Epsilon multiplicity, multiplicity=volume formula and analytic spread of family of ideals

TL;DR

This paper establishes a formula linking epsilon multiplicity with volume and explores the analytic spread of ideal filtrations in local rings, extending previous bounds and providing new insights.

Contribution

It proves that epsilon multiplicity of a filtration equals that of a related family, leading to a volume formula and analysis of the maximality of analytic spread.

Findings

Epsilon multiplicity coincides for related ideal families.

Derived a multiplicity=volume formula for epsilon multiplicity.

Investigated conditions for maximal analytic spread of filtrations.

Abstract

In an analytically unramified local ring $(R,\mathfrak m)$ of dimension $d\geq 1$, for a filtration of ideals $\mathfrak {I}=\{I_m\}_{m\in\mathbb N}$ satisfying $\mathfrak A(r)$ condition and for any $\mathfrak m$-primary ideal $K$, it is shown in $[18]$ that the epsilon multiplicity of the weakly graded family of ideals $\{(I_m:K)\}_{m\in\mathbb N}$ exists as a limit and it is bounded above by the epsilon multiplicity of $\mathfrak I$, $\epsilon(\mathfrak I)$. In this article, we first show that $\epsilon(\mathfrak I)$ coincides with the epsilon multiplicity of $\{(I_m:K)\}_{m\in\mathbb N}$ and this leads to the following: $(a)$ an expression for $\epsilon(\mathfrak I)$ as a limit of the epsilon multiplicities of other graded families of ideals and $(b)$ a multiplicity=volume formula for the epsilon multiplicity of an ideal $I$ in $R$. In the final part of the article, we investigate the maximality of the analytic spread of filtrations of ideals.