On topologies on the space of valuations and the valuative tree
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914442086711296 |
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| author | Manfredini, Vinicius Novacoski, Josnei de Souza, Caio Henrique Silva |
| author_facet | Manfredini, Vinicius Novacoski, Josnei de Souza, Caio Henrique Silva |
| contents | In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,Λ)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,Λ)$ and describe the supremum of an increasing family of valuations in a special subtree. We also view the valuative tree as a subset of the product $(Λ_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02561 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On topologies on the space of valuations and the valuative tree Manfredini, Vinicius Novacoski, Josnei de Souza, Caio Henrique Silva Commutative Algebra In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,Λ)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,Λ)$ and describe the supremum of an increasing family of valuations in a special subtree. We also view the valuative tree as a subset of the product $(Λ_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology. |
| title | On topologies on the space of valuations and the valuative tree |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2604.02561 |