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Hilbert function and fractional powers

Part of: Local rings and semilocal rings

Published online by Cambridge University Press:  26 February 2010

Juan Elias
Juan Elias
Affiliation:
Departament d'Algebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail:elias@mat.ub.es
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Extract

Let R be a Noetherian local ring with maximal ideal m and lull ring of fractions Q. In this paper we consider a numerical function EHI: ℤ → ℤ, where I is an m-primary ideal of R, that coincides with the Hilbert function HI for positive values and that takes account of the fractional powers of I for negative values. We focus our attention on the one-dimensional case. Among other results we characterize one-dimensional Gorenstein local rings by means of the symmetry of EHR in Theorem 2.1, we show that the extended Hilbert function is not determined by the Hilbert function in Example 2.2. and we generalize to m-primary ideals the upper bound for e1(m) given by Matlis for the maximal ideal.

MSC classification

Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Type
Research Article
Information
Mathematika , Volume 49 , Issue 1-2 , June 2002 , pp. 71 - 75
Copyright
Copyright © University College London 2002

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References

1.Bass, H.. On the ubiquity of Gorenstein rings. Math. Z., 82 (1963), 8–28.CrossRefGoogle Scholar
2.Elias, J.. Characterization of the Hilbert-Samuel polynomials of curve singularities. Compositio Math., 74 (1990). 135–155.Google Scholar
3.Elias, J.. The Hilbcrt function of m-primary ideals. Preprint (2001).Google Scholar
4.Elias, J.. On the deep structure of the blowing-up of curve singularities. Math. Proc. Camb. Phil. Soc., 131 (2001). 227–240.Google Scholar
5.Kirby, D.. The defect of a one-dimensional local ring. Mathematika, 6 (1959), 91–97.CrossRefGoogle Scholar
6.Kirby, D.. The reduction number of a one-dimensional local ring, J. London Math. Sot., 10 (1975). 471–481.CrossRefGoogle Scholar
7.Lipman, J.. Stable ideals and Arf rings. Amer. J. Math., 93 (1971), 649–685.CrossRefGoogle Scholar
8.Matlis, E.. 1-dimensional Cohen-Macaulay rings. L.N.M. Springer Verlag, 327 (1977).Google Scholar
9.Northcott, D.G.. The reduction number of a one-dimensional local ring. Mathematika, 6 (1959), 87–90.Google Scholar