The logarithmic $h$- and $v$-topologies

Doosung Park; Nikolai Opdan; Paul Arne {\O}stv{\ae}r

REVIEW 3 major objections 4 minor 34 references

The logarithmic $h$- and $v$-topologies

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper introduces logarithmic h- and v-topologies for fs log schemes and proves that log v-covers are exactly universally subtrusive morphisms, yielding log v-descent for log étale cohomology with torsion coefficients.

desk verdict The logv-cover theorem rests on a false pushout claim in Theorem 3.21, so the paper's central structural result is unproved; nevertheless the framework and intended descent theorem are worth refereeing carefully. read the letter →

arxiv 2608.06882 v1 pith:N4HKFWNO submitted 2026-08-07 math.AG

Nikolai Opdan , Doosung Park , Paul Arne {O}stv{ae}r This is my paper · ORCID

classification math.AG MSC 14A2114F42

keywords logarithmich-topologyv-topologylogvaluationringsvaluativemonoidsuniversallysubtrusivemorphismsétalecohomologymotivesdescent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper builds logarithmic versions of the h- and v-topologies for fine and saturated (fs) log schemes, which are the topologies that let singular varieties be treated as if they were smooth. Its central result is that a morphism of quasi-compact, quasi-separated fs log schemes is a log v-cover precisely when it is universally subtrusive, giving a logarithmic analogue of the v-topology. From this, the paper proves that log étale cohomology with torsion coefficients satisfies descent for log v-covers, so this cohomology becomes an lv-sheaf. The same framework is applied to logarithmic differential forms in characteristic zero and to the construction of stable categories of logarithmic h-motives.

What carries the argument

The central object is a log valuation ring $(V,P)$: $V$ is a valuation ring and $P$ is a valuative monoid, meaning that for every element of its group completion, either that element or its inverse lies in $P$. These rings lift specializations of points on log schemes, just as ordinary valuation rings do for schemes, and they give a clean definition of log v-covers through an extension lifting property. The load-bearing mechanism is Theorem 3.21: for any valuative monoid $P$ and any subdivision of fans $\Delta \to \Sigma$, the projection $\operatorname{Spec} P \times_\Sigma \Delta \to \operatorname{Spec} P$ is an isomorphism, which says the dual fan of a valuative monoid is too rigid to admit exotic subdivisions.

What would settle it

Find a valuative monoid $P$ and a subdivision of fans $\Delta \to \Sigma$ for which $\operatorname{Spec} P \times_\Sigma \Delta \to \operatorname{Spec} P$ is not an isomorphism, equivalently a subdivision for which every face $F_i$ of $P$ is a proper face; such an example would make Proposition 3.22 and the forward direction of Theorem 3.23 false. A concrete place to search is the value monoid of a valuation ring of rank at least two, where the dual fan might admit proper subdivisions.

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Extended reading notes

Core claim

The core discovery is that the valuative criterion for log schemes can be formulated using log valuation rings, each consisting of a valuation ring together with a valuative monoid mapping logarithmically into its multiplicative monoid. With this notion, a morphism of qcqs fs log schemes is a log v-cover exactly when it is universally subtrusive (Theorem 3.23), matching the scheme-theoretic characterization of v-covers. The proof rests on Theorem 3.21, which shows that the dual fan of a valuative monoid admits no nontrivial subdivisions, so the relevant fiber products are isomorphisms. Consequently, Theorem 4.10 shows that for a base scheme with trivial log structure and a torsion sheaf $G$, the functor $X \mapsto R\Gamma_{l\acute{e}t}(X,\alpha^*G)$ satisfies log v-descent.

Load-bearing premise

The argument rests on the claim that a valuative monoid's dual fan is too rigid to admit any nontrivial subdivision, so that every pullback $\operatorname{Spec} P \times_\Sigma \Delta \to \operatorname{Spec} P$ is an isomorphism; if that rigidity fails, the identification of dividing covers with log v-covers and the full equivalence in Theorem 3.23 collapse.

Editorial extensions

If this is right

The log v-topology on qcqs fs log schemes coincides with the universally subtrusive topology, so log v-covers form a class closed under base change and composition.
Log étale cohomology with torsion coefficients becomes a sheaf for the log v-topology, extending the classical fact that étale cohomology satisfies v-descent to logarithmic geometry.
Dividing covers and log blow-ups are log v-covers, so the new topology subsumes logarithmic birational modifications.
The constant torsion sheaf $\mathbb{Z}/n$ is representable in the $S^1$-stable category of logarithmic h-motives.
Under Conjectures 4.17 and 4.18, logarithmic differential forms on log smooth schemes are unchanged by log h-sheafification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the log v-descent theorem extends to integral coefficients or constructible complexes, it would yield logarithmic analogues of arc-descent with a tractable valuation-theoretic cover class.
The rigidity theorem suggests that valuative monoids behave like points in the fan world, which may imply that log v-covers are insensitive to certain log blow-up refinements.
For schemes with trivial log structure, the new log v-topology should recover the classical v-topology, making existing v-descent results a special case.
The conjectural invariance of log h-differentials points toward a definition of log de Rham cohomology on singular log schemes; the log point $\operatorname{pt}_{\mathbb{N},k}$ is a natural first test case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First: this is the right kind of paper to argue about, but the central structural claim is not proved. The proof of Theorem 3.21 contains a false monoid isomorphism, and the subsequent characterization of logv-covers rests on it. The paper is not ready as written.

What is genuinely useful: log valuation rings are a sensible notion, and Theorem 2.15, which lifts specializations by log valuation rings, is a clean contribution. The definitions of the log h- and v-topologies are the natural ones, and the torsion-cohomology descent statement in Theorem 4.10 is exactly the logarithmic analog one would want. The organization is transparent, the authors mark the conditional material in Section 4.2 honestly, and the heavy reliance on their own prior work is not a problem by itself since those are published external results.

Now the soft spots. In the proof of Theorem 3.21, after setting P_i := P ⊕_R Q_i, the text says "Observe that we have P^gp ≅ P_i^gp." That is false. Take R = N^2, P = N mapping to the a-axis, and Q = N mapping to the b-axis. The pushout monoid is trivial: the relation from φ(1,0) = ψ(1,0) forces 1_P = 0_Q, and the relation from φ(0,1) = ψ(0,1) forces 0_P = 1_Q, so the whole monoid collapses to its identity. Hence P_i^gp = 0 while P^gp = Z. This is not a minor typo: it underpins the identification P_{F_i} ≅ P_i and the gluing argument. Proposition 3.22 and the forward direction of Theorem 3.23 collapse with it, and the geometric meaning of logv-covers as universally subtrusive maps is lost. The example also suggests Theorem 3.21 as stated cannot hold, not merely that the proof needs patching.

The proof of Theorem 4.10 has a separate apparent problem. After invoking Rydh [30, Theorem 3.12], the factorization Y → Y ×_X X → X is just the identity, both in schemes and in log schemes, so it gives no reduction to the three cases listed. Perhaps a different fiber product or a different refinement was intended, but as written the descent proof does not work.

Salvage: the intended theorems may still be true, and the counterexample targets a bad proof step rather than a known external result. A repair of Theorem 3.21, perhaps by restricting to maximal cones and handling face overlaps separately, would likely restore Theorem 3.23, and the descent proof could then be fixed by a cleaner reduction. But the current version should not be accepted.

Who this is for: log geometers and motivic homotopy theorists. It deserves a serious referee because the definitions and intended theorems are important, but the referee should be told to focus on Theorem 3.21 and the reduction in Theorem 4.10. I would send it to peer review rather than desk reject.

Referee Report

3 major / 4 minor

Summary. The paper develops logarithmic analogues of Voevodsky's h-topology and Bhatt--Scholze's v-topology for quasi-compact quasi-separated fine saturated log schemes. It introduces log valuation rings, proves a log analogue of the fact that valuation rings lift specializations (Theorem 2.15), defines logv-covers via an extension property, and claims in Theorem 3.23 that logv-covers coincide with universally subtrusive morphisms. The paper then applies this to prove logv-descent for log étale cohomology with torsion coefficients (Theorem 4.10), studies the logh-sheafification of logarithmic differentials conditional on two conjectures (Section 4.2), and constructs stable logh-motives with representability of Z/n in S^1-stable logh-motives (Proposition 4.24).

Significance. If the main results are correct, the paper would establish a genuine logarithmic version of the v-topology and provide a powerful descent statement for torsion log étale cohomology, with consequences for logarithmic motives. The paper is clearly organized, gives careful definitions, and is honest about conditional results: Conjectures 4.17 and 4.18 are explicitly flagged, and Theorem 4.20 and Corollary 4.21 are stated as consequences of those conjectures. The paper relies on the authors' published prior work rather than restating claims from this manuscript, so there is no evident circularity. However, as detailed below, the proofs of Theorem 3.21 and of the descent theorem contain serious gaps that currently leave the central claims unsupported.

major comments (3)

[§3, Theorem 3.21] The proof of Theorem 3.21 is not valid. After defining P_i := P ⊕_R Q_i, the text asserts 'Observe that we have P^gp ≅ P_i^gp'. This assertion is false in general. For example, take R = N^2, P = N with (a,b) ↦ a, and Q = N with (a,b) ↦ b; then the pushout P ⊕_R Q is the trivial monoid, so (P_i)^gp = 0 while P^gp = Z. Consequently the subsequent identification P_{F_i} ≅ P_i and the gluing argument are unsupported. Moreover, the later appeal to the valuative criterion [26, Theorem II.1.6.3] supplies a lift only after an extension of the valuation monoid, not a lift of Spec P itself, and the proof does not explain why that suffices for the isomorphism claimed in (1). The statement of Theorem 3.21 may still be true, but the proof as written does not establish it. Since Theorem 3.21 is used in Proposition 3.22 and in the forward direction of Theorem 3.23, the identification of logv-covers with universally subtrusive morphisms is not currently proved.
[§4.1, proof of Theorem 4.10] The reduction in the proof of Theorem 4.10 after Lemma 4.12 is circular. The text factors f as Y →^g Y ×_X X →^h X and asserts that 'g is an isomorphism and h is strict'. In any category with fiber products, Y ×_X X is canonically isomorphic to Y, so g is the identity morphism and h = f; in particular, h is strict only if f was already strict. Thus the 'reduction' to the three cases 'quasi-compact open covering', 'strict proper surjective', and 'isomorphism' does not cover the general saturated morphism f under consideration. This step is load-bearing because it is what converts an arbitrary logv-cover into the cases handled by Lemmas 4.6--4.8. The descent theorem is therefore not proved as written.
[§3, Proposition 3.22 and Theorem 3.23] The converse direction of Theorem 3.23 and the proof of Proposition 3.22 both depend on Theorem 3.21. Proposition 3.22 asserts that every dividing cover is a logv-cover, and its proof uses Theorem 3.21 to obtain the commutative diagram (3). Theorem 3.23 then uses Proposition 3.22 in the forward direction to pass to a dividing cover and apply [18, Theorem 1.1]. Since Theorem 3.21 is unproved, both the statement that dividing covers are logv-covers and the equivalence in Theorem 3.23 are currently unsupported. The paper needs a correct proof of Theorem 3.21 or a clearly stated weakening that still suffices for the later arguments.

minor comments (4)

[Throughout] There are several typographical issues: 'Suppose se that' in the proof of Theorem 2.15, 'decent' in the Section 4.1 heading should be 'descent', 'We proof will be finished' in the proof of Theorem 3.21, and 'd≥0..' with a double period in Lemma 4.6.
[Example 3.7] The pullback square in Example 3.7 is hard to parse because the morphisms are not all labelled; please label the diagonal morphism and the log blow-up explicitly so that the reader can follow the claim that the pullback morphism is not surjective.
[Definition 2.9] The notation A_P is introduced but the sentence reads 'let A_P denote the log scheme associated to the log ring P → Z[P]'; it would be clearer to write 'let A_P denote the log scheme associated with the log ring...' for grammatical consistency.
[Proposition 2.7] The proof uses the notations P^+ and Q^+ without defining them; please define P^+ = P \ P^* (the non-units of a monoid) before using this notation.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: the logv/universal-subtrusiveness identification is proved internally; self-citations are to published external lemmas, and the serious gap in Theorem 3.21 is a correctness issue, not a circular reduction.

full rationale

No circular step is exhibited. Definition 3.16 defines logv-covers by a log valuation ring lifting property, and Theorem 3.23 is then proved by using Theorem 2.15 and Lemma 3.15 to move between universal subtrusiveness and that lifting property; the conclusion is not assumed in the definition. The self-citations that are load-bearing ([6, Lemma A.3.11] for the P=N case of Theorem 3.21, [6, Prop. A.11.5] in Proposition 3.22, [6, Theorem 7.5.4] in Theorem 4.20, [7, Prop. 3.2.18] in Proposition 4.24) are to published or preprint results with fixed assumptions that do not include the target statements, so under the stated rules they count as external evidence rather than circularity. Section 4.2 is explicitly conditional on Conjectures 4.17 and 4.18, so Theorem 4.20 is not a hidden restatement of its own input. I do flag a serious non-circular correctness gap: the proof of Theorem 3.21 asserts “Observe that we have P^gp ≅ P_i^gp” after defining P_i := P ⊕_R Q_i, which is false in general (e.g., R=N^2, P=N embedded in one factor, Q=N in the other gives P_i^gp=0). The subsequent conclusion SpecP ×_Σ Δ ≅ SpecP, and therefore Proposition 3.22 and the forward direction of Theorem 3.23, are not established by the written argument. That is a mathematical error or missing argument, not a reduction of the theorem to its own input; hence the circularity score is low.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

There are no fitted numerical parameters. The new objects introduced, log valuation rings and the lh/lv topologies, are definitions rather than postulated entities in the sense of new particles or forces. The conditional results in Section 4.2 rest on two unproved conjectures, listed above as paper-specific axioms.

assumptions (6)

standard math Valuative criterion for proper morphisms of monoschemes [26, Theorem II.1.6.3] applies to the lifting problem in Theorem 3.21.
The proof of Theorem 3.21 uses this criterion to lift the generic point of Spec P to the subdivision Δ; this is a background result from Ogus's monograph.
domain assumption Every finitely presented morphism of fs log schemes is, after a dividing cover on the target, integral [19, Theorem 1.1].
Used repeatedly in Theorem 3.23, Theorem 3.26, and the proof of Theorem 4.10 to reduce to integral morphisms.
domain assumption A quasi-compact morphism of schemes is subtrusive if it lifts specializations, per Rydh [30, Corollary 2.9, Remark 2.5].
Used in Theorem 3.23 to conclude that a logv-cover is subtrusive from specialization lifting alone.
domain assumption Nakayama's proper base change theorem and description of Kummer étale sheaves at log points [22, Theorem 5.1, Proposition 4.6].
Foundational input for Lemmas 4.5, 4.6, 4.8, 4.12 and for the proof of Theorem 4.10.
ad hoc to paper Conjecture 4.17: RΓ_lh(-, Ω^q_lh) is (P^m, P^m-1)-invariant for all m and q over a field of characteristic 0.
Unproved and paper-specific; assumed in Theorem 4.20.
ad hoc to paper Conjecture 4.18: For a smooth scheme with trivial log structure, RΓ(X, Ω^q) → RΓ_lh(X, Ω^q_lh) is a quasi-isomorphism.
Unproved and paper-specific; assumed in Theorem 4.20.

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Pith. "Pith review of The logarithmic $h$- and $v$-topologies." pith.science (2026). https://pith.science/paper/N4HKFWNO

@misc{pith2026260806882,
  author       = {Pith},
  title        = {Pith review of: The logarithmic $h$- and $v$-topologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4HKFWNO}},
  note         = {Machine review of arXiv:2608.06882}
}

abstract

We introduce the $h$- and $v$-topologies in the context of logarithmic geometry and discuss their applications to log \'etale cohomology, log differential forms, and log motives.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Artin, A

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work page 2021

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B. Bhatt and P. Scholze , Projectivity of the W itt vector affine G rassmannian , Inventiones mathematicae, 209 (2017), pp. 329--423

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F. Binda, T. Lundemo, D. Park, and P. A. Østvær , A H ochschild- K ostant- R osenberg theorem and residue sequences for logarithmic H ochschild homology , Advances in Mathematics, 435 (2023), p. 109354

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