On system of parameters, local intersection multiplicity and Bezout’s theorem
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- by Eduard Bod’a and Wolfgang Vogel PDF
- Proc. Amer. Math. Soc. 78 (1980), 1-7 Request permission
Abstract:
This paper provides effective methods for computing the local intersection multiplicity as the length of a well-defined ideal (see Theorem and Proposition 1). There are other ways of obtaining such an ideal (see [2], [9], [12], [18]) but ours is simpler because of our use of reducing systems of parameters. Applying these ideal theoretic methods we will give a new and simple proof of Bezout’s Theorem (see §4). Hence this proof again provides the connection between the different viewpoints which are treated in the work of Lasker-Macaulay-Gröbner and Severi-van der Waerden-Weil concerning the multiplicity theory.References
- Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
- Eduard Bod’a, Zur Berechnung von Schnittmultiplizitäten durch Längen, Math. Slovaca 28 (1978), no. 2, 173–179 (German, with English and Russian summaries). MR 526856
- Peter Schenzel, Ngô Viêt Trung, and Nguyễn Tụ’ Cu’ò’ng, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57–73 (German). MR 517641, DOI 10.1002/mana.19780850106 W. Gröbner, Moderne algebraische Geometrie, Springer-Verlag, Wien und Innsbruck, 1949.
- F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968. MR 0231816, DOI 10.1017/CBO9780511565922
- Bodo Renschuch and Wolfgang Vogel, Zum Nachweis arithmetischer Cohen-Macaulay Varietäten, Monatsh. Math. 85 (1978), no. 3, 201–210 (German, with English summary). MR 498618, DOI 10.1007/BF01534864
- Manfred Reufel, Beiträge zur Multiplizitäten- und Spezialisierungstheorie. I, Festband anlässlich des 65. Geburtstages von Ernst Peschl, Gesellsch. Math. Datenverarbeitung Bonn, Ber. No. 57, Gesellsch. Math. Datenverarbeitung, Bonn, 1972, pp. 177–201 (German). MR 0419436
- P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 4, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (French). MR 0072531 J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin and New York, 1965.
- Jürgen Stückrad and Wolfgang Vogel, Ein Korrekturglied in der Multiplizitätstheorie von D. G. Northcott und Anwendungen, Monatsh. Math. 76 (1972), 264–271 (German). MR 306178, DOI 10.1007/BF01322931
- Jürgen Stückrad and Wolfgang Vogel, Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie, J. Math. Kyoto Univ. 13 (1973), 513–528 (German). MR 335504, DOI 10.1215/kjm/1250523322
- Jürgen Stückrad and Wolfgang Vogel, Über das Amsterdamer Programm von W. Gröbner und Buchsbaum Varietäten, Monatsh. Math. 78 (1974), 433–445 (German). MR 498619, DOI 10.1007/BF01295487
- Jürgen Stückrad and Wolfgang Vogel, Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), no. 4, 727–746. MR 509072, DOI 10.2307/2373908
- Bartel L. van der Waerden, Eine Verallgemeinerung des Bézoutschen Theorems, Math. Ann. 99 (1928), no. 1, 497–541 (German). MR 1512464, DOI 10.1007/BF01459111
- André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
- D. J. Wright, A characterisation of multiplicity, Monatsh. Math. 79 (1975), 165–167. MR 429984, DOI 10.1007/BF01585674 O. Zariski and P. Samuel, Commutative algebra. Vol. II, Princeton Univ. Press, Princeton, N. J., 1962.
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 1-7
- MSC: Primary 14B99; Secondary 13H15
- DOI: https://doi.org/10.1090/S0002-9939-1980-0548071-3
- MathSciNet review: 548071