ with the same ultrafilter closure represent, as an intersection, the same integrally closed domain.