{"id":"a1ad512c-cc1c-4c9e-be6a-c5250e55dc78","arxiv_id":"2608.06882","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines log h- and v-topologies, identifies log v-covers with universally subtrusive morphisms, and proves lv-descent for log étale cohomology with torsion coefficients.","lead":"This paper introduces logarithmic versions of the h- and v-topologies for logarithmic schemes and proves that log étale cohomology with torsion coefficients satisfies descent for the new v-topology. If correct, it gives algebraic geometers a flexible way to transfer cohomological invariants from smooth spaces to singular and logarithmic ones.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.21's proof falsely asserts P^gp ≅ P_i^gp for all cones; the rigidity of valuative monoids and hence Theorem 3.23 are not established.","rationale":"The reader's weakest_assumption identifies Theorem 3.21, and I agree this is the most load-bearing step: it is the unique input that makes dividing covers lv-covers and gives the forward direction of the main equivalence in Theorem 3.23. The concrete false assertion in its proof means the theorem is unproven as written. A secondary gap in Theorem 4.10 (the factorization Y→Y×_X X→X is trivial and does not reduce logv-descent to the claimed three cases) further weakens the main application, but it is downstream and potentially repairable. For these reasons the reader's CONDITIONAL verdict is appropriate; the concerns are specific and falsifiable, not a rejection of the framework.","tokens_in":117,"tokens_out":21239,"duration_ms":531033,"concrete_test":"Test Theorem 3.21 with P=N, Σ=Spec N^2, and Δ the standard subdivision into cones cone(e1,e1+e2) and cone(e1+e2,e2), taking Spec N→Spec N^2 given by φ(a,b)=a. Compute Spec N ×_{Spec N^2} Δ explicitly: if the projection is an isomorphism, the theorem survives this case; then check whether the proof's assertion P^gp ≅ P_i^gp is used for the lower-dimensional face Q along (0,1), where P_i collapses. To probe non-discrete P, repeat with P = the lexicographic nonnegative elements of Z^2 and a map R→P with nontrivial kernel; an isomorphism would require the subdivision to induce no new points, which the current proof does not justify.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The main theorem identifying logv-covers with universally subtrusive morphisms rests on Theorem 3.21, which claims that for any valuative monoid P and subdivision Δ→Σ, the projection Spec P ×_Σ Δ → Spec P is an isomorphism. The proof of Theorem 3.21 contains an unjustified step: it sets P_i := P ⊕_R Q_i for the cones Q_i of Δ and asserts 'Observe that we have P^gp ≅ P_i^gp' for every i. This is false in general. Take R = N^2, P = N with φ(a,b)=a, and Q = N with ψ(a,b)=b (the face along the b-axis of a standard subdivision of N^2). Then the pushout P ⊕_R Q is the trivial monoid, so (P_i)^gp = 0 while P^gp = Z. The subsequent identification P_{F_i} ≅ P_i and the gluing argument depend on this assertion. The later invocation of 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 justify why the generic point lift suffices. Unless Theorem 3.21 is established by a correct argument (e.g., by restricting the assertion to maximal cones and handling face overlaps separately), Proposition 3.22 and the forward direction of Theorem 3.23 collapse, taking with them the geometric meaning of lv-covers.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":24794,"tokens_out":10058,"duration_ms":106022,"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":[{"comment":"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.","section":"§3, Theorem 3.21"},{"comment":"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.","section":"§4.1, proof of Theorem 4.10"},{"comment":"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.","section":"§3, Proposition 3.22 and Theorem 3.23"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Example 3.7"},{"comment":"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.","section":"Definition 2.9"},{"comment":"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.","section":"Proposition 2.7"}],"recommendation":"major_revision","confidential_remarks":"The two main proof gaps are substantial, but both may be repairable: Theorem 3.21 may admit a correct proof by a different argument, and the descent proof in Theorem 4.10 may be reorganized. The paper's heavy reliance on the authors' prior works [5]--[7] makes independent verification demanding; the editor may wish to seek an additional report focused on Section 4.1. Section 4.2 is explicitly conditional and should not be advertised as an unconditional theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nNow 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.\n\nThe 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.\n\nSalvage: 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.\n\nWho 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.","headline":"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.","tokens_in":25291,"tokens_out":8454,"would_cite":false,"duration_ms":90817,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A21","14F42"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["logarithmic h-topology","logarithmic v-topology","log valuation rings","valuative monoids","universally subtrusive morphisms","log étale cohomology","logarithmic motives","descent"],"falsifier":"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.","tokens_in":1802,"feed_emoji":"🪵","tokens_out":2300,"duration_ms":90722,"temperature":0.7,"pith_summary":"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.","feed_headline":"Log v-covers are exactly universally subtrusive maps","feed_subtitle":"A logarithmic v-topology makes torsion log étale cohomology a sheaf.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"The standard monograph on logarithmic algebraic geometry; it supplies valuative monoids, fans, monoschemes, the valuative criterion for proper monoschemes, and the log étale cohomology framework used throughout.","marker":"[26]"},{"why":"A foundational study of submersive and subtrusive morphisms of schemes; its valuative criterion is the scheme-theoretic statement that Theorem 3.23 extends to log schemes.","marker":"[30]"},{"why":"A result on integral morphisms and log blow-ups used to reduce proofs, including Theorem 3.23 and Theorem 4.10, to integral morphisms.","marker":"[19]"},{"why":"Another integrality reduction, cited in Theorem 3.23 to allow the use of fiber products in the category of log schemes.","marker":"[18]"},{"why":"Defines dividing covers and log Gysin sequences; it supplies the fact that dividing covers are log v-covers and the reduction to smooth log schemes in the study of differentials.","marker":"[6]"},{"why":"Nakayama's log étale cohomology provides proper base change, localization sequences, and the Kummer étale site used in the proof of Theorem 4.10.","marker":"[22]"},{"why":"A further log étale cohomology reference used for the equivalence between Kummer étale and log étale cohomology for torsion sheaves and for filtered colimit results.","marker":"[24]"},{"why":"The arc-topology proof of v-descent for étale cohomology is the scheme-theoretic template that Theorem 4.10 adapts to logarithmic geometry.","marker":"[2]"}],"fun_headline_variants":["Log v-covers: exactly universally subtrusive maps","Valuative criterion for log v-covers matches scheme case","Torsion log étale cohomology descends along log v-covers","Log v-descent for torsion sheaves from subtrusive maps"],"cache_read_input_tokens":27392,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Log v-covers: exactly universally subtrusive maps","Valuative criterion for log v-covers matches scheme case","Torsion log étale cohomology descends along log v-covers","Log v-descent for torsion sheaves from subtrusive maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000258,"raw_usage":{"total_tokens":1470,"prompt_tokens":724,"completion_tokens":746,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":340,"completion_tokens_details":{"reasoning_tokens":671}},"tokens_in":340,"tokens_out":746,"duration_ms":7590,"temperature":1.0,"reasoning_tokens":671,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:01:18.224057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ogus , Lectures on logarithmic algebraic geometry , Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2018","cited_arxiv_id":null,"evidence_quote":"The standard monograph on logarithmic algebraic geometry; it supplies valuative monoids, fans, monoschemes, the valuative criterion for proper monoschemes, and the log étale cohomology framework used throughout."},{"cited_title":"Rydh , Submersions and effective descent of \\'etale morphisms , Bulletin de la Soci\\'et\\'e Math\\'ematique de France, 138 (2010), pp","cited_arxiv_id":null,"evidence_quote":"A foundational study of submersive and subtrusive morphisms of schemes; its valuative criterion is the scheme-theoretic statement that Theorem 3.23 extends to log schemes."},{"cited_title":"831--843","cited_arxiv_id":null,"evidence_quote":"A result on integral morphisms and log blow-ups used to reduce proofs, including Theorem 3.23 and Theorem 4.10, to integral morphisms."},{"cited_title":"Kato , Log smooth deformation and moduli of log smooth curves , Internat","cited_arxiv_id":null,"evidence_quote":"Another integrality reduction, cited in Theorem 3.23 to allow the use of fiber products in the category of log schemes."},{"cited_title":"433 of Ast \\'e risque, Paris: Soci \\'e t \\'e Math \\'e matique de France (SMF), 2022","cited_arxiv_id":null,"evidence_quote":"Defines dividing covers and log Gysin sequences; it supplies the fact that dividing covers are log v-covers and the reduction to smooth log schemes in the study of differentials."},{"cited_title":"Nakayama , Logarithmic \\' e tale cohomology , Math","cited_arxiv_id":null,"evidence_quote":"Nakayama's log étale cohomology provides proper base change, localization sequences, and the Kummer étale site used in the proof of Theorem 4.10."},{"cited_title":"Math., 314 (2017), pp","cited_arxiv_id":null,"evidence_quote":"A further log étale cohomology reference used for the equivalence between Kummer étale and log étale cohomology for torsion sheaves and for filtered colimit results."},{"cited_title":"Bhatt and A","cited_arxiv_id":null,"evidence_quote":"The arc-topology proof of v-descent for étale cohomology is the scheme-theoretic template that Theorem 4.10 adapts to logarithmic geometry."}],"review_version":1}