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

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

2010-10-06

arxiv.org/abs/1010.1061v2

Mathematics

Commutative Algebra

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

Scientific paper

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.

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