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Fiber Cones of Ideals with Almost Minimal Multiplicity

Published online by Cambridge University Press:  11 January 2016

A. V. Jayanthan  and
J. K. Verma
A. V. Jayanthan
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad - 211019, India, jayan@mri.ernet.in
J. K. Verma
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai - 400076, India, jkv@math.iitb.ac.in
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Abstract

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Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi’s bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

Keywords

13H1013H1513C1513A02
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 177 , 2005 , pp. 155 - 179
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[1] Abhyankar, S., Local rings of high embedding dimension, Amer. J. Math., 89 (1967), 1073-1077.CrossRefGoogle Scholar
[2] Bruns, W. and Herzog, J., Cohen-Macaulay rings, Revised Edition, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1998.Google Scholar
[3] Chuai, J., Generalised parameter ideals in local C-M rings, Algebra Colloq., 3 (1996), no. 3,213–216.Google Scholar
[4] Cowsik, R. C. and Nori, M. V., Fibers of blowing up, J. Indian Math. Soc., 40 (1976), 217–222.Google Scholar
[5] Cortadellas, T. and Zarzuela, S., On the depth of the fiber cone of filtrations, J. Algebra, 198 (1997), no. 2, 428–445.Google Scholar
[6] D’Cruz, C., Raghavan, K. N. and Verma, J. K., Cohen-Macaulay fiber cones, Commutative Algebra, Algebraic Geometry and Computational Methods (Hanoi, 1996), Springer, Singapore (1999), pp. 233–246.Google Scholar
[7] D’Cruz, C., Verma, J. K., Hilbert series of fiber cones of ideals with almost minimal mixed multiplicity, J. Algebra, 251 (2002), 98–109.Google Scholar
[8] Elias, J., On the depth of the tangent cone and the growth of the Hilbert function, Trans. Amer. Math. Soc., 351 (1999), 4027–4042.Google Scholar
[9] Eisenbud, D. and Mazur, B., Evolutions, symbolic squares, and Fitting ideals, J. Reine Angew. Math., 488 (1997), 189–201.Google Scholar
[10] Goto, S., Cohen-Macaulayness and negativity of A-invariants in Rees algebras associated to m-primary ideals of minimal multiplicity, Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998), J. Pure Appl. Algebra, 152 (2000), no. 1–3, 93–107.Google Scholar
[11] Guerrieri, A. and Rossi, M. E., Hilbert coefficients of Hilbert filtrations, J. Algebra, 199 (1998), 40–61.Google Scholar
[12] Hübl, R., Evolutions and valuations associated to an ideal, Jour. Reine angew. Math., 517 (1999), 81–101.Google Scholar
[13] Hübl, R. and Huneke, C., Fiber cones and the integral closure of ideals, Collect. Math., 52 (2001), no. 1, 85–100.Google Scholar
[14] Huckaba, S., Reduction numbers for ideals of higher analytic spread, Math. Proc. Camb. Phil. Soc., 102 (1987), 49–57.Google Scholar
[15] Huckaba, S., 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.Google Scholar
[16] Hironaka, H., Resolution of singularities, Ann. of Math., 79 (1964), 109–326.Google Scholar
[17] Huckaba, S. and Marley, T., Hilbert coefficients and depth of associated graded rings, J. London Math. Soc. (2), 56 (1997), 64–76.Google Scholar
[18] Huneke, C., Hilbert functions and symbolic powers, Michigan Math. J., 34 (1987), no. 2, 293–318.Google Scholar
[19] Jayanthan, A. V., Singh, B. and Verma, J. K., Hilbert coefficients and depth of form rings, to appear, Comm. Algebra.Google Scholar
[20] Jayanthan, A. V. and Verma, J. K., Hilbert coefficients and depth of fiber cones, to appear, J. Pure and Applied Algebra.Google Scholar
[21] Johnston, B. and Verma, J. K., On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings, Math. Proc. Cambridge Philos. Soc., 111 (1992), 423–432.Google Scholar
[22] Northcott, D. G. and Rees, D., Reductions of ideals in local rings, Proc. Cambridge Philos. Soc., 50 (1954), 145–158.Google Scholar
[23] Ooishi, A., On the Gorenstein property of the associated graded ring and the Rees algebra of an ideal, J. Algebra, 155 (1993), 397–414.Google Scholar
[24] Rossi, M. E., Primary ideals with good associated graded ring, J. of Pure and Appl. Algebra, 145 (2000), 75–90.Google Scholar
[25] Rossi, M. E., A bound on the reduction number of a primary ideal, Proc. Amer. Math. Soc., 128 (2000), no. 5, 1325–1332.Google Scholar
[26] Rossi, M. E. and Valla, G., A conjecture of J. Sally, Comm. Algebra, 24 (1996), no. 13, 4249–4261.Google Scholar
[27] Rossi, M. E. and Valla, G., Cohen-Macaulay local rings of embedding dimension e + d - 3, Proc. London Math. Soc., 80 (2000), 107–126.CrossRefGoogle Scholar
[28] Sally, J. D., Cohen-Macaulay local rings of embedding dimension e + d - 2, J. Algebra, 83 (1983), 393–408.Google Scholar
[29] Sally, J. D., Tangent cones at Gorenstein singularities, Comp. Math., 40 (1980), 167–175.Google Scholar
[30] Shah, K., On the Cohen-Macaulayness of the Fiber Cone of an Ideal, J. Algebra, 143 (1991), 156–172.Google Scholar
[31] Singh, B., A numerical criterion for the permissibility of a blowing-up, Compositio Math., 33 (1976), 15–28.Google Scholar
[32] Valla, G., On form rings which are Cohen-Macaulay, J. Algebra, 58 (1979), 475–481.Google Scholar
[33] Valabrega, P. and Valla, G., Form rings and regular sequences, Nagoya Math. J., 72 (1978), 93–101.Google Scholar
[34] H.-Wang, J., On Cohen-Macaulay local rings with embedding dimension e+d- 2, J. Algebra, 190 (1997), no. 1, 226–240.Google Scholar
[35] Wang, H.-J., Hilbert coefficients and the associated graded rings, Proc. Amer. Math. Soc., 128 (1999), 963–973.Google Scholar
[36] Watanabe, J., The Dilworth number of Artin Gorenstein rings, Adv. Math., 76 (1989), no. 2, 194–199.Google Scholar