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A note on reductions of ideals with an application to the generalized Hilbert function

Published online by Cambridge University Press:  24 October 2008

D. G. Northcott  and
D. Rees
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1. Throughout this note Q will denote a local ring, m will denote its maximal ideal, q will denote a primary ideal belonging to m and k will denote the residue field Q/m. It will not be assumed that k is infinite, but we shall suppose that Q and k both have the same characteristic. Now let υ1, υ2 …,υd be a system of parameters contained in q, so that d = dim Q; then according to the definition given in (2) the ideal (υl υ2,…, υd) is a reduction of q if (υ1 υ2, …, υd) qm = qm+1 for at least one value of m. The use of the concept lies in the fact that such a reduction is, in a certain sense, a very good approximation to q itself; but the notion does, however, suffer from a minor disadvantage in that, if k is finite, q need not have any reductions. In §3 we shall generalize the notion of a reduction in such a way that we overcome this difficulty, and in such a way that the results concerning reductions obtained in (2) acquire some useful extensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 3 , July 1954 , pp. 353 - 359
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1)Northcott, D. G.Hilbert's function in a local ring. Quart. J. Math. (2), 4 (1953), 67–80.CrossRefGoogle Scholar
(2)Northcott, D. G. and Rees, D.Reductions of ideals in local rings. Proc. Camb. phil. Soc. 50 (1954), 145.CrossRefGoogle Scholar
(3)Samuel, P. La notion de multiplicité en algébre et en géométrie. Thesis (Paris, 1951).Google Scholar
(4)Samuel, P.Algèbre locale. Mémor. Sci. Math. 123 (1953).Google Scholar