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New Distinguished Classes of Spectral Spaces: A Survey

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Multiplicative Ideal Theory and Factorization Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 170))

Abstract

In the present survey paper, we present several new classes of Hochster’s spectral spaces “occurring in nature,” actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The general setting is the space of the semistar operations (of finite type), endowed with a Zariski-like topology, which turns out to be a natural topological extension of the space of the overrings of an integral domain, endowed with a topology introduced by Zariski. One of the key tool is a recent characterization of spectral spaces, based on the ultrafilter topology, given in Finocchiaro, Commun Algebra, 42:1496–1508, 2014, [15]. Several applications are also discussed.

The authors gratefully acknowledge partial support from INdAM, Istituto Nazionale di Alta Matematica.

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References

  1. D.D. Anderson, Star-operations induced by overrings. Commun. Algebra 16, 2535–2553 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. D.D. Anderson, S.J. Cook, Two star operations and their induced lattices. Commun. Algebra 28, 2461–2475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. D.F. Anderson, D.D. Anderson, Examples of star operations on integral domains. Commun. Algebra 18, 1621–1643 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley, Reading, 1969)

    MATH  Google Scholar 

  5. V. Barucci, D. Dobbs, M. Fontana, Conducive integral domains as pullbacks. Manuscr. Math. 54, 261–277 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bourbaki, Algèbre Commutative, Chap. 1–2 (Hermann, Paris, 1961)

    MATH  Google Scholar 

  7. P.-J. Cahen, Commutative torsion theory. Trans. Am. Math. Soc. 184, 73–85 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. P.-J. Cahen, K.A. Loper, F. Tartarone, Integer-valued polynomials and Prüfer \(v\)-multiplication domains. J. Algebra 226, 765–787 (2000)

    Google Scholar 

  9. C. Chevalley, Sur la théorie des variétés algébriques. Nagoya Math. J. 8, 1–43 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  10. C. Chevalley, H. Cartan, Schémas normaux; morphismes; ensembles constructibles. Séminaire Henri Cartan 8, Exp. No. 7, 1–10 (1955–1956)

    Google Scholar 

  11. D. Dobbs, M. Fontana, Kronecker function rings and abstract Riemann surfaces. J. Algebra 99, 263–274 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Dobbs, R. Fedder, M. Fontana, Abstract Riemann surfaces of integral domains and spectral spaces. Ann. Mat. Pura Appl. 148, 101–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Epstein, A guide to closure operations in commutative algebra, Progress in Commutative Algebra, vol. 2 (Walter de Gruyter, Berlin, 2012), pp. 1–37

    Chapter  Google Scholar 

  14. N. Epstein, Semistar operations and standard closure operations. Commun. Algebra 43, 325–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. C.A. Finocchiaro, Spectral spaces and ultrafilters. Commun. Algebra 42, 1496–1508 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. C.A. Finocchiaro, M. Fontana, K.A. Loper, Ultrafilter and constructible topologies on spaces of valuation domains. Commun. Algebra 41, 1825–1835 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. C.A. Finocchiaro, M. Fontana, K.A. Loper, The constructible topology on spaces of valuation domains. Trans. Am. Math. Soc. 365, 6199–6216 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. C.A. Finocchiaro, M. Fontana, D. Spirito, Spectral Spaces of Semistar Operations . J. Pure Appl. Algebra. 220, 2897–2913 (2016)

    Google Scholar 

  19. C.A. Finocchiaro, M. Fontana, D. Spirito, The Space of Inverse-Closed Subsets of a Spectral Space. (2016) (submitted)

    Google Scholar 

  20. C.A. Finocchiaro, M. Fontana, D. Spirito, On a Topological Version of Hilbert’s Nullstellensatz. J. Algebra (to appear)

    Google Scholar 

  21. C.A. Finocchiaro, D. Spirito, Some topological considerations on semistar operations. J. Algebra 409, 199–218 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Fontana, J. Huckaba, Localizing systems and semistar operations, in Non-Noetherian Commutative Ring Theory, ed. by Scott T. Chapman, Sarah Glaz (Kluwer Academic Publishers, Dordrecht, 2000), pp. 169–198

    Chapter  Google Scholar 

  23. M. Fontana, K.A. Loper, Kronecker function rings: a general approach, in Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), pp. 189–205. (Lecture Notes in Pure and Applied Mathematics, vol. 220 (Dekker, New York, 2001))

    Google Scholar 

  24. M. Fontana, K.A. Loper, Nagata rings, Kronecker function rings and related semistar operations. Commun. Algebra 31, 4775–4805 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Fontana, K.A. Loper, An historical overview of Kronecker function rings, Nagata rings, and related star and semistar operations, in Multiplicative Ideal Theory in Commutative Algebra: A Tribute to the Work of Robert Gilmer, ed. by J.W. Brewer, S. Glaz, W. Heinzer, B. Olberding (Springer, New York, 2006), pp. 169–187

    Chapter  Google Scholar 

  26. M. Fontana, K.A. Loper, The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring. Commun. Algebra 36, 2917–2922 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Fontana, K.A. Loper, Cancellation properties in ideal systems: a classification of e.a.b. semistar operations. J. Pure Appl. Algebra 213, 2095–2103 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Fontana, J. Huckaba, I. Papick, Prüfer domains (M. Dekker, New York, 1997)

    MATH  Google Scholar 

  29. M. Fontana, K.A. Loper, R. Matsuda, Cancellation properties in ideal systems: an e.a.b. not a.b. star operation. AJSE (Arabian Journal for Science and Engineering)–Mathematics 35, 45–49 (2010)

    MathSciNet  MATH  Google Scholar 

  30. P. Gabriel, La localisation dans les anneaux non commutatifs, in Séminaire Dubreil (sous la direction de P. Dubreil, M.-L. Dubreil-Jacotin, C. Pisot). Algèbre et théorie des nombres, 13 no. 1, Exposé No. 2 (1959–1960), 35 p

    Google Scholar 

  31. R. Gilmer, Multiplicative Ideal Theory. Queen’s Papers in Pure and Applied Mathematics, vol. I & II (Kingston, Ontario, Canada, 1968)

    Google Scholar 

  32. R. Gilmer, Multiplicative Ideal Theory (M. Dekker, New York, 1972)

    MATH  Google Scholar 

  33. A. Grothendieck, J. Dieudonné, Éléments de Géométrie Algébrique I, IHES 1960 (Springer, Berlin, 1970)

    Google Scholar 

  34. W. Heinzer, M. Roitman, Well-centered overrings of an integral domain. J. Algebra 272(2), 435–455 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Hochster, Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142, 43–60 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  36. M. Hochster, C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)

    MathSciNet  MATH  Google Scholar 

  37. O. Heubo-Kwegna, Kronecker function rings of transcendental field extensions. Commun. Algebra 38, 2701–2719 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Huckaba, Commutative Rings with Zero Divisors (M. Dekker, New York, 1988)

    MATH  Google Scholar 

  39. C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules, vol. 336, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 2006)

    MATH  Google Scholar 

  40. W. Krull, Idealtheorie (Springer, Berlin, 1935). (2nd edn. 1968)

    Google Scholar 

  41. W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche, I - II. Math. Z. 41, 545–577, 665–679 (1936)

    Google Scholar 

  42. W. Krull, Gesammelte Abhandlungen/Collected Papers, Hrsg. v. Paulo Ribenboim (Walter de Gruyter, Berlin, 1999)

    Google Scholar 

  43. T. Jech, Set Theory (Springer, New York, 1997). (1st edn. Academic Press, 1978)

    Book  MATH  Google Scholar 

  44. J. Lambek, Torsion Theories, Additive Semantics, and Rings of Quotients. Lecture Notes in Mathematics, vol. 177 (Springer, Berlin, 1971)

    Google Scholar 

  45. P. Maroscia, Sur les annéaux de dimension zéro. Rend. Acc. Naz. Lincei 56, 451–459 (1974)

    MathSciNet  MATH  Google Scholar 

  46. A. Okabe, R. Matsuda, Semistar operations on integral domains. Math. J. Toyama Univ. 17, 1–21 (1994)

    MathSciNet  MATH  Google Scholar 

  47. J.-P. Olivier, Anneaux absolument plats universels et épimorphismes à buts réduits, Sém P. Samuel, Algèbre Commutative, Année, Ex. N. 6 (1967/68)

    Google Scholar 

  48. J.-P. Olivier, Anneaux absolument plats universels et épimorphismes d’anneaux. C.R. Acad. Sci. Paris 266, 317–318 (1968)

    MathSciNet  MATH  Google Scholar 

  49. N. Schwartz, M. Tressl, Elementary properties of minimal and maximal points in Zariski spectra. J. Algebra 323, 698–728 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. B. Stenström, Rings and Modules of Quotients. Lecture Notes in Math, vol. 237 (Springer, Berlin, 1971)

    Google Scholar 

  51. J.C. Vassilev, Structure on the set of closure operations of a commutative ring. J. Algebra 321, 2737–2753 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. O. Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions. Bull. Am. Math. Soc 50, 683–691 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  53. O. Zariski, P. Samuel, Commutative Algebra, vol. II (Van Nostrand, Princeton, 1960)

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi “Roma Tre”, 00146, Rome, Italy

    Marco Fontana & Dario Spirito

  2. Institute of Analysis and Number Theory, University of Technology, Kopernikusgasse 24, 8010, Graz, Austria

    Carmelo A. Finocchiaro

Authors
  1. Carmelo A. Finocchiaro
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  2. Marco Fontana
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  3. Dario Spirito
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Correspondence to Marco Fontana .

Editor information

Editors and Affiliations

  1. Dept of Math & Stat, Sam Houston State University, Huntsville, Texas, USA

    Scott Chapman

  2. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  3. Institut für Mathematik und Wissenschaftliches Rechnen, Universität Graz, Graz, Austria

    Alfred Geroldinger

  4. Dept. Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, USA

    Bruce Olberding

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Finocchiaro, C.A., Fontana, M., Spirito, D. (2016). New Distinguished Classes of Spectral Spaces: A Survey. In: Chapman, S., Fontana, M., Geroldinger, A., Olberding, B. (eds) Multiplicative Ideal Theory and Factorization Theory. Springer Proceedings in Mathematics & Statistics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-38855-7_5

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