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Über Deformationen von analytischen Abbildungskeimen

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Abstract

The notion of deformations of germs of k-analytic mappings generalizes the one of deformations of germs of k-analytic spaces. Using algebraic terms, we prove:

  1. 1.

    The morphism f: A→B of analytic algebras is rigid, iff it is infinitesimally rigid. Moreover, this is equivalent to ExA (B,B)=0. This theorem generalizes a result of SCHUSTER [11].

  2. 2.

    Let A be a regular analytic algebra. Then f is rigid iff there exists a rigid analytic algebra Bo such that f is equivalent to the canonic injection A→A⊗Bo.

  3. 3.

    If f is “almost everywhere” rigid or smooth, then the injection Ext lB (ΩB|A, Bn)→ExA(B, Bn) is an isomorphism.

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Literatur

  1. ARTIN, M.: On the Solutions of Analytic Equations. Inventiones math. 5, 277–291 (1968).

    Google Scholar 

  2. GRAUERT, H. u. H. KERNER: Deformationen von Singularitäten komplexer Räume. Math. Ann. 153, 236–260 (1964).

    Google Scholar 

  3. GRAUERT, H., REMMERT, R., RIEMENSCHNEIDER, O.: Analytische Stellenalgebren, 1. Aufl. Berlin-Heidelberg-New York: Springer 1971.

    Google Scholar 

  4. GROTHENDIECK, A.: Eléments de géométrie algébrique, chap. OIV. Publ. math. I.H.E.S. 20 (1964).

  5. GROTHENDIECK, A.: Techniques de construction en géométrie analytique, VI et VII. In: Sém. H. Cartan 1960/61.

  6. HOUZEL, C.: Géométrie analytique locale I–IV. In: Sém. H. Cartan 1960/61.

  7. KERNER, H.: Zur Theorie der Deformationen komplexer Räume. Math. Z. 103, 389–398 (1968).

    Google Scholar 

  8. KRAMM, B.: Deformationen von k-analytischen Abbildungskeimen. Diss. Frankfurt/M 1972.

  9. KRAMM, B., SCHUSTER, H. W.: Über Produkte starrer analytischer Raumkeime. In Vorbereitung.

  10. SCHLESSINGER, M.: Functors of Artin Rings. Trans. AMS 130, 208–222 (1968).

    Google Scholar 

  11. SCHUSTER, H.W.: Deformationen analytischer Algebren. Inventiones math. 6, 262–274 (1968)

    Google Scholar 

  12. SCHUSTER, H.W.: Infinitesimale Erweiterungen komplexer Räume. Comment. Math. Helv., Vol. 45, (3) 265–286 (1970)

    Google Scholar 

  13. STORCH, U.: Über starre analytische Algebren. Math. Ann. 179, 53–62 (1968).

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Seminar der Universität, Robert-Mayer-Straße 6-10, D 6000, Frankfurt/M.

    Bruno Kramm

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  1. Bruno Kramm
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Kramm, B. Über Deformationen von analytischen Abbildungskeimen. Manuscripta Math 10, 163–189 (1973). https://doi.org/10.1007/BF01475040

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  • DOI: https://doi.org/10.1007/BF01475040

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