Fonction asymptotique de Samuel des sections hyperplanes et multiplicit\'e

TL;DR

This paper investigates the asymptotic Samuel function in local noetherian rings, focusing on its behavior under hyperplane sections and its relation to mixed multiplicities, providing new insights into asymptotic invariants.

Contribution

It introduces a detailed study of the asymptotic Samuel function's behavior under hyperplane sections, extending the theory of mixed multiplicities to this context.

Findings

The asymptotic Samuel function exhibits specific limiting behaviors under hyperplane sections.

The study reveals connections between the asymptotic Samuel function and mixed multiplicities.

Results provide new tools for analyzing local ring invariants.

Abstract

Let $(A,\mathfrak{m}_A,k)$ be a local noetherian ring and $I$ an $\mathfrak{m}_A$-primary ideal. The asymptotic Samuel function (with respect to $I$) $\bar{v}_I$ $:$ $A\longrightarrow \mathbb{R}\cup {+\infty}$ is defined by $\bar{v}_I(x)=lim_{k \to \+infty}\frac{ord_I(x^k}{k}$, $\forall x \in A$. Similary, one defines for another ideal $J$, $\bar{v}_I(J)$ as the minimum of $\bar{v}_I(x)$ as $x$ varies in $J$. Of special interest is the rational number $\bar{v}_I(\mathfrak{m}_A)$. We study the behavior of the Asymptotic Samuel Function (with respect to $I$) when passing to hyperplanes sections of $A$ as one does for the theory of mixed multiplicities.