Differential and symbolic powers of ideals
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908287369216000 |
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| author | De Stefani, Alessandro Grifo, Eloísa Jeffries, Jack |
| author_facet | De Stefani, Alessandro Grifo, Eloísa Jeffries, Jack |
| contents | We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Differential and symbolic powers of ideals De Stefani, Alessandro Grifo, Eloísa Jeffries, Jack Commutative Algebra Algebraic Geometry We characterize symbolic powers of prime ideals in polynomial rings over any field in terms of $\mathbb{Z}$-linear differential operators, and of prime ideals in polynomial rings over complete discrete valuation rings with a $p$-derivation $δ$ in terms of $\mathbb{Z}$-linear differential operators and of $δ$. This extends previous work of the same authors, as it allows the removal of separability hypotheses that were otherwise necessary. The absence of separability and the fact that modules of $\mathbb{Z}$-linear differential operators are typically not finitely generated require the introduction of new techniques. As a byproduct, we extend a characterization of symbolic powers due to Cid-Ruiz which also holds in the nonsmooth case. Finally, we produce an example of an unramified discrete valuation ring that has no $p$-derivations. |
| title | Differential and symbolic powers of ideals |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2503.21754 |