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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The constructible topology on spaces of valuation domains


Authors: Carmelo A. Finocchiaro, Marco Fontana and K. Alan Loper
Journal: Trans. Amer. Math. Soc. 365 (2013), 6199-6216
MSC (2010): Primary 13A18, 13F05, 13G05
Published electronically: March 25, 2013
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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\vert 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\vert 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.


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Additional Information

Carmelo A. Finocchiaro
Affiliation: Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo San Leonardo Murialdo, 1, 00146 Rome, Italy
Email: carmelo@mat.uniroma3.it

Marco Fontana
Affiliation: Dipartimento di Matematica, Università degli Studi “Roma Tre”, Largo San Leonardo Murialdo, 1, 00146 Rome, Italy
Email: fontana@mat.uniroma3.it

K. Alan Loper
Affiliation: Department of Mathematics, Ohio State University, Newark, Ohio 43055
Email: lopera@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05741-8
PII: S 0002-9947(2013)05741-8
Received by editor(s): October 7, 2010
Received by editor(s) in revised form: March 29, 2011, and August 11, 2011
Published electronically: March 25, 2013
Additional Notes: During the preparation of this paper, the first two authors were partially supported by a research grant PRIN-MiUR
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.