Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A $d$-dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

Author: Sam Huckaba
Journal: Proc. Amer. Math. Soc. 124 (1996), 1393-1401
MSC (1991): Primary 13D40, 13A30, 13H10
MathSciNet review: 1307529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A $d$-dimensional version is given of a $2$-dimensional result due to C. Huneke. His result produced a formula relating the length $\lambda (I^{n+1}/JI^{n})$ to the difference $P_{I}(n+1)-H_{I}(n+1)$, where $I$ is primary for the maximal ideal of a $2$-dimensional Cohen-Macaulay local ring $R$, $J$ is a minimal reduction of $I$, $H_{I}(n)=\lambda (R/I^{n})$, and $P_{I}(n)$ is the Hilbert-Samuel polynomial of $I$. We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of $I$. We also include a characterization, in terms of the Hilbert coefficients of $I$, of the condition $depth(G(I))\geq d-1$.

References [Enhancements On Off] (What's this?)

  • [BH] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. CMP 94:05
  • [G] A. Guerrieri, On the depth of certain graded rings associated to an ideal, Ph.D. Dissertation, Purdue University (1993).
  • [Hun] C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293--318. MR 89b:13037
  • [KM] D. Kirby and H. A. Mehran, A note on the coefficients of the Hilbert-Samuel polynomial for a Cohen-Macaulay module, J. London Math. Soc. (2) 25 (1982), 449--457. MR 84a:13022
  • [K] K. Kubota, On the Hilbert-Samuel function, Tokyo J. Math. 8 (1985), 439--448. MR 87f:13023a
  • [M1] T. Marley, Hilbert functions of ideals in Cohen-Macaulay local rings, Ph.D. Dissertation, Purdue University (1989).
  • [M2] T. Marley, The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. (2) 40 (1989), 1--8. MR 90m:13026
  • [N] M. Nagata, Local Rings, Kreiger, Huntington and New York, 1975. MR 57:301
  • [Na] M. Narita, A note on the coefficients of Hilbert characteristic functions in semi-regular local rings, Proc. Cambridge Philos. Soc. 59 (1963), 269--275. MR 26:3734
  • [No] D. G. Northcott, A note on the coefficients of the abstract Hilbert function, J. London Math. Soc. 35 (1960), 209--214. MR 22:1599
  • [NR] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145--158. MR 15:596a
  • [O] A. Ooishi, $\Delta $-genera and sectional genera of commutative rings, Hiroshima Math. J. 17 (1987), 361--372. MR 89f:13033
  • [S1] J. D. Sally, Hilbert coefficients and reduction number $2$, J. Algebraic Geom. 1 (1992), 325--333. MR 93b:13026
  • [S2] J. D. Sally, Ideals whose Hilbert function and Hilbert polynomial agree at $n=1$, J. Algebra 157 (1993), 534--547. MR 94d:13016
  • [Sw] I. Swanson, A note on analytic spread, Comm. Algebra 22 (1994), 407--411. MR 95b:13007
  • [VV] P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14010
  • [V] W. V. Vasconcelos, Hilbert functions, analytic spread, and Koszul homology, Contemporary Math. 159 (1994), 401--422. MR 95a:13006
  • [Wu] Y. Wu, Reduction numbers and Hilbert polynomials of ideals in higher dimensional
    Cohen-Macaulay local rings
    , Math. Proc. Cambridge Philos. Soc. 111 (1992), 47--56. MR 92g:13020
  • [ZS] O. Zariski and P. Samuel, Commutative Algebra, vol. 2, Springer-Verlag, New York, 1960. MR 52:10706

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13D40, 13A30, 13H10

Retrieve articles in all journals with MSC (1991): 13D40, 13A30, 13H10

Additional Information

Sam Huckaba
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-3027

Keywords: Hilbert-Samuel polynomial, depth, associated graded ring, Cohen-Macaulay
Received by editor(s): June 8, 1994
Received by editor(s) in revised form: November 8, 1994
Additional Notes: The author was partially supported by the NSA (#MDA904-92-H-3040).
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society