Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909878826565632 |
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| author | Polstra, Thomas |
| author_facet | Polstra, Thomas |
| contents | We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain essentially of finite type over a field, there exists a constant $C$ so that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mbox{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property Polstra, Thomas Commutative Algebra Algebraic Geometry 13H15 (Primary) 13A18, 13A30, 14C17 We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain essentially of finite type over a field, there exists a constant $C$ so that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mbox{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$. |
| title | Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property |
| topic | Commutative Algebra Algebraic Geometry 13H15 (Primary) 13A18, 13A30, 14C17 |
| url | https://arxiv.org/abs/2406.00759 |