Ultrafilter and Constructible Topologies on Spaces of Valuation Domains

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 pipe 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 pipe A). We extend results regarding distinguished spectral topologies on spaces of valuation domains. © 2013 Copyright Taylor and Francis Group, LLC.