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

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.