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Multiplicities and Rees valuations. (English) Zbl 1216.13016
The purpose of this article is to establish connections between the Rees valuations of an ideal $I$ and various multiplicities of $I$ such as the $j$-multiplicity and the $\varepsilon$-multiplicity. Let $(R,\mathfrak m)$ be a Noetherian local ring and $I$ an ideal of $R$. It is well known that when $I$ is $\mathfrak m$-primary then $j(I)=e(I)$, where $e(I)$ and $j(I)$ denote the usual and $j$-multiplicity of $I$. The authors are concerned with non $\mathfrak m$-primary ideals with maximal analytic spread, namely $d$. They show that in this case there exist positive integers $d_i(I,\nu_i)$ such that $j(I)=\sum_i d_i(I,\nu_i)\cdot\nu_i(I)$, where the $\nu_i$ are the Rees valuations of $I$ centered on $\mathfrak m$ such that each $\nu_i$ is an $\mathfrak m$-valuation. This formula generalizes a multiplicity formula due to Rees in the case of $\mathfrak m$-primary ideals. The $\varepsilon$-multiplicity of an ideal $I$ is defined to be $\varepsilon(I)=\limsup_{n\to\infty}\frac{d!}{n^d}\cdot\lambda(\Gamma_{\mathfrak m}(R/I^n))$, where $\Gamma_{\mathfrak m}$ is the zero-th local cohomology functor. They show that this multiplicity is invariant up to integral closure. Furthermore, $\varepsilon(I)\ne 0$ if and only if $I$ has maximal analytic spread. Finally, in the appendix they discuss the one-to-one correspondence between the Rees valuations of $I$ centered on $\mathfrak m$ and the Rees valuations of $I\widehat R$ centered on $\mathfrak m\widehat R$, where $\widehat R$ is the completion of $R$.

MSC:
13H15Multiplicity theory and related topics
13A30Associated graded rings of ideals and related topics
13J10Complete rings, completion
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