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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.01480 |
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| _version_ | 1866918207356403712 |
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| author | Polstra, Thomas |
| author_facet | Polstra, Thomas |
| contents | 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 $φ: F^{e+e_0}_*R \to R$ such that $φ(F^{e+e_0}_*x) \notin I$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong $F$-regularity and the Uniform Symbolic Topology Property Polstra, Thomas Commutative Algebra 13A15, 13A35 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 $φ: F^{e+e_0}_*R \to R$ such that $φ(F^{e+e_0}_*x) \notin I$. |
| title | Strong $F$-regularity and the Uniform Symbolic Topology Property |
| topic | Commutative Algebra 13A15, 13A35 |
| url | https://arxiv.org/abs/2411.01480 |