The Briançon-Skoda Theorem and Coefficient Ideals for Non m-Primary Ideals

Abstract

We generalize a Briançon-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 $w\ge -1$, $\overline{I^{\ell +w}}\subseteq J^{w+1}\mathfrak{a}(I,J),$ 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.

5 pages. To appear in Proc. Amer. Math. Soc