A remark on symbolic powers

Abstract

By an easy application of Skoda's theorem on ideal generation, a non-local version of the Briancon-Skoda theorem is obtained. In particular, the symbolic powers $I^{(p)}$ of a zero dimensional radical ideal $I$ generated by $r$ holomorphic functions on an $n$-dimensional Stein manifold are shown to satisfy $I^{(p+q)}\subset I^p$ for $q=\min\{n,r-1\}$ and all natural $p$, which contributes to the so-called containment problem.