On the depth of the tangent cone and the growth of the Hilbert function
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Abstract:
For a $d-$dimensional Cohen-Macaulay local ring $(R, \mathbf {m})$ we study the depth of the associated graded ring of $R$ with respect to an $\textbf { m}$-primary ideal $I$ in terms of the Vallabrega-Valla conditions and the length of $I^{t+1}/JI^{t}$, where $J$ is a $J$ minimal reduction of $I$ and $t\ge 1$. As a corollary we generalize Sally’s conjecture on the depth of the associated graded ring with respect to a maximal ideal to $\mathbf {m}$-primary ideals. We also study the growth of the Hilbert function.References
- Shreeram Shankar Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073–1077. MR 220723, DOI 10.2307/2373418
- Blancafort, C. Hilbert functions of graded algebras over Artinian rings. Pure and Applied Alg., 125:55–78, 1998.
- Cristina Blancafort, Hilbert functions of graded algebras over Artinian rings, J. Pure Appl. Algebra 125 (1998), no. 1-3, 55–78. MR 1600008, DOI 10.1016/S0022-4049(96)00124-7
- Juan Elías, Characterization of the Hilbert-Samuel polynomials of curve singularities, Compositio Math. 74 (1990), no. 2, 135–155. MR 1047736
- Juan Elías, The conjecture of Sally on the Hilbert function for curve singularities, J. Algebra 160 (1993), no. 1, 42–49. MR 1237076, DOI 10.1006/jabr.1993.1176
- Juan Elías, Roller coaster curve singularities, J. Algebra 168 (1994), no. 3, 864–867. MR 1293630, DOI 10.1006/jabr.1994.1259
- Juan Elias, The regularity index and the depth of the tangent cone of curve singularities, Japan. J. Math. (N.S.) 22 (1996), no. 1, 51–68. MR 1394372, DOI 10.4099/math1924.22.51
- Paul Eakin and Avinash Sathaye, Prestable ideals, J. Algebra 41 (1976), no. 2, 439–454. MR 419428, DOI 10.1016/0021-8693(76)90192-7
- Anna Guerrieri, On the depth of the associated graded ring of an $m$-primary ideal of a Cohen-Macaulay local ring, J. Algebra 167 (1994), no. 3, 745–757. MR 1287068, DOI 10.1006/jabr.1994.1210
- Anna Guerrieri, On the depth of the associated graded ring, Proc. Amer. Math. Soc. 123 (1995), no. 1, 11–20. MR 1211580, DOI 10.1090/S0002-9939-1995-1211580-9
- Surender K. Gupta and Leslie G. Roberts, Cartesian squares and ordinary singularities of curves, Comm. Algebra 11 (1983), no. 2, 127–182. MR 688046, DOI 10.1080/00927878308822842
- William Heinzer, David Lantz, and Kishor Shah, The Ratliff-Rush ideals in a Noetherian ring, Comm. Algebra 20 (1992), no. 2, 591–622. MR 1146317, DOI 10.1080/00927879208824359
- Sam Huckaba and Thomas Marley, Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc. (2) 56 (1997), no. 1, 64–76. MR 1462826, DOI 10.1112/S0024610797005206
- Sam Huckaba, A $d$-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1393–1401. MR 1307529, DOI 10.1090/S0002-9939-96-03182-6
- Huckaba, S. On associated graded rings having almost maximal depth. Comm. Algebra 26:967–976 (1998).
- Shiroh Itoh, Hilbert coefficients of integrally closed ideals, J. Algebra 176 (1995), no. 2, 638–652. MR 1351629, DOI 10.1006/jabr.1995.1264
- Joseph Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649–685. MR 282969, DOI 10.2307/2373463
- M. E. Rossi and G. Valla, A conjecture of J. Sally, Comm. Algebra 24 (1996), no. 13, 4249–4261. MR 1414582, DOI 10.1080/00927879608825812
- Judith D. Sally and Wolmer V. Vasconcelos, Stable rings, J. Pure Appl. Algebra 4 (1974), 319–336. MR 409430, DOI 10.1016/0022-4049(74)90012-7
- Judith D. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ. 17 (1977), no. 1, 19–21. MR 450259, DOI 10.1215/kjm/1250522807
- Judith D. Sally, Numbers of generators of ideals in local rings, Marcel Dekker, Inc., New York-Basel, 1978. MR 0485852
- Judith D. Sally, Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra 56 (1979), no. 1, 168–183. MR 527163, DOI 10.1016/0021-8693(79)90331-4
- Judith D. Sally, Superregular sequences, Pacific J. Math. 84 (1979), no. 2, 465–481. MR 568663
- Judith D. Sally, Stretched Gorenstein rings, J. London Math. Soc. (2) 20 (1979), no. 1, 19–26. MR 545198, DOI 10.1112/jlms/s2-20.1.19
- Judith D. Sally, Good embedding dimensions for Gorenstein singularities, Math. Ann. 249 (1980), no. 2, 95–106. MR 578716, DOI 10.1007/BF01351406
- Judith D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 (1980), no. 2, 167–175. MR 563540
- Judith D. Sally, Cohen-Macaulay local rings of embedding dimension $e+d-2$, J. Algebra 83 (1983), no. 2, 393–408. MR 714252, DOI 10.1016/0021-8693(83)90226-0
- Judith D. Sally, Hilbert coefficients and reduction number $2$, J. Algebraic Geom. 1 (1992), no. 2, 325–333. MR 1144442
- Balwant Singh, Effect of a permissible blowing-up on the local Hilbert functions, Invent. Math. 26 (1974), 201–212. MR 352097, DOI 10.1007/BF01418949
- Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101. MR 514892
- Giuseppe Valla, On form rings which are Cohen-Macaulay, J. Algebra 58 (1979), no. 2, 247–250. MR 540637, DOI 10.1016/0021-8693(79)90159-5
- Wolmer V. Vasconcelos, Hilbert functions, analytic spread, and Koszul homology, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 401–422. MR 1266195, DOI 10.1090/conm/159/01520
- Vaz Pinto, M.T.R. Structure of Sally modules and Hilbert functions. PhD thesis, Rutgers University, 1995.
- Hsin-Ju Wang, On Cohen-Macaulay local rings with embedding dimension $e+d-2$, J. Algebra 190 (1997), no. 1, 226–240. MR 1442154, DOI 10.1006/jabr.1996.6894
Additional Information
- Juan Elias
- Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
- MR Author ID: 229646
- ORCID: 0000-0003-3053-1542
- Email: elias@cerber.mat.ub.es
- Received by editor(s): June 24, 1997
- Published electronically: April 20, 1999
- Additional Notes: Partially supported by DGICYT PB94-0850
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1491860