On the depth of the tangent cone and the growth of the Hilbert function

Elias, Juan

doi:10.1090/S0002-9947-99-02278-3

On the depth of the tangent cone and the growth of the Hilbert function
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Trans. Amer. Math. Soc. 351 (1999), 4027-4042 Request permission

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