On the depth of the tangent cone
and the growth of the Hilbert function
Author:
Juan Elias
Journal:
Trans. Amer. Math. Soc. 351 (1999), 4027-4042
MSC (1991):
Primary 13A30, 13D40, 13H10
DOI:
https://doi.org/10.1090/S0002-9947-99-02278-3
Published electronically:
April 20, 1999
MathSciNet review:
1491860
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: For a dimensional Cohen-Macaulay local ring
we study the depth of the associated graded ring of
with respect to an
-primary ideal
in terms of the Vallabrega-Valla conditions and the length of
, where
is a
minimal reduction of
and
. As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to
-primary ideals. We also study the growth of the Hilbert function.
- [Abh67] Abhyankar, S.S. Local rings of high embedding dimension. Amer. J. Math., 89:1073-1077, 1967. MR 36:3775
- [Bla95] Blancafort, C. Hilbert functions of graded algebras over Artinian rings. Pure and Applied Alg., 125:55-78, 1998.
- [BN96] Blancafort, C. and Nollet, S. Hilbert polynomials over Artinian local rings. Preprint, 1996. MR 98m:13023
- [Eli90] Elias, J. Characterization of the Hilbert-Samuel polynomials of curve singularities. Compositio Math., 74:135-155, 1990. MR 91h:13016
- [Eli93a] Elias, J. The conjecture of Sally on the Hilbert function for curve singularities. J. of Algebra, 160(1):42-49, 1993. MR 94j:13018
- [Eli94c] Elias, J. Roller Coaster Curve Singularities. J. of Algebra, 168(3):864-867, 1994. MR 95e:13013
- [Eli96] Elias, Juan. The regularity index and the depth of the tangent cone of curve singularities. Japan J. Math., 22(1):51-68, 1996. MR 97f:13004
- [ES76] Eakin, P. and Sathaye, A. Prestable ideals. J. of Algebra, 41:439-454, 1976. MR 54:7449
- [Gue94] Guerrieri, A. On the depth of the associated graded ring of an m-primary ideal of a Cohen-Macaulay local ring. J. of Algebra, 167:745-757, 1994. MR 95h:13004
- [Gue95] Guerrieri, A. On the depth of the associated graded ring. Proc. A.M.S., 123:11-20, 1995. MR 95c:13002
- [GR83] Gupta, S.K. and Roberts, L.G. Cartesian squares and ordinary singularities of curves. Comm. in Algebra, 11(2):127-182, 1983. MR 84h:13036
- [HLS92] Heinzer, W., Lantz, D., and Shah, K. The Ratliff-Rush ideals in a Noetherian ring. Comm. in Algebra, 20(2):591-622, 1992. MR 93c:13002
- [HM94] Huckaba, S. and Marley, T. Hilbert coefficients and the depths of associated graded rings. J. London Math. Soc., 56:64-76, 1997. MR 98i:13028
- [Huc96] Huckaba, S. A d-dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients. Proc. A.M.S., 124:1393-1401, 1996. MR 96g:13018
- [Huc97] Huckaba, S. On associated graded rings having almost maximal depth. Comm. Algebra 26:967-976 (1998). CMP 98:08
- [Ito95] Itoh, S. Hilbert coefficients on integrally closed ideals. J. of Algebra, 176:638-652, 1995. MR 96g:13019
- [Lip71] Lipman, J. Stable ideals and Arf rings. Amer. J. of Math., 93:649-685, 1971. MR 44:203
- [RosV96a] Rossi, M.E. and Valla, G. On a conjecture of Sally. Comm. in Algebra, 24:4249-4261, 1996. MR 97j:13021
- [SV74] Sally, J. and Vasconcelos, W.V. Stable rings. J. Pure and Appl. Alg., 4:319-336, 1974. MR 53:13185
- [Sal77] Sally, J. On the associated graded ring of a local Cohen-Macaulay ring. J. Math. Kyoto Univ., 17:19-21, 1977. MR 56:8555
- [Sal78] Sally, J. Number of generators of ideals in local rings. Lec. Notes in Pure and Appl. Math., 35, Marcel Dekker, New York, 1978. MR 58:5654
- [Sal79a] Sally, J. Cohen-Macaulay local rings of maximal embedding dimension. J. of Algebra, 56:168-183, 1979. MR 80e:14022
- [Sal79b] Sally, J. Super-regular sequences. Pacific J. Math., 84:465-481, 1979. MR 81m:13024
- [Sal79c] Sally, J.D. Stretched Gorenstein rings. J. London Math. Soc., 20(2):19-26, 1979. MR 80k:14006
- [Sal80b] Sally, J. Good embedding dimensions for Gorenstein singularities. Math. Ann., 249:95-106, 1980. MR 82c:13031
- [Sal80a] Sally, J. Tangent cones at Gorenstein singularities. Compositio Mathematica, 40(2):167-175, 1980. MR 81e:14004
- [Sal83]
Sally, J. Cohen-Macaulay local rings of embedding dimension
. J. of Algebra, 83:393-408, 1983. MR 85c:13017
- [Sal92] Sally, J. Hilbert coefficients and reduction number 2. J. Algebraic Geometry, (1):325-333, 1992. MR 93b:13026
- [Sin74] Singh, B. Effect of a permisible blowing-up on the local Hilbert function. Inv. Math., 26:201-212, 1974. MR 50:4584
- [VV78] Vallabrega, P. and Valla, G. Form rings and regular sequences. Nagoya Math. J., 72:93-101, 1978. MR 80d:14010
- [Val79] Valla, G. On form rings which are Cohen-Macaulay. J. of Algebra, pages 247-250, 1979. MR 80h:13025
- [Vas94] Vasconcelos, W.V. Hilbert functions, analytic spread, and Koszul homology. Contemp. Math., 159:401-422, 1994. MR 95a:13006
- [Vaz95] Vaz Pinto, M.T.R. Structure of Sally modules and Hilbert functions. PhD thesis, Rutgers University, 1995.
- [Wan97]
Wang, H. On Cohen-Macaulay local rings with embedding dimension
. J. of Algebra, 190:226-240, 1997. MR 98d:13027
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Additional Information
Juan Elias
Affiliation:
Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
Email:
elias@cerber.mat.ub.es
DOI:
https://doi.org/10.1090/S0002-9947-99-02278-3
Received by editor(s):
June 24, 1997
Published electronically:
April 20, 1999
Additional Notes:
Partially supported by DGICYT PB94-0850
Article copyright:
© Copyright 1999
American Mathematical Society