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 $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ 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 introduced by Shah 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 $n$.