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

Abstract

In an analytically unramified local ring $(R,\mathfrak{m})$ of dimension $d\ge 1$, 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.

12 pages, comments are welcome