Reduction numbers of equimultiple ideals

Abstract

Let (A, m) be a local ring and Z and an ideal of A. In this paper, all local rings are assumed to have infinite residue fields. An ideal .Z 5 Z is called a reduction of Z if Zflfl = Z”J for some nonnegative integer n. A reduction J is called a minimal reduction if it does not properly contain a reduction of I. These notions were introduced by Northcott and Rees [16]. They proved that minimal reductions of Z do always exist, and every minimal reduction of Z is minimally generated by E(Z):= dim 0, k 0 P/ml” which is called the anQ~y~~c spread of 1. If J is a minimal reduction of I, we define the reduction number of I with respect to J, denoted by r,(Z), to be the last nonnegative integer n such that I”+’ = Z”J. The reduction number of Z is defined by r{Z) = min{r.,(Z); J is a minimal reduction of Z 3. Reduction numbers have been proven to be very useful in studying the Cohen-Macaulay (abbr. C-M} and Gorenstein property of the associated graded ring G(Z) = @,I>OZn/Zn+l and the Rees algebra A[Zt] = a,, z 0 Z”t” of I. It was initiated by Sally, Goto and Shimoda, and is intensively studied now by various authors (see, e.g., [l] for references). Therefore it is of great interest to study properties of reduction numbers: see 14, 10,12-14, 19,231. The main aim of this paper is to give upper bounds for the reduction numbers of equimultiple ideals. Recall that ht Z 1, easy