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a-Transforms of Local Rings and a Theorem on Multiplicities of Ideals

Published online by Cambridge University Press:  24 October 2008

D. Rees
D. Rees
Affiliation:
Department of MathematicsUniversity of Exeter
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Extract

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A. A reduction of a is an ideal b contained in a which satisfies ar+1 = arb for all sufficiently large r. This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m, and that a is m-primary. It is well known (Samuel (10)) that the length of the ideal an is, for large values of n equal to Pa(n) where Pa(n) is a polynomial in n whose degree d is equal to the dimension of A. If we write the coefficient of nd in Pa(n) in the form e(a)/d!, e(a) is a positive integer termed the multiplicity of a. If now b is a reduction of a, then b is also m-primary, and e(b) = e(a).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 1 , January 1961 , pp. 8 - 17
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Bhattacharya, P. B., The Hilbert function of two ideals. Proc. Camb. Phil. Soc. 53 (1957), 568–75.CrossRefGoogle Scholar
(2)Lech, C., On the associativity formula for multiplicities. Ark. Mat. 3 (1956), 301–14.CrossRefGoogle Scholar
(3)Nagata, M., On the chain problem of prime ideals. Nagoya Math. J. 10 (1956), 51–64.CrossRefGoogle Scholar
(4)Northcott, D. G., On homogeneous ideals. Proc. Glasgow Math. Ass. 2 (1955), 105–11.CrossRefGoogle Scholar
(5)Northcott, D. G., and Rees, D., Reductions of ideals in local rings. Proc. Camb. Phil. Soc. 50 (1954), 145–58.CrossRefGoogle Scholar
(6)Northcott, D. G., and Rees, D., A note on the reduction of ideals with an application to the generalized Hilbert function. Proc. Camb. Phil. Soc. 50 (1954), 353–59.CrossRefGoogle Scholar
(7)Rees, D., Valuations associated with ideals. II. J. Lond. Math. Soc. 31 (1956), 221–8.CrossRefGoogle Scholar
(8)Rees, D., Valuations associated with alocal ring. II. J. Lond. Math. Soc. 31 (1956), 228–35.CrossRefGoogle Scholar
(9)Rees, D., A note on form rings and ideals. Mathematika, 4 (1957), 51–60.CrossRefGoogle Scholar
(10)Samuel, P., Algébre locale. Mem. Sci. Math. 123; Paris, 1953.Google Scholar