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.
@article{bwmeta1.element.bwnjournal-article-cmv75z2p167bwm,
author = {Krzysztof Nowak},
title = {On the intersection multiplicity of images under an etale morphism},
journal = {Colloquium Mathematicae},
volume = {78},
year = {1998},
pages = {167-174},
zbl = {0927.14010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv75z2p167bwm}
}
Nowak, Krzysztof. On the intersection multiplicity of images under an etale morphism. Colloquium Mathematicae, Tome 78 (1998) pp. 167-174. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv75z2p167bwm/
[000] [1] A. B. Altman and S. L. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math. 146, Springer, 1970. | Zbl 0215.37201
[001] [2] C. Chevalley, Intersections of algebraic and algebroid varieties, Trans. Amer. Math. Soc. 57 (1945), 1-85. | Zbl 0063.00841
[002] [3] W. Fulton, Intersection Theory, Springer, Berlin, 1984. | Zbl 0541.14005
[003] [4] A. Grothendieck and J. A. Dieudonné, Éléments de Géométrie Algébrique, Springer, Berlin, 1971.
[004] [5] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970.
[005] [6] D. Mumford, Algebraic Geometry I. Complex projective varieties, Springer, Berlin, 1976. | Zbl 0356.14002
[006] [7] M. Nagata, Local Rings, Interscience Publishers, New York, 1962.
[007] [8] K. J. Nowak, Flat morphisms between regular varieties, Univ. Iagel. Acta Math. 35 (1997), 243-246. | Zbl 0967.14014
[008] [9] K. J. Nowak, A proof of the criterion for multiplicity one, ibid., 247-250.
[009] [10] P. Samuel, Algèbre locale, Mémorial Sci. Math. 123, Gauthier-Villars, Paris, 1953.
[010] [11] P. Samuel, Méthodes d'algèbre abstraite en géométrie algébrique, Ergeb. Math. Grenzgeb. 4, Springer, Berlin, 1955. | Zbl 0067.38904
[011] [12] F. Severi, Über die Grundlagen der algebraischen Geometrie, Abh. Math. Sem. Hamburg Univ. 9 (1933), 335-364. | Zbl 59.0604.02
[012] [13] A. Weil, Foundations of Algebraic Geometry, Amer. Math. Soc. Colloq. Publ. 29, 1962. | Zbl 0168.18701
[013] [14] O. Zariski and P. Samuel, Commutative Algebra, Vols. I and II, Van Nostrand, Princeton, 1958, 1960. | Zbl 0081.26501