A bound on the reduction number of a primary ideal
Author:
M. E. Rossi
Journal:
Proc. Amer. Math. Soc. 128 (2000), 13251332
MSC (1991):
Primary 14M05; Secondary 13H10
Published electronically:
October 5, 1999
MathSciNet review:
1670423
Fulltext PDF Free Access
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Abstract: Let be a local ring of positive dimension and let be an primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is CohenMacaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.
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 [CPV]
 A. Corso, C. Polini, M. Vaz Pinto, Sally Modules and associated graded rings, Communications in Algebra 26 (8) (1998), 26892708. CMP 98:15
 [E]
 J. Elias, On the depth of the tangent cone and the growth of the Hilbert function, to appear in Trans. Amer. Math. Soc. CMP 98:07
 [GR]
 A. Guerrieri, M. E. Rossi, Hilbert coefficients for Hilbert filtrations, J. Algebra 199 (1998), 4061. MR 98i:13027
 [HM]
 S. Huckaba, T. Marley, Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc. 56 (1997), 6476. MR 98i:13028
 [H]
 C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293318. MR 89b:13037
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 S. Itoh, Hilbert coefficients of integrally closed ideals, J. Algebra 176 (1995), 638652. MR 96g:13019
 [N]
 D. G. Northcott, A note on the coefficients of the abstract Hilbert function, J. London Math. Soc. 35 (1960), 209214. MR 22:1599
 [O]
 A. Ooishi, genera and sectional genera of commutative rings, Hiroshima Math. J. 17 (1987), 361372. MR 89f:13033
 [RR]
 L. J. Ratliff, D. Rush, Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), 929934. MR 58:22034
 [R]
 M. E. Rossi, Primary ideals with good associated graded ring, to appear in J. of Pure and Applied Algebra.
 [RV]
 M. E. Rossi, G. Valla, A conjecture of J. Sally, Communications in Algebra 24 (13) (1996), 42494261. MR 97j:13021
 [S]
 I. Swanson, A note on the analytic spread, Communications in Algebra 22, 2 (1994), 407411. MR 95b:13007
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 J. Sally, Bounds for numbers of generators for CohenMacaulay ideals, Pacific J. Math. 63 (1976), 517520. MR 53:13208
 [S2]
 J. Sally, CohenMacaulay local ring of embedding dimension , J. Algebra 83 (1983), 325333. MR 85c:13017
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 J. Sally, Hilbert coefficients and reduction number 2, J. Algebraic Geom. 1 (1992), 325533. MR 93b:13026
 [V]
 G. Valla, On form rings which are CohenMacaulay, J. Algebra 58 (1979), 247250. MR 80h:13025
 [VV]
 P. Valabrega, G. Valla, Form rings and regular sequences, Nagoya Math. J. 72(2) (1978), 475481. MR 80d:14010
 [VW]
 W. Vasconcelos, Cohomological Degrees of graded modules, Six Lectures on Commutative algebra, Progress in Math. 166 , Birkhauser, Boston (1998), 345392. CMP 99:02
 [W]
 H. Wang, On CohenMacaulay local rings with embedding dimension , J. Algebra 190 (1997), 226240. MR 98d:13027
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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/S0002993999053939
PII:
S 00029939(99)053939
Keywords:
CohenMacaulay 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
