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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

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