Uniform equivalence of symbolic and adic topologies

Abstract

Let $(R,m)$ be a local ring. We study the question of when there exists a positive integer $h$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(hn)}$ is contained in $P^n$, for all $n\geq1$. We show that such an $h$ exists when $R$ is a reduced isolated singularity such that $R$ either contains a field of positive characteristic and $R$ is $F$-finite or $R$ is essentially of finite type over a field of characteristic zero.