On the intersection multiplicity of images under an etale morphism

Krzysztof Nowak

Colloquium Mathematicae (1998)

  • Volume: 75, Issue: 2, page 167-174
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Nowak, Krzysztof. "On the intersection multiplicity of images under an etale morphism." Colloquium Mathematicae 75.2 (1998): 167-174. <http://eudml.org/doc/210535>.

@article{Nowak1998,
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.},
author = {Nowak, Krzysztof},
journal = {Colloquium Mathematicae},
keywords = {intersection multiplicity; multiplicity of ideals in a semilocal ring; etale morphisms; unramified morphisms; algebraic varieties; étale morphism},
language = {eng},
number = {2},
pages = {167-174},
title = {On the intersection multiplicity of images under an etale morphism},
url = {http://eudml.org/doc/210535},
volume = {75},
year = {1998},
}

TY - JOUR
AU - Nowak, Krzysztof
TI - On the intersection multiplicity of images under an etale morphism
JO - Colloquium Mathematicae
PY - 1998
VL - 75
IS - 2
SP - 167
EP - 174
AB - 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.
LA - eng
KW - intersection multiplicity; multiplicity of ideals in a semilocal ring; etale morphisms; unramified morphisms; algebraic varieties; étale morphism
UR - http://eudml.org/doc/210535
ER -

References

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  1. [1] A. B. Altman and S. L. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math. 146, Springer, 1970. Zbl0215.37201
  2. [2] C. Chevalley, Intersections of algebraic and algebroid varieties, Trans. Amer. Math. Soc. 57 (1945), 1-85. Zbl0063.00841
  3. [3] W. Fulton, Intersection Theory, Springer, Berlin, 1984. Zbl0541.14005
  4. [4] A. Grothendieck and J. A. Dieudonné, Éléments de Géométrie Algébrique, Springer, Berlin, 1971. 
  5. [5] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. 
  6. [6] D. Mumford, Algebraic Geometry I. Complex projective varieties, Springer, Berlin, 1976. Zbl0356.14002
  7. [7] M. Nagata, Local Rings, Interscience Publishers, New York, 1962. 
  8. [8] K. J. Nowak, Flat morphisms between regular varieties, Univ. Iagel. Acta Math. 35 (1997), 243-246. Zbl0967.14014
  9. [9] K. J. Nowak, A proof of the criterion for multiplicity one, ibid., 247-250. 
  10. [10] P. Samuel, Algèbre locale, Mémorial Sci. Math. 123, Gauthier-Villars, Paris, 1953. 
  11. [11] P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1955. Zbl0067.38904
  12. [12] F. Severi, Über die Grundlagen der algebraischen Geometrie, Abh. Math. Sem. Hamburg Univ. 9 (1933), 335-364. Zbl59.0604.02
  13. [13] A. Weil, Foundations of Algebraic Geometry, Amer. Math. Soc. Colloq. Publ. 29, 1962. Zbl0168.18701
  14. [14] O. Zariski and P. Samuel, Commutative Algebra, Vols. I and II, Van Nostrand, Princeton, 1958, 1960. Zbl0081.26501

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