Equimultiple coefficient ideals

Abstract

Let (R, m) be a quasi-unmixed local ring and I an equimultiple ideal of R of analytic spread s. In this paper, we introduce the equimultiple coefficient ideals. Fix k ∈ {1, ..., s}. The largest ideal L containing I such that e i (I p ) = e i (L p ) for each i ∈ {1, ..., k} and each minimal prime p of I is called the k-th equimultiple coefficient ideal denoted by I k . It is a generalization of the coefficient ideals firstly introduced by Shah [S] for the case of m-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring G I (R) satisfies the S 1 condition if and only if I n = (I n ) 1 for all n. * Work partially supported by CAPES-Brazil 10056/12-2. † Work partially supported by FAPESP-Brazil 2012/20304-1 and CNPq-Brazil 303682/2012-4.