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Degree Functions in Local Rings

Published online by Cambridge University Press:  24 October 2008

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

Let Q be a local domain of dimension d with maximal ideal m and let q be an m-primary ideal. Then we define the degree function dq(x) to be the multiplicity of the ideal , where x; is a non-zero element of m. The degree function was introduced by Samuel (5) in the case where q = m. The function dq(x) satisfies the simple identity

The main purpose of this paper is to obtain a formula

where vi(x) denotes a discrete valuation centred on m (i.e. vi(x) ≥ 0 if x ∈ Q, vi(x) > 0 if x ∈ m) of the field of fractions K of Q. The valuations vi(x) are assumed to have the further property that their residue fields Ki have transcendence degree d − 1 over k = Q/m. The symbol di(q) denotes a non-negative integer associated with vi(x) and q which for fixed q is zero for all save a finite set of valuations vi(x).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 1 , January 1961 , pp. 1 - 7
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Nagata, M., The theory of multiplicities in general local rings. Proceedings of the International Symposium on Algebraic Number Theory (Tokyo-Nikko, 1955), 191–226.Google Scholar
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(3)Rees, D., A note on analytically unramified local rings. J. Lond. Math. Soc. (to appear).Google Scholar
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(5)Samuel, P., Sur les variétés algébroïdes. Ann. Inst. Fourier, 2 (1950), 147–60.CrossRefGoogle Scholar