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On the number of generators of Cohen-Macaulay ideals

Authors: Clare D'Cruz and J. K. Verma
Journal: Proc. Amer. Math. Soc. 128 (2000), 3185-3190
MSC (1991): Primary 13H10, 13D40
Published electronically: June 7, 2000
MathSciNet review: 1676364
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Abstract | References | Similar Articles | Additional Information


Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.

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  • [A] S. S. Abhyankar, Local rings of high embedding dimension, Amer. J. Math. 89 (1967), 1073-1077. MR 36:3775
  • [Ak] Y. Akizuki, Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz, Jap. J. Math. 14 (1938), 85-102.
  • [B] P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Camb. Philos. Soc. 53 (1957), 568-575. MR 19:727b
  • [BED] M. Boratynski, D. Eisenbud and D. Rees, On the number of generators of ideals in local Cohen-Macaulay rings, J. Algebra 57 (1979), 77-81. MR 82k:13021
  • [C] I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950), 27-42. MR 11:413g
  • [DGV] L. R. Doering, Tor Gunston and W. Vasconcelos, Cohomological degrees and Hilbert functions of graded modules, Amer. J. Math. 120 (1998), 493-504. CMP 98:13
  • [KV] D. Katz, and J. K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202 (1989), 111-128. MR 90i:13024
  • [NR] D. G. Northcott, and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 15:596a
  • [R1] D. Rees, ${\cal A}$- Transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Philos. Soc. 57 (1961), 8-17. MR 22:9521
  • [R2] D. Rees, Multiplicities, Hilbert functions and degree functions, In Commutative Algebra: Durham 1981, London Mathematical Society Lecture Notes 72 (ed. R. Y. Sharp), Cambridge Univ. Press (1983), 170-178. MR 84j:13001
  • [R3] D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29 (1984), 397-414. MR 86e:13023
  • [R4] D. Rees, Estimates for the minimum number of generators for Cohen-Macaulay ideals , preprint.
  • [RS] D. Rees, and R. Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc.(2)18 (1978), 449-463. MR 80e:13009
  • [S] J. D. Sally, Bounds for the number of generators of Cohen-Macaulay ideals, Pacific J. Math. 63(1976), 517-520.
  • [T1] B. Teissier, Cycles èvanescents, section planes, et conditions de Whitney, Singularitiés à Cargése, 1972. Astérisque 7-8 (1973), 285-362. MR 51:10682
  • [T2] B. Teissier, Sur une inégalité à la Minkowski pour les multiplicités, appendix to : D. Eisenbud and H. Levine, an algebraic formula for the degree of $C^{\infty}$ map-germ Ann. Math. 106 (1977), 19-44.
  • [V] G. Valla, Generators of ideals and multiplicities, Comm. Algebra 15(1981), 1541-1549.

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Additional Information

Clare D'Cruz
Affiliation: SPIC Mathematical Institute, 92 G. N. Chetty Road, T. Nagar, Chennai 600 017, India

J. K. Verma
Affiliation: Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India

Keywords: Cohen-Macaulay ring, Cohen-Macaulay ideal, number of generators of ideals, multiplicities, mixed multiplicities
Received by editor(s): October 7, 1998
Received by editor(s) in revised form: January 7, 1999
Published electronically: June 7, 2000
Additional Notes: Presented at the first national meeting of commutative algebra and algebraic geometry held at the Institute of Astrophysics, Kodaikanal, India, March 1998.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

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