Skip to main content
SpringerLink
Log in

Das zariskische Diskriminantenkriterium und die Fortsetzung von Derivationen

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let A be a reduced equidimensional local analytic algebra and let R⊂A be a regular local “parametrization” of A. Then the Zariski discriminant criterion can be stated as follows: If A is a simple extension of R, i.e. A=R[x] for a certain x, and if the (reduced) discriminant locus S in R of A is smooth, then A is “lipschitz-meromorphically” trivial along S; this means that every derivation of R leaving S invariant can be extended to the relative saturation ÃR of A over R.- In this paper quite generally (i.e. not only for the case of a simple extension) the following question is considered: Which conditions should a derivation of R satisfy in order that it leaves invariant the ring ÃR?

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literatur

  1. ABHYANKAR,S.: On the valuations centered in a local domain. Amer. Journ. Math.78, 321–348 (1956)

    Google Scholar 

  2. BÖGER,E.: Zur Theorie der Saturation bei analytischen Algebren. Math. Ann.211, 119–143 (1974)

    Google Scholar 

  3. BÖGER,E.: Über die Gleichheit von absoluter und relativer Lipschitz-Saturation bei analytischen Algebren. manuscripta math.16, 229–249 (1975)

    Google Scholar 

  4. BRIANÇON,J. GALLIGO,A., GRANGER,M.: Deformations equisingulières des germes de courbes gauches reduites. Departement de Mathématiques, Université de Nice, Decembre 1978

  5. Buchweitz,R.-O., GREUEL,G.-M.: The Milnor number and deformations of complex curve singularities. IHES, Bures-sur-Yvettes, Fevrier 1979

    Google Scholar 

  6. DRAPER,R., FISCHER,K.: Derivations into the integral closure. Department of Math., George Mason University, ca. 1979

    Google Scholar 

  7. LIPMAN,J.: Relative Lipschitz-Saturation. Amer. Journ. Math.97, 791–813 (1975)

    Google Scholar 

  8. PHAM, F., TEISSIER,B.: Fractions lipschitziennes d'une algèbre analytique complexe et saturation de Zariski. Centre de Mathém. de l'Ecole Polytechnique, Paris, Juni 1969, Nr. M 17.0669

    Google Scholar 

  9. REGEL,M., SCHEJA,G.: Fortsetzung von Derivationen bei zyklischen Erweiterungen. Vortrag in Reinhardsbrunn (1978), erscheint in Nova acta Leopoldina, Halle

  10. SCHEJA,G., STORCH,U.: Fortsetzung von Derivationen. Journ. of Algebra54, 353–365 (1978)

    Google Scholar 

  11. ZARISKI,O.: Studies in equisingularity I. Equivalent singularities of plane algebroid curves. Amer. Journ. Math.87, 507–536 (1965)

    Google Scholar 

  12. ZARISKI,O.: Studies in equisingularity II. Equisingulatity in codimension 1 (and characteristic zero). Amer. Journ. Math.87, 972–1006 (1965)

    Google Scholar 

  13. ZARISKI,O.: Studies in equisingularity III. Saturation of local rings and equisingularity. Amer. Journ. Math.90, 961–1023 (1968)

    Google Scholar 

  14. ZARISKI,O.: General theory of saturation and of saturated local rings III. Saturation in arbitrary dimension and, in particular, saturation of algebroid hypersurfaces. Amer. Journ. Math.97, 415–502 (1975)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ruhr-Universität Bochum, Universitätsstr. 150, D 4630, Bochum 1

    Erwin Böger

Authors
  1. Erwin Böger
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Böger, E. Das zariskische Diskriminantenkriterium und die Fortsetzung von Derivationen. Manuscripta Math 36, 67–81 (1981). https://doi.org/10.1007/BF01174813

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01174813

Search

Navigation