A Less Restrictive Brian\c{c}on-Skoda Theorem with Coefficients

TL;DR

This paper improves the Briançon-Skoda theorem for certain F-rational local rings by establishing new containment results involving integral closures of powers of ideals, under less restrictive conditions.

Contribution

It extends existing theorems by providing less restrictive conditions for containment of integral closures in F-rational and Gorenstein rings.

Findings

Integral closure of I^ell contained in J I_{ell-1} under new conditions

In Gorenstein rings, integral closure of I^{ell-1} contained in J

Improves upon previous results by Aberbach and Huneke

Abstract

The Brian\c{c}on-Skoda theorem in its many versions has been studied by algebraists for several decades. In this paper, under some assumptions on an F-rational local ring $(R,\m)$, and an ideal $I$ of $R$ of analytic spread $\ell$ and height $g < \ell$, we improve on two theorems by Aberbach and Huneke. Let $J$ be a reduction of $I$. We first give results on when the integral closure of $I^\ell$ is contained in the product $J I_{\ell-1}$, where $I_{\ell-1}$ is the intersection of the primary components of $I$ of height $\leq \ell-1$. In the case that $R$ is also Gorenstein, we give results on when the integral closure of $I^{\ell-1}$ is contained in $J$.