Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T19:43:02.414Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction numbers for ideals of higher analytic spread

Published online by Cambridge University Press:  24 October 2008

Sam Huckaba
Sam Huckaba
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Let (R, M) be a commutative Noetherian local ring having an identity, and assume the residue field R/M is infinite. If I is an ideal in R, recall that an ideal J contained in I is called a reduction of I if JIn = In + 1 for some non-negative integer n. A reduction of J of I is called a minimal reduction of I if it does not properly contain another reduction of I. Reductions (and minimal reductions) were introduced and studied by Northcott and Rees[8]. If J is a reduction of I we define the reduction number of I with respect to J, denoted rJ(I), to be the smallest non-negative integer n such that JIn = In + 1 (note that rJ(I) = 0 if and only if J = I). The reduction number of I (sometimes referred to as the reduction exponent) is defined as r(I) = min{rj(I)|J ⊆ I is a minimal reduction of I}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 1 , July 1987 , pp. 49 - 57
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Eakin, P. and Sathaye, A.. Prestable ideals. J. Algebra 41 (1976), 439–454.CrossRefGoogle Scholar
[2] Huckaba, S.. Reduction numbers and ideals of analytic spread one. Thesis, Purdue University (1986).CrossRefGoogle Scholar
[3] Huckaba, S.. Reduction numbers for ideals of analytic spread one. J. Algebra. (To appear.)Google Scholar
[4] Huneke, C.. On the symmetric and Rees algebra of an ideal generated by a d-sequence. J. Algebra 62 (1980), 268–275.CrossRefGoogle Scholar
[5] Lipman, J.. Stable ideals and Arf rings. Amer. J. Math. 97 (1975), 791–813.CrossRefGoogle Scholar
[6] Lipman, J. and Tessier, B.. Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals. Michigan Math. J. 28 (1981), 97–116.CrossRefGoogle Scholar
[7] Micali, A.. Sur les algèbres universelles. Ann. Inst. Fourier, 14 (1964), 33–88.CrossRefGoogle Scholar
[8] Northcott, D. G. and Rees, D.. Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145–158.CrossRefGoogle Scholar
[9] Sally, J. D.. Tangent cones at Gorenstein singularities. Compositio Math. 40 (1980), 167–175.Google Scholar
[10] Sally, J. D.. Reductions, local cohomology and Hilbert functions of local rings. Commutative Algebra: Durham 1981, London Math. Soc. Lecture Note Series, vol. 72 (Cambridge University Press, 1982), 231–241.Google Scholar
[11] Sally, J. D.. Cohen-Macaulay local rings of embedding dimension e + d − 2. J. Algebra 83 (1983), 393–408.CrossRefGoogle Scholar
[12] Sally, J. D. and Vasconcelos, W.. Stable rings. J. Pure Appl. Algebra 4 (1974), 319–336.CrossRefGoogle Scholar
[13] Trung, Ngo Viet. Reduction exponent and degree bound for the defining equations of graded rings. Preprint (1986).CrossRefGoogle Scholar