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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A bound on the reduction number
of a primary ideal


Author: M. E. Rossi
Journal: Proc. Amer. Math. Soc. 128 (2000), 1325-1332
MSC (1991): Primary 14M05; Secondary 13H10
Published electronically: October 5, 1999
MathSciNet review: 1670423
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Abstract: Let $ ( A,{\cal M})$ be a local ring of positive dimension $d $ and let $ I $ be an $ \cal M $-primary ideal. We denote the reduction number of $I$ by $r(I)$, which is the smallest integer $r$ such that $I^{r+1}=JI^r$ for some reduction $J$ of $I.$ In this paper we give an upper bound on $ r(I) $ in terms of numerical invariants which are related with the Hilbert coefficients of $I $ when $A $ is Cohen-Macaulay. If $ d=1 $, it is known that $ r(I) \le e(I) -1 $ where $ e(I) $ denotes the multiplicity of $I. $ If $ d \le 2, $ in Corollary 1.5 we prove $ r(I) \le e_1(I) - e(I) + \lambda (A/I) + 1 $ where $e_1(I) $ is the first Hilbert coefficient of $I.$ From this bound several results follow. Theorem 1.3 gives an upper bound on $ r(I)$ in a more general setting.


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Additional Information

M. E. Rossi
Affiliation: Dipartimento di Matematica, Universita’ di Genova, Via Dodecaneso 35, 16146- Genova, Italy
Email: rossim@dima.unige.it

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05393-9
PII: S 0002-9939(99)05393-9
Keywords: Cohen-Macaulay local ring, primary ideals, reduction number, minimal reduction, associated graded ring, Hilbert function, Hilbert coefficients
Received by editor(s): July 6, 1998
Published electronically: October 5, 1999
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society