The constructible topology on spaces of valuation domains

DOI10.1090/S0002-9947-2013-05741-8zbMATH Open1297.13007arXiv1206.3521MaRDI QIDQ2855925

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Publication date: 23 October 2013

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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.

Full work available at URL: https://arxiv.org/abs/1206.3521

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