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The Hilbert function of two ideals

Published online by Cambridge University Press:  24 October 2008

P. B. Bhattacharya
P. B. Bhattacharya
Affiliation:
24 Earl Street Cambridge
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Extract

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 3 , July 1957 , pp. 568 - 575
Copyright
Copyright © Cambridge Philosophical Society 1957

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References

REFERENCES

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(3)Rees, D.Two classical theorems of ideal theory. Proc. Comb. Phil. Soc. 52 (1956), 155–7.CrossRefGoogle Scholar
(4)Samuel, P.Algèbre locale (Paris, 1953).Google Scholar
(5)Van Der Waerden, B. L.On Hilbert's function, series of composition of ideals and a generalization of the theorem of Bezout. Proc. K. Akad. Wet. Amst. 31 (1928), 749–70.Google Scholar