Taiwanese Journal of Mathematics

Castelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Ideals

Cao Huy Linh

Full-text: Open access

Abstract

Let $A$ be a noetherian local ring of dimension $d \geq 1$ and $\operatorname{depth}(A) \geq d-1$. In this paper, we study the non-positivity for the Hilbert coefficients of parameter ideals in the ring $A$. Moreover, we establish a bound for the Castelnuovo-Mumford regularity of associated graded ring of $A$ with respect to parameter ideal in terms of the first Hilbert coefficient and the dimension.

Article information

Source
Taiwanese J. Math., Volume 23, Number 5 (2019), 1115-1131.

Dates
Received: 3 November 2018
Revised: 15 January 2019
Accepted: 21 January 2019
First available in Project Euclid: 30 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1548817227

Digital Object Identifier
doi:10.11650/tjm/190106

Mathematical Reviews number (MathSciNet)
MR4012372

Zentralblatt MATH identifier
07126941

Subjects
Primary: 13D45: Local cohomology [See also 14B15] 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 14B15: Local cohomology [See also 13D45, 32C36]

Keywords
Hilbert coefficients the depth of associated graded rings parameter ideals Castelnuovo-Mumford regularity postulation number

Citation

Linh, Cao Huy. Castelnuovo-Mumford Regularity and Hilbert Coefficients of Parameter Ideals. Taiwanese J. Math. 23 (2019), no. 5, 1115--1131. doi:10.11650/tjm/190106. https://projecteuclid.org/euclid.twjm/1548817227


Export citation

References

  • C. Blancafort, On Hilbert functions and cohomology, J. Algebra 192 (1997), no. 1, 439–459.
  • M. Brodmann and C. H. Linh, Castelnuovo-Mumford regularity, postulation numbers and relation types, J. Algebra 419 (2014), 124–140.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.
  • J. Elias, Depth of higher associated graded rings, J. London Math. Soc. (2) 70 (2004), no. 1, 41–58.
  • ––––, On the last Hilbert-Samuel coefficient of isolated singularities, J. Algebra 394 (2013), 285–295.
  • S. Goto and K. Ozeki, Uniform bounds for Hilbert coefficients of parameters, in: Commutative Algebra and its Connections to Geometry, 97–118, Contemp. Math. 555, Amer. Math. Soc., Providence, RI, 2011.
  • L. T. Hoa, Reduction numbers and Rees algebras of powers of an ideal, Proc. Amer. Math. Soc. 119 (1993), no. 2, 415–422.
  • ––––, Reduction numbers of equimultiple ideals, J. Pure Appl. Algebra 109 (1996), no. 2, 111–126.
  • S. Huckaba and T. Marley, Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc. (2) 56 (1997), no. 1, 64–76.
  • C. H. Linh, Upper bound for the Castelnuovo-Mumford regularity of associated graded modules, Comm. Algebra 33 (2005), no. 6, 1817–1831.
  • ––––, Castelnuovo-Mumford regularity and degree of nilpotency, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 3, 429–437.
  • C. H. Linh and N. V. Trung, Uniform bounds in generalized Cohen-Macaulay rings, J. Algebra 304 (2006), no. 2, 1147–1159.
  • C. H. Linh and V. D. Trung, Hilbert coefficients and the depth of associated graded rings with respect to parameter ideals, to appear in Vietnam Journal of Mathematics.
  • M. Mandal, B. Singh and J. K. Verma, On some conjectures about the Chern numbers of filtrations, J. Algebra 325 (2011), 147–162.
  • T. Marley, The reduction number of an ideal and the local cohomology of the associated graded ring, Proc. Amer. Math. Soc. 117 (1993), no. 2, 335–341.
  • L. Mccune, Hilbert coefficients of parameter ideals, J. Commut. Algebra 5 (2013), no. 3, 399–412.
  • A. Ooishi, Genera and arithmetic genera of commutative rings, Hiroshima Math. J. 17 (1987), no. 1, 47–66.
  • M. E. Rossi, N. V. Trung and G. Valla, Castelnuovo-Mumford regularity and extended degree, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1773–1786.
  • M. E. Rossi and G. Valla, Hilbert Functions of Filtered Modules, Lecture Notes of the Unione Matematica Italiana 9, Springer-Verlag, Berlin, 2010.
  • A. Saikia and K. Saloni, Bounding Hilbert coefficients of parameter ideals, J. Algebra 501 (2018), 328–344.
  • N. V. Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), no. 2, 229–336.
  • W. V. Vasconcelos, The Chern coefficients of local rings, Michigan Math. J. 57 (2008), 725–743.