Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T00:52:44.675Z Has data issue: false hasContentIssue false
  • English
  • Français

Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals

Published online by Cambridge University Press:  16 March 2010

LÊ TUÂN HOA  and
TRÂN NAM TRUNG
LÊ TUÂN HOA
Affiliation:
Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam. e-mail: [email protected] and [email protected]
TRÂN NAM TRUNG
Affiliation:
Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam. e-mail: [email protected] and [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits also exist.

As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 149 , Issue 2 , September 2010 , pp. 229 - 246
Copyright
Copyright © Cambridge Philosophical Society 2010

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]Cutkosky, D., Ein, L. and Lazarsfeld, R.Positivity and complexity of ideal sheaves. Math. Ann. 321 (2001), no. 2, 213–234.CrossRefGoogle Scholar
[2]Cutkosky, D., Herzog, J., Trung, N. V.Asymptotic behavior of the Castelnuovo–Mumford regularity. Compositio Math. 118 (1999), 243–261.CrossRefGoogle Scholar
[3]Eisenbud, D.Commutative algebra with a view toward algebraic geometry. Graduate Texts in Math. (Springer-Verlag, 1995).Google Scholar
[4]Giang, D. H. and Hoa, L. T. On local cohomology of a tetrahedral curve. Acta Math. Vietnam. (to appear); ArXiv: 0910.0919.Google Scholar
[5]Goto, S. and Watanabe, K.On graded rings, I. J. Math. Soc. Japan. 30 (1978), 179–213.CrossRefGoogle Scholar
[6]Herzog, J., Hibi, T. and Trung, N. V.Symbolic powers of monomial ideals and vertex cover algebra. Adv. Math. 210 (2007), 304–322.CrossRefGoogle Scholar
[7]Herzog, J., Hoa, L. T. and Trung, N. V.Asymptotic linear bounds for the Castelnuovo–Mumford regularity. Trans. Amer. Math. Soc. 354 (2002), 1793–1809.CrossRefGoogle Scholar
[8]Hoa, L. T. and Hyry, E.On local cohomology and Hilbert function of powers of ideals. Manuscripta Math. 112 (2003), 77–92.CrossRefGoogle Scholar
[9]Hoa, L. T. and Trung, N. V.On the Castelnuovo–Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229 (1998), 519–537.CrossRefGoogle Scholar
[10]Hoa, L. T. and Trung, T. N.Castelnuovo–Mumford regularity of sums of powers of polynomial ideals. Comm. Algebra 36 (2008), 806–820.CrossRefGoogle Scholar
[11]Katzman, M.The complexity of Frobenius powers of ideals. J. Algebra 203 (1998), 211–225.Google Scholar
[12]Kodiyalam, V.Asymptotic behavior of Castelnouvo-Mumford regularity. Proc. Amer. Math. Soc. 128 (2000), 407–411.Google Scholar
[13]Reid, L., Roberts, L. G. and Vitulli, M. A.Some results on normal homogeneous ideals. Comm. Algebra 31 (2003), 4485–4506.CrossRefGoogle Scholar
[14]Schrijver, A., Theory of linear and integer programming. (John Wiley & Sons, Ltd., 1986).Google Scholar
[15]Stanley, R. P.Combinatorics and Commutative Algebra (Birkhauser, 1996).Google Scholar
[16]Takayama, Y.Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 48 (96) (1987), 327–344.Google Scholar
[17]Trung, N. V.Gröbner bases, local cohomology and reduction number. Proc. Amer. Math. Soc. 129 (2001), 9–18.CrossRefGoogle Scholar
[18]Trung, N. V. and Wang, H. J.On the asymptotic linearity of Castelnuovo–Mumford regularity. J. Pure Appl. Algebra 201 (2005), 42–48.CrossRefGoogle Scholar
[19]Trung, T. N.Regularity index of Hilbert function of powers of ideals. Proc. Amer. Math. Soc. 137 (2009), 2169–2174.CrossRefGoogle Scholar