The Brian\c{c}on-Skoda Theorem and Coefficient Ideals for Non m-Primary Ideals

TL;DR

This paper extends the Briançon-Skoda theorem to non m-primary ideals in regular local rings, establishing new inclusions involving coefficient ideals and integral closures, thus broadening its applicability.

Contribution

It generalizes the Briançon-Skoda theorem to non m-primary ideals, providing new containment results involving coefficient ideals in regular local rings.

Findings

Proves a generalized Briançon-Skoda type inclusion for non m-primary ideals.

Establishes the role of coefficient ideals in integral closure containment.

Extends known results from m-primary to more general ideals.

Abstract

We generalize a Brian\c{c}on-Skoda type theorem first studied by Aberbach and Huneke. With some conditions on a regular local ring $(R,\m)$ containing a field, and an ideal $I$ of $R$ with analytic spread $\ell$ and a minimal reduction $J$, we prove that for all $w \geq -1$, $ \bar{I^{\ell+w}} \subseteq J^{w+1} \mathfrak{a} (I,J),$ where $\mathfrak{a}(I,J)$ is the coefficient ideal of $I$ relative to $J$, i.e. the largest ideal $\mathfrak{b}$ such that $I\mathfrak{b}=J\mathfrak{b}$. Previously, this result was known only for $\m$-primary ideals.