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On topologies on the space of valuations and the valuative tree

Vinicius Manfredini, Josnei Novacoski, Caio Henrique Silva de Souza

Abstract

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.

On topologies on the space of valuations and the valuative tree

Abstract

In this paper, we discuss topological aspects of the space of valuations and the valuative tree . We present a relation between the weak tree topology and the Scott topology in 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 and prove that it is closed if we consider the natural product topology.

Paper Structure

This paper contains 22 sections, 36 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The set $\mathbb{V}$
  4. Topologies on $\Lambda$-trees
  5. $\Lambda$-trees
  6. The weak tree topology
  7. Key polynomials and tangent directions
  8. The Scott topology
  9. The valuative tree $\mathcal{T}_{\rm sme}$
  10. Construction of $\mathbb{R}^{\mathbb{I}}$, $\mathbb{R}_{\rm sme}$ and $\Gamma_{\rm sme}$.
  11. Quasi-cuts
  12. The subtree $\mathcal{T}_{\rm sme}$
  13. Upper bounds and suprema in $\mathcal{T}(v,\Lambda)$
  14. Upper bounds of increasing families of valuations
  15. Supremum of an increasing family of valuations
  16. ...and 7 more sections

Key Result

Lemma 2.5

Let $\mu, \nu$ be equivalent valuations. Then: $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Interval representation in the valuative tree
  • Figure 2: Representation of Lemma \ref{['lema1']} in the valuative tree.
  • Figure 3: Representations of the elements of $\mathcal{T}_{\mu}$ in the valuative tree.

Theorems & Definitions (84)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 74 more