Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-19T06:03:08.970Z Has data issue: false hasContentIssue false
  • English
  • Français

On equimultiplicity

Published online by Cambridge University Press:  24 October 2008

M. Herrmann  and
U. Orbanz
M. Herrmann
Affiliation:
University of Köln
U. Orbanz
Affiliation:
University of Köln
Get access Rights & Permissions [Opens in a new window]

Extract

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).

In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 2 , March 1982 , pp. 207 - 213
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Böger, E.Eine Verallgemeinerung eines Multiplizitätensatzes von D. Rees. J. of Algebra 12 (1969), 207–215.Google Scholar
(2)Grothendieck, A. and Dieudonné, J.Elements de géométrie algébrique, IV/3, Publ. IHES no. 28, Bures-sur-Yvettes 1966.Google Scholar
(3)Hironaka, H.Normal cones in analytic Whitney stratifications, Publ. IHES 36, 127–138, Bures-sur-Yvettes 1969.Google Scholar
(4)Herrmann, M. and Orbanz, U.Faserdimension von Aufblasungen lokaler Ringe und Äquimultiplizität. J. Math. Kyoto Univ. 20 (1980), 651–659.Google Scholar
(5)Herrman, M., Schmidt, R. und Vogel, W.Theorie der normalen Flachheit (Leipzig 1977).Google Scholar
(6)Lipman, J.Equimultiplicity, reduction, and blowing up. (Preprint 1980.)Google Scholar
(7)Matsumura, H.Commutative Algebra (New York 1970).Google Scholar
(8)Northcott, D. G.Lessons on rings, modules, and multiplicities (Cambridge University Press, 1968).CrossRefGoogle Scholar
(9)Northcott, D. G. and Rees, D.Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145–158.CrossRefGoogle Scholar
(10)Northcott, D. G. and Rees, D.A note on reductions of ideals with application to the generalized Hubert function. Proc. Cambridge Phil. Soc. 50 (1954), 353–359.Google Scholar
(11)Orbanz, U. Multiplicities and Hubert functions under blowing up, to appear in Manuscripta Mathematica.Google Scholar
(12)Ratliff, L. J.On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals (II). Amer. J. Math. 92 (1970), 99–144.CrossRefGoogle Scholar
(13)Ratliff, L. J.On quasi-unmixed local rings and the altitude formula. Amer. J. Math. 87 (1965), 278–284.Google Scholar
(14)Teissier, B.Résolution simultanée et cycles évanescents. Lecture Notes in Mathematics, no. 777 (Springer-Verlag, Berlin-Heidelberg-New York, 1980), pp. 82–146.Google Scholar