Download PDF - On the intersection multiplicity of images under an etale morphismAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On the intersection multiplicity of images under an etale morphism

Authors 1

Affiliations

  1. Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Abstract

We present a formula for the intersection multiplicity of the images of two subvarieties under an etale morphism between smooth varieties over a field k. It is a generalization of Fulton's Example 8.2.5 from [3], where a strong additional assumption has been imposed. In a special case where the base field k is algebraically closed and a proper component of the intersection is a closed point, intersection multiplicity is an invariant of etale morphisms. This corresponds with analytic geometry where intersection multiplicity is an invariant of local biholomorphisms.

Keywords

ENG
intersection multiplicity, multiplicity of ideals in a semilocal ring, etale morphisms, unramified morphisms, algebraic varieties

Bibliography

  1. A. B. Altman and S. L. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math. 146, Springer, 1970.
  2. C. Chevalley, Intersections of algebraic and algebroid varieties, Trans. Amer. Math. Soc. 57 (1945), 1-85.
  3. W. Fulton, Intersection Theory, Springer, Berlin, 1984.
  4. A. Grothendieck and J. A. Dieudonné, Éléments de Géométrie Algébrique, Springer, Berlin, 1971.
  5. H. Matsumura, Commutative Algebra, Benjamin, New York, 1970.
  6. D. Mumford, Algebraic Geometry I. Complex projective varieties, Springer, Berlin, 1976.
  7. M. Nagata, Local Rings, Interscience Publishers, New York, 1962.
  8. K. J. Nowak, Flat morphisms between regular varieties, Univ. Iagel. Acta Math. 35 (1997), 243-246.
  9. K. J. Nowak, A proof of the criterion for multiplicity one, ibid., 247-250.
  10. P. Samuel, Algèbre locale, Mémorial Sci. Math. 123, Gauthier-Villars, Paris, 1953.
  11. P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1955.
  12. F. Severi, Über die Grundlagen der algebraischen Geometrie, Abh. Math. Sem. Hamburg Univ. 9 (1933), 335-364.
  13. A. Weil, Foundations of Algebraic Geometry, Amer. Math. Soc. Colloq. Publ. 29, 1962.
  14. O. Zariski and P. Samuel, Commutative Algebra, Vols. I and II, Van Nostrand, Princeton, 1958, 1960.
Colloquium Mathematicum
1998/75/2
Export to PBN, Opens in new tab
Pages:
167-174
Main language of publication
English
Received
1997-03-28
Published
1998
Exact and natural sciences