Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-29T08:20:22.599Z Has data issue: false hasContentIssue false

Castelnuovo–Mumford regularity and Degree of nilpotency

Published online by Cambridge University Press:  01 May 2007

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

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
Copyright
Copyright © Cambridge Philosophical Society 2007

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]Andrews, D. G. and McIntyre, M. E.. On wave-action and its relatives. J. Fluid Mech. 89 (1978), 647664.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), 493504.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), 13931401.CrossRefGoogle Scholar
[5]Linh, C. H.. Upper bound for Castelnuovo–Mumford regularity of associated graded modules. Comm. Algebra 33 (6) (2005), 18171831.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), 18.CrossRefGoogle Scholar
[7]Ooishi, A.. Genera and arithmetic genera of commutative rings. Hiroshima Math. J. 17 (1987), 4766.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, 17731786.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) 99114.CrossRefGoogle Scholar
[11]Schwartz, N.. Bounds for the postulation numbers of Hilbert functions. J. Algebra 193 (1997), 581615.CrossRefGoogle Scholar
[12]Srinivas, V. and Trivedi, V.. On the Hilbert functions of a Cohen–Macaulay ring. J. Algebraic Geom. 6 (1997), 733751.Google Scholar
[13]Trivedi, V.. Hilbert functions, Castelnuovo–Mumford regularity and uniform Artin-Rees numbers. Manuscripta Math. 94 (1997), 543558.CrossRefGoogle Scholar
[14]Trivedi, V.. Finiteness of Hilbert functions for generalized Cohen–Macaulay modules. Comm. Algebra. 29 (2) (2001), 805813.CrossRefGoogle Scholar
[15]Trung, N. V.. Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (1987), 229234.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), 28132832.CrossRefGoogle Scholar
[17]Vasconcelos, W.. The homological degree of module. Trans. Amer. Math. Soc. 350 (1998), 11671179.CrossRefGoogle Scholar
[18]Vasconcelos, W.. Cohomological degrees of graded modules. Six lectures on commutative algebra (Bellaterra, 1996). Progr. Math. 166 (1998) 345392.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), 964973.CrossRefGoogle Scholar