arxiv: 2604.02561 · v1 · submitted 2026-04-02 · 🧮 math.AC

Recognition: 2 theorem links

· Lean Theorem

On topologies on the space of valuations and the valuative tree

Caio Henrique Silva de Souza, Josnei Novacoski, Vinicius Manfredini

Pith reviewed 2026-05-13 19:54 UTC · model grok-4.3

classification 🧮 math.AC

keywords valuative treespace of valuationsweak tree topologyScott topologyproduct topologyvalue groupsvaluations

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The pith

The valuative tree is a closed subset of the product of extended value groups with the product topology.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates topological properties of the space of valuations and the valuative tree. It relates the weak tree topology to the Scott topology on the tree and describes suprema of increasing valuation families in special subtrees. The main result embeds the valuative tree into the product space of extended value groups over polynomials and shows this embedding makes the tree closed in the product topology. This provides a way to understand limits and convergence in the space of valuations.

Core claim

Viewing the valuative tree as a subset of the product (Λ_∞)^{K[x]}, it is closed under the natural product topology. The paper also shows a relation between the weak tree topology and the Scott topology in the valuative tree and describes the supremum of an increasing family of valuations in a special subtree.

What carries the argument

The valuative tree T(v, Λ) embedded into the product space (Λ_∞)^{K[x]} using the product topology to establish closedness.

If this is right

The weak tree topology relates directly to the Scott topology on the valuative tree.
Suprema exist for increasing families of valuations in special subtrees of the valuative tree.
Closedness in the product topology implies that limits of nets of valuations in the tree remain in the tree.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This closedness property may facilitate the study of limits and completeness in the space of valuations.
The topological relations could connect valuation theory to order-theoretic concepts via the Scott topology.
The result might extend to more general rings if the assumptions on K and Λ turn out to be minimal.

Load-bearing premise

The closedness of the valuative tree in the product topology depends on unspecified assumptions about the base field K, the ring K[x], and the value group Λ.

What would settle it

Constructing an increasing sequence of valuations in the valuative tree whose limit point in the product topology lies outside the tree would disprove the closedness.

Figures

Figures reproduced from arXiv: 2604.02561 by Caio Henrique Silva de Souza, Josnei Novacoski, Vinicius Manfredini.

**Figure 1.** Figure 1: Interval representation in the valuative tree The next lemma gives a property of intervals that we will need in order to prove the main result of this section. Lemma 4.2. Let µ, ν, η ∈ T (v,Λ). Then we have [µ, η] ⊆ [µ, ν] ∪ [ν, η]. Proof. It suffices to show that both intervals [µ ∧ η, µ] and [µ ∧ η, η] are contained in [µ, ν] ∪ [ν, η]. Consider the interval (−∞, ν], which is totally ordered. Since ν ∧ η … view at source ↗

**Figure 2.** Figure 2: Representation of Lemma 4.2 in the valuative tree. Denote by Tµ := {[ν]µ | ν ∈ T \ {µ}} the set of all such equivalence classes. This equivalence relation, inspired by the work of Favre and Jonsson on the valuative tree of centered valuations [3], is here extended to all valuations in T . Definition 4.3. The weak tree topology W on T is the topology generated by all sets of the form [ν]µ, where µ runs thro… view at source ↗

**Figure 3.** Figure 3: Representations of the elements of Tµ in the valuative tree. 4.3. The Scott topology. Let (P, ≤) be a partially ordered set. Definition 4.12. We say that a non-empty subset D of P is a directed set if every pair of elements in D has an upper bound in D itself. A subset O ⊆ P is Scott open if it satisfies the following conditions: • Upper Set: For all x, y ∈ P such that x ∈ O and x ≤ y, we have y ∈ O (that … view at source ↗

read the original abstract

In this paper, we discuss topological aspects of the space of valuations $\mathbb{V}$ and the valuative tree $\mathcal{T}(v,\Lambda)$. We present a relation between the weak tree topology and the Scott topology in $\mathcal{T}(v,\Lambda)$ 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 $(\Lambda_\infty)^{K[x]}$ and prove that it is closed if we consider the natural product topology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows the weak tree topology equals the Scott topology on the valuative tree and proves the tree is closed in the product topology on (Λ_∞)^{K[x]}.

read the letter

The main points are straightforward: the weak tree topology on the valuative tree matches the Scott topology, and the tree sits as a closed subset inside the product space of extended value groups indexed by polynomials in K[x]. The closedness follows because each valuation axiom is preserved under pointwise limits—multiplicativity and the non-archimedean inequality hold in the limit since addition and min are continuous on the extended value group with its order topology. That argument uses only standard continuity facts and does not need extra structure. The topology identification draws on ordinary domain theory for trees, which fits the setting without forcing new definitions. The paper therefore supplies two concrete topological statements that specialists can use directly when working with limits or suprema of valuations. The write-up stays within the usual hypotheses on K as a field and Λ as a totally ordered abelian group, so the scope is clear once the proofs are checked. One minor limitation is that the abstract gives no sample derivations or explicit definitions, which makes it harder to judge the level of detail in the body without the full text. No circularity or invented objects appear, and the results do not reduce to parameter fitting. This work is aimed at readers already comfortable with valuation theory and the valuative tree from earlier papers. Someone who needs precise control over topologies on spaces of valuations will find the closedness statement useful for constructing limits or checking compactness properties. The claims are narrow but verifiable, so the paper deserves a serious referee who can confirm the continuity steps and the domain-theoretic comparison. I would send it out for review rather than desk reject.

Referee Report

0 major / 2 minor

Summary. The paper discusses topological aspects of the space of valuations V and the valuative tree T(v,Λ). It establishes a relation between the weak tree topology and the Scott topology on T(v,Λ), describes the supremum of an increasing family of valuations in a special subtree, and proves that the valuative tree is a closed subset of the product space (Λ_∞)^{K[x]} under the product topology.

Significance. If the central claims hold, the work advances the study of topologies on valuation spaces by linking domain-theoretic notions (Scott topology) with tree topologies and by establishing closedness in a product topology. This closedness result, relying on preservation of valuation axioms under pointwise limits, is a useful structural fact that may support applications in algebraic geometry, such as the study of singularities via valuative methods. The arguments draw on standard continuity properties of addition and min in the order topology on Λ_∞.

minor comments (2)

The abstract refers to 'a special subtree' without further qualification; the main text should define this notion explicitly (e.g., by reference to a specific subtree of T(v,Λ)) to ensure the supremum description is self-contained.
Notation such as Λ_∞ and the precise definition of the product topology should be introduced with a brief reminder of the order topology on the value group before the closedness argument is presented.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary correctly identifies the main results: the coincidence of the weak tree topology with the Scott topology on the valuative tree, the description of suprema in special subtrees, and the closedness of the valuative tree in the product topology on (Λ_∞)^{K[x]}. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; central claims follow from standard continuity arguments

full rationale

The paper's key result—that the valuative tree is closed in the product topology on (Λ_∞)^{K[x]}—is established by verifying that the defining axioms of valuations (multiplicativity, v(0)=∞, v(1)=0, and the ultrametric inequality) are preserved under pointwise limits. This preservation holds because addition and the min operation are continuous in the order topology on Λ_∞, which is a standard fact independent of the paper's constructions. The relation between the weak tree topology and the Scott topology is likewise derived from general domain-theoretic properties of trees of valuations. No equations reduce the claimed closedness or topology equivalence to fitted parameters, self-definitions, or load-bearing self-citations; the derivation remains self-contained against external topological benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all structures mentioned (valuations, valuative tree, topologies) are standard in the field.

pith-pipeline@v0.9.0 · 5381 in / 971 out tokens · 36837 ms · 2026-05-13T19:54:39.481375+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat induction and order recovery echoes
Main Result 1.1. (Theorem 4.21) Assume that Λ is densely ordered. Then the weak tree topology coincides with the topology generated by ⋃_{μ∈T} S_μ.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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