The constructible topology on spaces of valuation domains

Abstract:

We consider properties and applications of a compact, Hausdorff topology called the “ultrafilter topology” defined on an 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 $\operatorname {Zar}(K|A)$ of all valuation domains, having $K$ as the 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 $\operatorname {Zar}(K|A)$. 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.