Equimultiple Coefficient Ideals

TL;DR

This paper introduces equimultiple coefficient ideals in quasi-unmixed local rings, generalizing Shah's coefficient ideals, and explores their properties and applications, including conditions for the associated graded ring to satisfy Serre's $S_1$ condition.

Contribution

It defines the $k$-th equimultiple coefficient ideal, extending Shah's concept to a broader class of ideals, and investigates their algebraic properties and implications.

Findings

The associated graded ring $G_{I}(R)$ satisfies $S_1$ if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

Introduces the $k$-th equimultiple coefficient ideal as a maximal ideal with specific multiplicity properties.

Provides applications linking these ideals to properties of the associated graded ring.

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 $n$.