Strong $F$-regularity and the Uniform Symbolic Topology Property

Abstract

We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain $R$. Let $R$ be a normal domain of prime characteristic $p>0$ that is $F$-finite or essentially of finite type over an excellent local ring. Assume there exists a finite extension $R\to S$ so that the non-strongly $F$-regular locus of $\mathrm{Spec}(S)$ consists only of isolated points, then there exists a constant $C$ such that for all ideals $I \subseteq R$ and $n \in \mathbb{N}$, the symbolic power $I^{(Cn)}$ is contained in the ordinary power $I^n$. In other words, $R$ enjoys the Uniform Symbolic Topology Property. Moreover, if $R$ is $F$-finite and strongly $F$-regular, then $R$ enjoys a property that is proven to be stronger: there exists a constant $e_0 \in \mathbb{N}$ such that for any ideal $I \subseteq R$ and all $e \in \mathbb{N}$, if $x \in R \setminus I^{[p^e]}$, then there exists an $R$-linear map $\varphi: F^{e+e_0}_*R \to R$ such that $\varphi(F^{e+e_0}_*x) \notin I$.