Groups of invertible ideals of one-dimensional Prüfer domains as groups of integer-valued functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912970854891520 |
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| author | Spirito, Dario |
| author_facet | Spirito, Dario |
| contents | Let $G$ be a one-dimensional $\ell$-subgroup of the group $\mathcal{F}(X,\mathbb{Z})$ of integer-valued functions on a set $X$. We show that $G$ is free under some hypothesis on the spectrum of $G$ and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of a Prüfer domain $D$: in particular, we introduce the class of \emph{dd-domains} as the class of Prüfer domains having a set $X$ that is dense in $\mathrm{Spec}(D)$ (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Prüfer domains for which $\mathrm{Inv}(D)$ is isomorphic (as an $\ell$-group) to a subgroup of $\mathcal{F}(X,\mathbb{Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Groups of invertible ideals of one-dimensional Prüfer domains as groups of integer-valued functions Spirito, Dario Commutative Algebra Let $G$ be a one-dimensional $\ell$-subgroup of the group $\mathcal{F}(X,\mathbb{Z})$ of integer-valued functions on a set $X$. We show that $G$ is free under some hypothesis on the spectrum of $G$ and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of a Prüfer domain $D$: in particular, we introduce the class of \emph{dd-domains} as the class of Prüfer domains having a set $X$ that is dense in $\mathrm{Spec}(D)$ (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Prüfer domains for which $\mathrm{Inv}(D)$ is isomorphic (as an $\ell$-group) to a subgroup of $\mathcal{F}(X,\mathbb{Z})$. |
| title | Groups of invertible ideals of one-dimensional Prüfer domains as groups of integer-valued functions |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2603.16322 |