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Hilbert-Samuel function and Grothendieck group

Published online by Cambridge University Press:  20 January 2009

Koji Nishida
Koji Nishida
Affiliation:
Department of Mathematics and Informatics, Graduate School of Science and Technology, Chiba University, Yayoi-cho 1–33, Inage-ku, Chiba-shi 263, Japan
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Abstract

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Let (A, m) be a Noetherian local ring such that the residue field A/m is infinite. Let I be arbitrary ideal in A, and M a finitely generated A-module. We denote by ℓ(I, M) the Krull dimension of the graded module ⊕n≥0InM/mInM over the associated graded ring of I. Notice that ℓ(I, A) is just the analytic spread of I. In this paper, we define, for 0 ≤ i ≤ ℓ = ℓ(I, M), certain elements ei(I, M) in the Grothendieck group K0(A/I) that suitably generalize the notion of the coefficients of Hilbert polynomial for m-primary ideals. In particular, we show that the top term eℓ (I, M), which is denoted by eI(M), enjoys the same properties as the ordinary multiplicity of M with respect to an m-primary ideal.

Keywords

Hilbert-Samuel functionGrothendieck groupmultiplicityanalytic spreadregular local ringGorenstein local ring
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 73 - 94
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Auslander, M. and Buchsbaum, D., Codimension and multiplicity, Ann. Math. 68 (1958), 625–657.CrossRefGoogle Scholar
2.Bourbaki, N., Algèbre commutative (Hermann, Paris, 1961–1965).Google Scholar
3.Cowsik, R. and Nori, S., On the fibres of blowing up, J. Indian Math. 40 (1976), 217–222.Google Scholar
4.Fraser, M., Multiplicities and Grothendieck groups, Trans. Am. Math. Soc. 136 (1969), 77–92.Google Scholar
5.Goto, S. and Shimoda, Y., On the Gorensteinness of Rees and form rings of almost complete intersections, Nagoya Math. J. 92 (1983), 69–88.Google Scholar
6.Herrmann, M., Ikeda, S. and Orbanz, U., Equimultiplicity and blowing-up (Springer, 1988).CrossRefGoogle Scholar
7.Huneke, C., The theory of d-sequences and powers of ideals, Adv. Math. 46 (1982), 249–279.Google Scholar
8.Huneke, C., On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. Algebra. 62 (1980), 268–275.Google Scholar
9.Herzog, J. and Kunz, E., Der kanonishe Modul eines Cohen–Macaulay-Rings, Lecture Notes in Mathematics, vol. 238 (Springer, 1971).CrossRefGoogle Scholar
10.Lipman, J., Equimultiplicity, reduction, and blowing up, Lecture Notes in Pure and Applied Mathematics, vol. 68, pp. 111–147 (Dekker, New York, 1982).Google Scholar
11.Nagata, M., Local rings (Interscience, 1962).Google Scholar
12.Northcott, D. G. and Rees, D., Reductions of ideals in local rings, Proc. Camb. Phil. Soc. 50 (1954), 145–158.Google Scholar
13.Samuel, P., La notion de multiplicité en algèbre et en géométrie algébrique, J. Math. Pure Appl. 30 (1951), 159–274.Google Scholar
14.Serre, J. P., Algèbre locale-multiplicités, Lecture Notes in Mathematics, vol. 11 (Springer, 1965).Google Scholar
15.Smoke, W., Dimension and multiplicity for graded algebras, J. Algebra 21 (1972), 149–173.Google Scholar
16.Stückrad, J., Grothendieck–Gruppen abelscher Kategorien und Multiplizitäten, Math. Nachr. 62 (1974), 5–26.Google Scholar
17.Yoshino, Y., Cohen–Macaulay modules over Cohen–Macaulay rings, Mathematical Society Lecture Note Series, vol. 146 (1990).Google Scholar