Equimultiple Coefficient Ideals

Abstract

Let $(R,\mathfrak{m})$ 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 $k\in \{1,...,s\}.$ The largest ideal $L$ containing $I$ such that $e_i(I_{\mathfrak{p}})=e_i(L_{\mathfrak{p}})$ for each $i\in \{1,...,k\}$ and each minimal prime $\mathfrak{p}$ 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 \cite{S} for the case of $\mathfrak{m}$-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 $n$.