Geometric configuration of integrally closed Noetherian domains
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908799009292288 |
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| author | Chang, Gyu Whan Peruginelli, Giulio |
| author_facet | Chang, Gyu Whan Peruginelli, Giulio |
| contents | In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over $\mathbb{Z}_{(p)}$ for some prime $p \in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in $\mathbb{C}_p$, the completion of the algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of $\mathbb{C}_p$ to its valuation domain $\mathbb{O}_p$, as $p\in\mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_22314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric configuration of integrally closed Noetherian domains Chang, Gyu Whan Peruginelli, Giulio Commutative Algebra Number Theory 13F05, 13B22, 13E05, 13B25, 13F20 In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over $\mathbb{Z}_{(p)}$ for some prime $p \in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in $\mathbb{C}_p$, the completion of the algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of $\mathbb{C}_p$ to its valuation domain $\mathbb{O}_p$, as $p\in\mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. |
| title | Geometric configuration of integrally closed Noetherian domains |
| topic | Commutative Algebra Number Theory 13F05, 13B22, 13E05, 13B25, 13F20 |
| url | https://arxiv.org/abs/2601.22314 |