Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916826125959168 |
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| author | Baron, David Loepp, S. |
| author_facet | Baron, David Loepp, S. |
| contents | Given a finite poset $X$, we find necessary and sufficient conditions for there to exist a local Noetherian UFD $A$ and a saturated embedding of posets $ϕ: X \longrightarrow \mbox{Spec}(A)$ such that $\dim(X)=\dim(A)$. The conditions imposed on $X$ in our characterization are remarkably mild, demonstrating that there is a large class of finite posets that can be embedded into the spectrum of a local Noetherian UFD of the same dimension as $X$ in a way that preserves saturated chains. We also show that given any finite poset $Y$, there exists a semi-local quasi-excellent ring $S$ and a saturated embedding $ψ: Y \longrightarrow \mbox{Spec}(S)$ such that if $z$ is a minimal element of $Y$, then $ψ(z)$ is a minimal prime ideal of $S$ and the coheight of $ψ(z)$ is the same as the length of the longest chain in $Y$ that starts at $z$ and ends at a maximal element of $Y$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs Baron, David Loepp, S. Commutative Algebra Combinatorics 13F15, 13E05, 13F40 (Primary) 13J10, 06A11 (Secondary) Given a finite poset $X$, we find necessary and sufficient conditions for there to exist a local Noetherian UFD $A$ and a saturated embedding of posets $ϕ: X \longrightarrow \mbox{Spec}(A)$ such that $\dim(X)=\dim(A)$. The conditions imposed on $X$ in our characterization are remarkably mild, demonstrating that there is a large class of finite posets that can be embedded into the spectrum of a local Noetherian UFD of the same dimension as $X$ in a way that preserves saturated chains. We also show that given any finite poset $Y$, there exists a semi-local quasi-excellent ring $S$ and a saturated embedding $ψ: Y \longrightarrow \mbox{Spec}(S)$ such that if $z$ is a minimal element of $Y$, then $ψ(z)$ is a minimal prime ideal of $S$ and the coheight of $ψ(z)$ is the same as the length of the longest chain in $Y$ that starts at $z$ and ends at a maximal element of $Y$. |
| title | Dimension-Preserving Saturated Embeddings of Finite Posets into the Spectra of Noetherian UFDs |
| topic | Commutative Algebra Combinatorics 13F15, 13E05, 13F40 (Primary) 13J10, 06A11 (Secondary) |
| url | https://arxiv.org/abs/2507.03574 |