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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The constructible topology on spaces of valuation domains
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by Carmelo A. Finocchiaro, Marco Fontana and K. Alan Loper PDF
Trans. Amer. Math. Soc. 365 (2013), 6199-6216 Request permission

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.
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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
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6199-6216
  • MSC (2010): Primary 13A18, 13F05, 13G05
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05741-8
  • MathSciNet review: 3105748