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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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(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