A FORMULA FOR REDUCTION NUMBER OF AN IDEAL RELATIVE TO A NOETHERIAN MODULE

A FORMULA FOR REDUCTION NUMBER OF AN IDEAL RELATIVE TO A NOETHERIAN MODULE

Let (A, m) be a Noetherian local ring with infinite residue field and E be a finitely generated d dimensional Cohen-Macaulay A-module. Let b be an ideal of A such that htEb = 0 and λ(b, E) = 1. Assume that bp = 0 for all p ∈ Min(E/bE). Let r(b, E) > 0. We show that if Gb(E) is Cohen-Macaulay, then r(b, E) = a(Gb(E)) + 1.

___

  • M. Brodmann and R. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge University Press, Cambridge, 1998.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993.
  • C. D’Cruz, V. Kodiyalam and J. K. Verma, Bounds on the a-invariant and reduction numbers of ideals, J. Algebra, 274 (2004) 594-601.
  • S. Goto and S. Huckaba, On graded rings associated to analytic deviation one ideals, Amer. J. Math., 116 (1994) 905-919.
  • S. Gote and K. Watanabe, Graded rings I, J. Math. Soc. Japan, 30 (1978) 179-213.
  • M. Hermann, J. Ribbe and S. Zarzula, On the Gorenstein property of Rees and form rings of powers of ideals, Trans. Amer. Math. Soc., 342(2) (1994), 631-643.
  • M. Hermann, S. Ikeda and U. Orbanz, Equimultiplicity and Blowing Up, Springer-Verlag, New york, 1988.
  • L. T. Hoa, Reduction numbers and Rees algebras of powers of an ideal, Proc. Amer. Math. Soc., 119 (1993) 415-422.
  • L. T. Hoa, Reduction numbers of equimultiple ideals, J. Pure Appl. Algebra, 109 (1996) 111-126.
  • T. Marley, The reduction number of an ideal and the local cohomology of the associated graded ring, Proc. Amer. Math. Soc., 117 (1993) 335-341.
  • D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Camb. Phil. Soc., 50 (1954) 145-158.
  • J. Sally, On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ., 17 (1977) 19-21.
  • J. Sally, Reductions, local cohomology and Hilbert functions of local rings, Com- mutative algebra: Durham 1981, London Math. Soc. Lecture Note Ser.,vol. 72, Cambridge University Press, 1982, pp. 393-408.
  • N. V. Trung, Reduction exponents and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc., 101 (1987) 229-236.
  • N. V. Trung, Towards a theory of generalized Cohen-Macaulay modules, Nagoya Math. J., 102 (1986) 1-49. Naser Zamani Faculty of Science,
  • University of Mohaghegh Ardabili
  • P.O.Box 179, Ardabil, Iran
  • e-mail: naserzaka@yahoo.com