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Note on analytic spread and asymptotic sequences

Published online by Cambridge University Press:  24 October 2008

L. J. Ratliff Jr
L. J. Ratliff Jr
Affiliation:
University of California, Riverside
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Extract

Since the foundational paper (10) by Northcott and Rees in 1954 there have been quite a few papers concerning reductions of ideals and the analytic spread of an ideal. One particular line of investigation concerning the analytic spread l(I) of an ideal I in a local ring (R, M) was begun in 1972 by Burch in (5), where it was shown that l(I) ≤ altitude R – min (grade R/In; n ≥ 1). This result was sharpened in 1980–81 by Brodmann in three papers, (2, 3, 4). Therein he showed that the sets {grade R/In; n ≥ 1} and {grade In−1/In; n ≥ 1} stabilize for all large n, and calling the stable values t and t*, respectively, it holds that t ≤ t* and l(I) ≤ altitude R – t* when I is not nilpotent. He then gave a case (involving R being quasi-unmixed) when equality holds. In 1981 in (20) Rees used two new approaches to Burch's inequality, and he proved two nice results which may both be stated as: l(I) ≤ altitude R – s(I) with equality holding when R is quasi-unmixed; here, s(I) = min {height P; P is a minimal prime divisor of (M, u) R[tI, u]}– 1 (in the first theorem), and s(I) is the length of a maximal asymptotic sequence over I (in the second theorem).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 1 , January 1983 , pp. 49 - 55
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

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