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A note on the coefficients of Hilbert characteristic functions in semi-regular local rings

Published online by Cambridge University Press:  24 October 2008

Masao Narita
Masao Narita
Affiliation:
Department of Mathematics, International Christian University, Mitaka, Tokyo, Japan
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Extract

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:

where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 2 , April 1963 , pp. 269 - 275
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Northcott, D. G., The coefficients of the abstract Hilbert functions. J. London Math. Soc. 35 (1960), 209–214.Google Scholar
(2)Kirby, D., The defect of a one-dimensional local ring. Mathematika, 6 (1959), 91–99.Google Scholar