Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On some properties of (fc)-sequences of ideals in local rings

Author: Duong Quôc Viêt
Journal: Proc. Amer. Math. Soc. 131 (2003), 45-53
MSC (2000): Primary 13A15; Secondary 13H15
Published electronically: May 15, 2002
MathSciNet review: 1929022
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The paper characterizes the length of maximal sequences satisfying conditions (i) and (ii) of (FC)-sequences, and proves some properties of (FC)-sequences, such as a bound on their lengths. As a consequence we get some results for mixed multiplicities and multiplicities of Rees rings of equimultiple ideals. We also prove that if $I$ is an ideal of positive height and $x_1, x_2, \ldots,x_p$ is an arbitrary maximal sequence in $I$ satisfying conditions (i) and (ii) of (FC)-sequences, then $(x_1,x_2, \ldots, x_p)$ is a reduction of $I.$

References [Enhancements On Off] (What's this?)

  • 1. A. Auslander and D. Buchsbaum, Codimension and multiplicity, Ann. Math. 68 (1958), 625-657. MR 20:6414; MR 21:5658
  • 2. P.B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge Phil. Soc. 53(1957), 568-575. MR 19:727b
  • 3. M. Herrmann, E. Hyry, J. Ribbe and Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197, 311-341(1997). MR 98k:13006
  • 4. D. Katz and J.K. Verma, Extended Rees algebras and mixed multiplicities, Math. Z. 202(1989), 111-128. MR 90i:13024
  • 5. D. Katz and J.K. Verma, On the multiplicity of blow-ups associated to almost complete intersection space curves, Comm. Algebra 22(2)(1994), 721-734. MR 94k:13003
  • 6. D. G. Northcott and D. Rees, Reduction of ideals in local rings, Proc. Cambridge Phil. Soc. 50 (1954), 145-158. MR 15:596a
  • 7. D. Rees, $\mathfrak{a}$-transforms of local rings and a theorem on multiplicities of ideals, Proc. Cambridge Phil. Soc. 57(1961), 8-17. MR 22:9521
  • 8. D. Rees, Hilbert functions and pseudorational local rings of dimension two, J. London Math. Soc. 24(1981), 467-479. MR 83d:13032
  • 9. D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. 29(1984), 397-414. MR 86e:13023
  • 10. I. Swanson, Mixed multiplicities, joint reductions and quasi-unmixed local rings, J. London Math. Soc. 48(1993), 1-14. MR 94d:13027
  • 11. B. Teissier, Cycles èvanescents, sections planes, et conditions de Whitney, Singularities à Cargése, 1972. Astérisque 7-8(1973), 285-362. MR 51:10682
  • 12. N.V. Trung, Reduction exponents and degree bound for the defining equation of graded rings, Proc. Amer. Math. Soc. 101(1987), 229-234.
  • 13. J.K. Verma, Rees algebras and mixed multiplicities, Proc. Amer. Math. Soc. 104(1988), 1036-1044. MR 89d:13018
  • 14. J. K. Verma, Rees algebras with minimal multiplicity, Comm. Algebra 17(12)(1988), 2999-3024. MR 91b:13032
  • 15. J. K. Verma, Multigraded Rees algebras and mixed multiplicities, J. Pure Appl. Algebra. 77(1992), 219-228. MR 93e:13005
  • 16. D. Q. Viet, Mixed multiplicities of arbitrary ideals in local rings, Comm. Algebra 28(8)(2000), 3803-3821. MR 2001f:13036

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A15, 13H15

Retrieve articles in all journals with MSC (2000): 13A15, 13H15

Additional Information

Duong Quôc Viêt
Affiliation: Department of Mathematics, Hanoi University of Technology, Dai Co Viet, Hanoi, Vietnam

Keywords: (FC)-sequence, mixed multiplicity, multiplicity of Rees ring, reduction, equimultiple ideal
Received by editor(s): April 9, 2001
Received by editor(s) in revised form: August 16, 2001
Published electronically: May 15, 2002
Additional Notes: The author was partially supported by the National Basic Research Program
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society