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**

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

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

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