The Briançon-Skoda theorem and coefficient ideals for non-$\mathfrak {m}$-primary ideals

Abstract:

We generalize a Briançon-Skoda type theorem first studied by Aberbach and Huneke. With some conditions on a regular local ring $(R,\mathfrak {m})$ containing a field, and an ideal $I$ of $R$ with analytic spread $\ell$ and a minimal reduction $J$, we prove that for all $w \geq -1$, $\overline {I^{\ell +w}} \subseteq J^{w+1} \mathfrak {a} (I,J),$ 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 $\mathfrak {m}$-primary ideals.