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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

Extract

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
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), 6780.CrossRefGoogle Scholar
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(3)Samuel, P. La notion de multiplicité en algébre et en géométrie. Thesis (Paris, 1951).Google Scholar
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