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.

Erroneous Corollaries 3.2 and 3.4 on symbolic powers of radical positive dimensional ideals are removed. The author is grateful to Lawrence Ein, Tai Ha, and Robert Lazarsfeld for pointing out the errors