Equimultiple Coefficient Ideals

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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 eᵢ(Iₚ)=eᵢ(Lₚ) for each i ∈ {1,...,k} and each minimal prime p of I is called the k-th equimultiple coefficient ideal denoted by Iₖ. 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 Gᵢ(R) satisfies the S₁ condition if and only if Iⁿ=(Iⁿ)₁ for all n.