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Castelnuovo–Mumford regularity and Degree of nilpotency

Published online by Cambridge University Press:  01 May 2007

CAO HUY LINH
CAO HUY LINH*
Affiliation:
Department of Mathematics, College of Education, Hue University, 34 Le Loi, Hue City, Vietnam.
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Abstract

In this paper we show that the Castelnuovo–Mumford regularity of the associated graded module with respect to an m-primary ideal I is effectively bounded by the degree of nilpotency of I. From this it follows that there are only a finite number of Hilbert-Samuel functions for ideals with fixed degree of nilpotency.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 429 - 437
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Andrews, D. G. and McIntyre, M. E.. On wave-action and its relatives. J. Fluid Mech. 89 (1978), 647–664.CrossRefGoogle Scholar
[2]Brodmann, M. and Sharp, R. Y.. Local cohomology - An Algebraic Introduction with Geometric Applications (Cambridge University Press, 1998).CrossRefGoogle Scholar
[3]Doering, L. R., Gunston, T. and Vasconcelos, W.. Cohomological degrees and Hilbert functions of graded modules. Amer. J. Math. 120 (1998), 493–504.CrossRefGoogle Scholar
[4]Huckaba, S.. A d-dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients. Proc. Amer. Math. Soc. 124 (1996), 1393–1401.CrossRefGoogle Scholar
[5]Linh, C. H.. Upper bound for Castelnuovo–Mumford regularity of associated graded modules. Comm. Algebra 33 (6) (2005), 1817–1831.CrossRefGoogle Scholar
[6]Marley, T.. The coefficients of the Hilbert polynomial and the reduction number of an ideal. J. London Math. Soc. (2) 40 (1989), 1–8.CrossRefGoogle Scholar
[7]Ooishi, A.. Genera and arithmetic genera of commutative rings. Hiroshima Math. J. 17 (1987), 47–66.Google Scholar
[8]Rossi, M. E., Trung, N. V. and Valla, G.. Castelnuovo–Mumford regularity and extended degree. Trans. Amer. Math. Soc. 355 (2003), no. 5, 1773–1786.CrossRefGoogle Scholar
[9]Rossi, M. E., Trung, N. V. and Valla, G.. Castelnuovo–Mumford regularity and finiteness of Hilbert functions, In: Corso, A. et al. (eds), Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects. Lecture Notes Pure Appl. Math. 244, (CRC Press, 2005).Google Scholar
[10]Rossi, E., Valla, G. and Vasconcelos, W.. Maximal Hilbert functions. Results in Math. 39 (2001) 99–114.CrossRefGoogle Scholar
[11]Schwartz, N.. Bounds for the postulation numbers of Hilbert functions. J. Algebra 193 (1997), 581–615.CrossRefGoogle Scholar
[12]Srinivas, V. and Trivedi, V.. On the Hilbert functions of a Cohen–Macaulay ring. J. Algebraic Geom. 6 (1997), 733–751.Google Scholar
[13]Trivedi, V.. Hilbert functions, Castelnuovo–Mumford regularity and uniform Artin-Rees numbers. Manuscripta Math. 94 (1997), 543–558.CrossRefGoogle Scholar
[14]Trivedi, V.. Finiteness of Hilbert functions for generalized Cohen–Macaulay modules. Comm. Algebra. 29 (2) (2001), 805–813.CrossRefGoogle Scholar
[15]Trung, N. V.. Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (1987), 229–234.CrossRefGoogle Scholar
[16]Trung, N. V.. The Castelnuovo–Mumford regularity of the Rees algebra and the associated graded ring. Trans. Amer. Math. Soc. 350 (1998), 2813–2832.CrossRefGoogle Scholar
[17]Vasconcelos, W.. The homological degree of module. Trans. Amer. Math. Soc. 350 (1998), 1167–1179.CrossRefGoogle Scholar
[18]Vasconcelos, W.. Cohomological degrees of graded modules. Six lectures on commutative algebra (Bellaterra, 1996). Progr. Math. 166 (1998) 345–392.Google Scholar
[19]Vasconcelos, W.. Integral Closure (Springer Press, 2005).Google Scholar
[20]Wang, H. J.. Hilbert coeficients and the associated graded rings. Proc. Amer. Math. Soc. 128 (1999), 964–973.CrossRefGoogle Scholar