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On the Fibers of Analytic Mappings

  • Jürgen Bingener
  • Hubert Flenner
Part of the The University Series in Mathematics book series (USMA)

Abstract

Let f: XS be a morphism of complex or real spaces, and P a property of homomorphisms of local rings. Consider the set ℙ(f) of points x∈X for which the induced map of local rings O S,f(x) O X,x has property P. In this chapter we give a criterion for ℙ(f) being constructible (resp., Zariski open) in X. Moreover, we verify this criterion for a wide class of properties P.

Keywords

Prime Ideal Local Ring Finite Type Noetherian Ring Fiber Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    W. Adkins, A. Andreotti, and J. V. Leahy, Weakly normal complex spaces, Atti Accad. Naz. Lincei (1980).Google Scholar
  2. 2.
    A. Andreotti and F. Norguet, La convexité holomorphe dans l’espace analytique des cycles d’un variété algébrique, Ann. Scuola Norm. Sup. Pisa 21, 31–82 (1967).MATHMathSciNetGoogle Scholar
  3. 3.
    L. Avramov, Homology of local flat extensions and complete intersection defects, Math. Ann. 228, 27–37 (1977).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. Banica, Un théorème concernant les familles analytiques d’espaces complexes, Rev. Roumaine Math. Pures Appl. 10, 1515–1520 (1973).MathSciNetGoogle Scholar
  5. 5.
    C. Banica, Le lieu réduit et le lieu normal d’un morphisme, preprint, Bucharest (1976).Google Scholar
  6. 6.
    C. Banica and M. Stoia, Gorenstein points of a flat morphism of complex spaces, preprint, Bucharest (1979).Google Scholar
  7. 7.
    J. Bingener and H. Flenner, Constructible and quasiconstructible sheaves on analytic spaces, Abh. Math. Sem. U. Hamburg 54, 119–139 (1984).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Bingener and H. Flenner, Variation of the divisor class group, J. Reine Angew. Math. 351, 20–41 (1984).MATHMathSciNetGoogle Scholar
  9. 9.
    J. Bingener and U. Storch, Zur Berechnung der Divisorenklassengruppen kompletter lokaler Ringe, Leopoldina Symposion Singularitäten—Singularities, Nova Acta Leopoldina 52, Neue Folge 240, 7–63 (1981).MathSciNetGoogle Scholar
  10. 10.
    N. Bourbaki, Algèbre Commutative, Hermann, Paris, (1961–1967).Google Scholar
  11. 11.
    H. Cartan, Seminaire 1960/61, 2ième éd., corrigée.Google Scholar
  12. 12.
    C. Cumino, S. Greco, and M. Manaresi, Bertini theorems for weak normality, Compositio Math. 48, 351–362 (1983).MATHMathSciNetGoogle Scholar
  13. 13.
    G. Fischer, Complex Analytic Geometry, Lecture Notes in Math., Vol. 538, Springer-Verlag, Berlin and New York (1976).MATHGoogle Scholar
  14. 14.
    H. Flenner, Die Sätze von Bertini für lokale Ringe, Math. Ann. 229, 97–111 (1977).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. M. Fossum, The divisor class group of a Krull domain, Ergeb. Math. Grenzgeb., Bd. 74, Springer-Verlag, Berlin and New York (1973).MATHCrossRefGoogle Scholar
  16. 16.
    J. Frisch, Points de platitude d’un morphisme d’espaces analytiques complexes, Invent. Math. 4, 118–138(1967).MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Fujiki, Closedness of the Douady spaces of compact Kahler spaces, Publ. RIMS Kyoto Univ. 14, 1–52 (1978).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    H. Grauert and H. Kerner, Deformationen von Singularitäten komplexer Räume, Math. Ann. 153, 236–260 (1964).MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    S. Greco and M. G. Marinari, Nagata’s criterion and openness of loci for Gorenstein and complete intersection, Math. Z. 160, 207–216 (1978).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    A. Grothendieck, Revêtements Etales et Groupe Fondamental (SGA 1), Lecture Notes in Math, Vol. 224, Springer-Verlag, Berlin and New York (1971).MATHGoogle Scholar
  21. 21.
    A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, Ins t. Hautes Etudes Sci. Publ. Math. 4,8,11,17,20,24,28,32 (1960–1967).Google Scholar
  22. 22.
    R. Hartshorne, Residues and Duality, Lecture Notes in Math., Vol. 20, Springer-Verlag, Berlin and New York (1966).Google Scholar
  23. 23.
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. 79, 109–326 (1964).MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    H. Hironaka, Subanalytic sets, in Number Theory, Algebraic Geometry and Commutative Algebra, in honour of Y. Akizuki, pp. 453–493, Kinokuniya, Tokyo (1973).Google Scholar
  25. 25.
    W. Kexel, Zariski-Offenheit von eigentlichen, flachen holomorphen Abbildungen, Dissertation, Bayreuth (1977).Google Scholar
  26. 26.
    R. Kiehl, Note zu der Arbeit von J. Frisch: “Points de platitude d’un morphisme d’espace analytiques complexes,” Invent. Math. 4, 139–141 (1967).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. Kiehl, Analytische Familien affinoider Algebren, Sitzungsber.-Bayer Heidelberger Akad. Wiss. Math.-Nat. Kl., 25–49 (1968).Google Scholar
  28. 28.
    R. Kiehl and E. Kunz, Vollständige Durchschnitte und p-Basen, Arch. Math. 16, 348–362 (1965).MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    M. Lejeune-Jalabert and B. Teissier, Normal cones and sheaves of relative jets, Compositio Math. 28, 305–331 (1974).MATHMathSciNetGoogle Scholar
  30. 30.
    D. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, in Lecture Notes in Math., Vol. 670, pp. 140–186, Springer-Verlag, Berlin and New York (1978).Google Scholar
  31. 31.
    M. Manaresi, Some properties of weakly normal varieties, Nagoya Math. J.11, 61–74 (1980).MathSciNetGoogle Scholar
  32. 32.
    M. Manaresi, Sard and Bertini type theorems for complex spaces, Ann. Mat. Pura Appl. 131, 265–279 (1982).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    H. Matsumura, Commutative Algebra, 2nd ed., Benjamin/Cummings, Reading MA (1980).MATHGoogle Scholar
  34. 34.
    M. Nagata, Local Rings, Wiley, New York (1962).MATHGoogle Scholar
  35. 35.
    M. Raynaud, Anneaux Locaux Henséliens, Lecture Notes in Math., Vol. 169, Springer-Verlag, Berlin and New York (1970).Google Scholar
  36. 36.
    R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133, 328–370 (1957).MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    O. Riemenschneider, Über die Anwendung algebraischer Methoden in der Deformationstheorie komplexer Räume, Math. Ann. 187, 40–55 (1970).MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    C. Rotthaus, On the approximation property of excellent rings, Invent. Math. 88, 39–63 (1987).MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    J. B. Sancho de Salas, Semicontinuity for the local Hilbert function, Math. Z. 194, 217–225 (1987); Correction, Math. Z. 196, 301 (1987).MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    G. Scheja, Differentialmoduln lokaler analytischer Algebren, Schriften. Math. Inst. U. Freiburg. LUE. No. 2 (1970).Google Scholar
  41. 41.
    G. Scheja and U. Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann. 197, 137–170 (1972).MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    G. Scheja and U. Storch, Lokale Verzweigungstheorie, Schriften. Math. Inst. U. Freiburg. LUE. No. 5 (1974).Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Jürgen Bingener
    • 1
  • Hubert Flenner
    • 2
  1. 1.Fakultät für MathematikUniversitätRegensburgGermany
  2. 2.Mathematisches InstitutGeorg-August-UniversitätGöttingenGermany

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