Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2013-05741-8
Published electronically: March 25, 2013
MathSciNet review: 3105748
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. Paul-Jean Cahen, Alan Loper, and Francesca Tartarone, Integer-valued polynomials and Prüfer $ v$-multiplication domains, J. Algebra 226 (2000), 765-787. MR 1752759 (2001i:13025)
  • 2. Claude Chevalley and Henri Cartan, Schémas normaux; morphismes; ensembles constructibles, Séminaire Henri Cartan 8 (1955-1956), Exp. No. 7, 1-10.
  • 3. David E. Dobbs, Richard Fedder, and Marco Fontana, Abstract Riemann surfaces of integral domains and spectral spaces. Ann. Mat. Pura Appl. 148 (1987), 101-115. MR 932760 (89f:14002)
  • 4. David E. Dobbs and Marco Fontana, Kronecker Function Rings and Abstract Riemann Surfaces, J. Algebra 99 (1986), 263-274. MR 836646 (87e:14001)
  • 5. James Dugundji, Topology, Allyn and Bacon, Boston, 1966. MR 0193606 (33:1824)
  • 6. Alice Fabbri, Kronecker function rings of domains and projective models, Ph.D. Thesis, Università degli Studi ``Roma Tre'', 2010.
  • 7. Carmelo A. Finocchiaro, Marco Fontana and K. Alan Loper, Ultrafilter and constructible topologies on spaces of valuation domains, Comm. Algebra (to appear).
  • 8. Marco Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123 (1980), 331-335. MR 581935 (81j:13001)
  • 9. Marco Fontana and James Huckaba, Localizing systems and semistar operations, in ``Non-Noetherian commutative ring theory'', 169-197, Math. Appl., 520, Kluwer Acad. Publ., Dordrecht, 2000. MR 1858162 (2002k:13001)
  • 10. Marco Fontana and K. Alan Loper: Kronecker function rings: a general approach, in Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), 189-205, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001. MR 1836601 (2002h:13029)
  • 11. Marco Fontana and Alan Loper, A Krull-type theorem for the semistar integral closure of an integral domain. Commutative algebra. AJSE, Arab. J. Sci. Eng. Sect. C Theme Issues 26 (2001), no. 1, 89-95. MR 1843459 (2002e:13019)
  • 12. Marco Fontana and K. Alan Loper, Cancellation properties in ideal systems: a classification of e.a.b. semistar operations, J. Pure Appl. Algebra 213 (2009), 2095-2103. MR 2533308 (2010d:13002)
  • 13. Marco Fontana and K. Alan Loper, The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring, Comm. Algebra 36 (2008), 2917-2922. MR 2440291 (2009d:13001)
  • 14. J. Fresnel and M. van der Put, Géométrie analytique rigide et applications, Progress in Mathematics 18, Birkhäuser, Basel, 1981. MR 644799 (83g:32001)
  • 15. Robert Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972. MR 0427289 (55:323)
  • 16. Alexander Grothendieck et Jean Dieudonné, Éléments de Géométrie Algébrique I, Springer, Berlin, 1970.
  • 17. Melvin Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43-60. MR 0251026 (40:4257)
  • 18. Franz Halter-Koch, Kronecker function rings and generalized Integral closures. Comm. Algebra 31 (2003), 45-59. MR 1969212 (2004e:13004)
  • 19. Olivier Kwegna Heubo, Kronecker function rings of transcendental field extensions, Comm. Algebra 38 (2010), 2701-2719. MR 2730273 (2011k:13006)
  • 20. Roland Huber, Bewertungsspektrum und rigide Geometrie, Regensburger Mathematische Schriften, vol. 23, Universität Regensburg, Fachbereich Mathematik, Regensburg, 1993. MR 1255978 (95c:32036)
  • 21. Roland Huber and Manfred Knebusch, On valuation spectra, in ``Recent advances in real algebraic geometry and quadratic forms: proceedings of the RAGSQUAD year'', Berkeley, 1990-1991, Contemp. Math. 155, Amer. Math. Soc. Providence RI, 1994. MR 1260707 (95f:13002)
  • 22. Franz-Viktor Kuhlmann, Places of algebraic fields in arbitrary characteristic, Advances Math. 188 (2004), 399-424. MR 2087232 (2005h:12014)
  • 23. Bruce Olberding, Holomorphy rings of function fields, in ``Multiplicative ideal theory in commutative algebra'', 331-347, Springer, New York, 2006. MR 2265818 (2008f:13035)
  • 24. Bruce Olberding, Irredundant intersections of valuation overrings of two-dimensional Noetherian domains, J. Algebra 318 (2007), 834-855. MR 2371974 (2009b:13057)
  • 25. Bruce Olberding, Overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings, J. Pure Appl. Algebra 212 (2008), 1797-1821. MR 2400744 (2009c:13008)
  • 26. Bruce Olberding, On Matlis domains and Prüfer sections of Noetherian domains, in ``Commutative Algebra and its Applications, Proceedings of the Fifth International Fez Conference on Commutative Algebra and Applications'', Fez, Morocco, 2008 (Edited by M. Fontana, S.-E. Kabbaj, B. Olberding, and I. Swanson), Walter de Gruyter, Berlin, New York 2009, pages 321-332. MR 2640313 (2011c:13039)
  • 27. Niels Schwartz, Compactification of varieties, Ark. Mat. 28 (1990), 333-370. MR 1084021 (92i:14022)
  • 28. Niels Schwartz and Marcus Tressl, Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra 323 (2010), 698-728. MR 2574858 (2010k:13006)
  • 29. John Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257-269. MR 0306196 (46:5323)
  • 30. Oscar Zariski, The reduction of singularities of an algebraic surface, Ann. Math. 40 (1939), 639-689. MR 0000159 (1:26d)
  • 31. Oscar Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50 (1944), 683-691. MR 0011573 (6:186b)
  • 32. Oscar Zariski and Pierre Samuel, Commutative Algebra, Volume 2, Springer Verlag, Graduate Texts in Mathematics 29, New York, 1975 (First Edition, Van Nostrand, Princeton, 1960).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13A18, 13F05, 13G05

Retrieve articles in all journals with MSC (2010): 13A18, 13F05, 13G05


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: https://doi.org/10.1090/S0002-9947-2013-05741-8
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.

American Mathematical Society