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Ratliff-Rush ideal and reduction numbers. (English) Zbl 1391.13019
Let \((R,\mathfrak{m})\) be a Cohen-Macaulay local ring of positive dimension \(d\) and infinite residue field. Let \(I\) be an \(\mathfrak{m}\)-primary ideal and \(J\) a minimal reduction of \(I\). The author shows that \(\widetilde{r_{J}(I)} \leq r_{J}(I)\). This answer to a question that made by M. E. Rossi and I. Swanson [Contemp. Math. 331, 313–328 (2003; Zbl 1089.13501), Question 4.6]. The author shows that every minimal reduction of \(I\) can be generated by a tame supercial sequence of \(I\) and uses inductive arguments for to show the most result.

MSC:
13C14 Cohen-Macaulay modules
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Software:
Macaulay2
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References:
[1] Bruns, W.; Herzog, J., Cohen-Macaulay rings, (1998), Cambridge University Press, Cambridge, UK
[2] Caruth, A., A modules relation associated with the Artin-Rees lemma, Commun. Algebra, 21, 3545-3555, (1993) · Zbl 0787.16019
[3] Dinh, T. T.; Rossi, M. E.; Trung, N. V., Castelnuovo-Mumford Regularity and Ratliff-Rush Closure
[4] Elias, J., On the computation of the ratliff–rush closure, J. Symbolic Comput., 37, 717-725, (2004) · Zbl 1137.13310
[5] Grayson, D. R.; Stillman, M. E., Macaulay 2, a Software System for Research in Algebraic Geometry
[6] Heinzer, W.; Lantz, D.; Shah, K., The ratliff–rush ideals in a Noetherian ring, Commun. Algebra, 20, 591-622, (1992) · Zbl 0747.13002
[7] Huckaba, S.; Marley, T., Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc., 56, 64-76, (1997) · Zbl 0910.13008
[8] Huneke, K. C.; Swanson, I., Integral Closure of Ideals, Rings, and Modules, (2006), Cambridge University Press, Cambridge, UK · Zbl 1117.13001
[9] Kaplansky, I., R-sequences and homological dimension, Nagoya Math. J., 20, 195-199, (1962) · Zbl 0106.25702
[10] Marley, T., The reduction number of an ideal and the local cohomology of the associated graded ring, Proc. Amer. Math. Soc., 117, 335-341, (1993) · Zbl 0772.13006
[11] McAdam, S., Asymptotic Prime Divisors, (1983), Springer-Verlag, Berlin · Zbl 0529.13001
[12] Northcott, D. G.; Rees, D., Reduction of ideals in local rings, Proc. Cambridge Philos. Soc., 50, 145-158, (1954) · Zbl 0057.02601
[13] Puthenpurakal, T. J., Ratliff–rush filtration, regularity and depth of higher associated graded modules I, J. Pure Appl. Algebra, 208, 159-176, (2007) · Zbl 1106.13003
[14] Ratliff, L. J.; Rush, D., Two notes on reductions of ideals, Indiana Univ. Math. J., 27, 929-934, (1978) · Zbl 0368.13003
[15] Rossi, M. E.; Swanson, I., Notes on the behaviour of the ratliff–rush filtration, Contemp. Math., 331, 313-328, (2003) · Zbl 1089.13501
[16] Swanson, I., A note on analytic spread, Commun. Algebra, 22, 407-411, (1994) · Zbl 0796.13006
[17] Tajarod, R.; Zakeri, H., Comparison of certain complexes of modules of generalized fractions and čech complexes, Commun. Algebra, 35, 4032-4041, (2007) · Zbl 1131.13014
[18] Trung, N. V., Reduction exponent and dgree bound for the defining equations of graded rings, Proc. Amer. Math. Soc., 101, 229-236, (1987) · Zbl 0641.13016
[19] Trung, N. V., The Castelnuovo regularity of the Rees algebra and the associated graded ring, Trans. Amer. Math. Soc., 35, 2813-2832, (1998) · Zbl 0899.13002
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