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On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$

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Pith's one-line read The paper establishes logarithmic p-adic Hodge theory for proper log smooth rigid analytic varieties, proving Hodge–Tate and Hodge–log de Rham degeneration via a new logarithmic B_dR^+ cohomology theory.

desk verdict Substantial and mostly well-built log generalization of the Bosco/DLLZ23a program, but the load-bearing log Hodge–Tate lemma is delegated to a one-line 'similar' proof and needs to be supplied before the degenerations can be trusted. read the letter →

arxiv 2501.16916 v2 pith:GWMTVP4I submitted 2025-01-28 math.NT math.AG

Xinyu Shao This is my paper · ORCID
classification math.NTmath.AG MSC 14F3014G2214F40
keywords p-adicHodgetheorylogrigidanalyticvarietiesKummerpro-étalecohomologyB_dRHodge–TatespectralsequencedeRhamperiodsheavescondensedmathematics
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 extends p-adic Hodge theory from smooth rigid analytic varieties to log smooth ones, which carry a well-behaved boundary structure. Its main claim is that for a proper log smooth rigid analytic variety X over the p-adic completion C, the Hodge–log de Rham spectral sequence and the Hodge–Tate spectral sequence both degenerate at their first page, and that the logarithmic B_dR^+ cohomology it defines is a finite free B_dR^+-module. This yields a logarithmic de Rham–étale comparison theorem, filling the missing logarithmic analogue of the classical rigid-analytic comparison. If correct, it provides a foundation for semistable and syntomic comparison theorems in p-adic geometry.

What carries the argument

The central object is the logarithmic B_dR^+ cohomology RΓ_{B_dR^+}(X) := RΓ_{ét,cond}(X, Lη_t Rλ_* B_dR^+), where Lη_t is the décalage functor that adjusts the Hodge filtration. The key structural input is the logarithmic Hodge–Tate isomorphism R^i λ_* Ô_{X_{prokét}} ≃ $Ω^{{i,log}}$_X(-i), stated as Lemma 5.16 and asserted to be analogous to a known lemma in the smooth case. The splitting is achieved by choosing a log smooth B_dR^+/$t^{2}$-lift of X and proving that the lift gives a torsor whose class splits the Hodge–Tate map, so that Rν_*Ô_X decomposes into shifted logarithmic differentials; the degeneration theorems then follow by dimension counting.

What would settle it

Verify Lemma 5.16 explicitly for the one-dimensional toric log smooth example X = Spa(C⟨N⟩, O_C⟨N⟩), the log unit disk with log structure at the origin: if the natural map from M^gp_X to $R^{1}$λ_*Ô_{X_{prokét}}(1)▼ is not an isomorphism, or if the higher direct images differ from $Ω^{{i,log}}$_X(-i), Theorem 5.20 collapses. Alternatively, compute the Hodge–Tate spectral sequence for a proper log smooth curve over C and check whether it degenerates at E_2; a counterexample would directly falsify the theorem.

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Extended reading notes

Core claim

The paper introduces a logarithmic B_dR^+ cohomology RΓ_{B_dR^+}(X) for log smooth rigid analytic varieties over C, defined by applying the décalage functor Lη_t to Rλ_*B_dR^+ on the Kummer pro-étale site. Theorem 5.19 shows that after inverting t this cohomology is canonically isomorphic to H^i_{két}(X,Q_p)⊗_{Q_p}B_dR, and that it agrees with the known comparison isomorphism when X descends to a discretely valued field. For X proper and log smooth, Theorem 5.20 proves that the Hodge–log de Rham spectral sequence $E_1^{{ij}}$=H^j(X,$Ω^{{i,log}}$_X) degenerates at E_1, that the Hodge–Tate spectral sequence $E_2^{{ij}}$=H^j(X,$Ω^{{i,log}}$_X)(-j) degenerates at E_2, and that H^i_{B_dR^+}(X) is finite free over B_dR^+. The proof obtains a splitting of Rν_*Ô_X as ⊕_i $Ω^{{i,log}}$_X(-i)[-i] using a log smooth B_dR^+/$t^{2}$-lift of X, then counts dimensions via the primitive comparison theorem.

Load-bearing premise

The proof of the main degeneration theorem rests on Lemma 5.16, which states that the higher direct images of the structure sheaf along the Kummer pro-étale site are exactly the logarithmic differentials with Tate twists; the paper does not prove this lemma, only saying it is analogous to a known result, and if the logarithmic analogue fails the degeneration argument no longer goes through.

Editorial extensions

If this is right

  • For proper log smooth rigid varieties, the log de Rham–étale comparison (Theorem 5.19) holds, giving a filtered isomorphism between Kummer étale cohomology with B_dR coefficients and the new B_dR^+ cohomology.
  • The Hodge–log de Rham spectral sequence degenerates at E_1, so logarithmic de Rham cohomology of proper log smooth X is finite-dimensional and admits a Hodge decomposition in the p-adic sense.
  • The Hodge–Tate spectral sequence degenerates at E_2, producing a Hodge–Tate decomposition for log smooth proper rigid analytic varieties.
  • H^i_{B_dR^+}(X) is finite free over B_dR^+, so Theorem 5.13 describes B_dR^+ cohomology as a deformation of log de Rham cohomology that specializes to it modulo t.
  • When the log structure comes from a strictly normal crossing divisor D, the logarithmic B_dR^+ cohomology of X is isomorphic to the B_dR^+ cohomology of the open complement U = X - D, connecting log and ordinary pro-étale cohomology.

Reading between the lines

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

  • Supplying a full proof of Lemma 5.16 would likely allow the same framework to cover almost proper and punctured rigid spaces, extending the logarithmic Riemann–Hilbert correspondence beyond the proper case.
  • The splitting of Rν_*Ô_X via a B_dR^+/t^2-lift is probably independent of the chosen lift up to homotopy; one can test whether the induced Hodge–Tate filtration on H^k_{két}(X,Q_p) is independent of that choice.
  • The same Lη_t machinery should define logarithmic syntomic cohomology, and the paper announces this as a sequel; a natural test is whether the syntomic comparison holds for proper log smooth X without the discretely valued descent assumption.
  • A concrete check for a proper log smooth curve over C (for instance, the projective line with log structure at several points) would verify that the Hodge–Tate degeneration recovers known computations of log de Rham cohomology, testing the dimension-counting step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper develops logarithmic p-adic cohomology for log smooth rigid analytic varieties. In the first part, it computes the Kummer pro-étale cohomology of B_dR and B_dR^+ for log smooth rigid analytic varieties over a discretely valued field, with coefficients in a filtered module with integrable log connection (Theorem 4.6), generalizing Bosco's theorem to the log setting. In the second part, for X over C, it introduces a logarithmic B_dR^+-cohomology RΓ_{B_dR^+}(X) by applying the Lη_f functor to Rλ_*B_dR^+ (Definition 5.7), proves that this theory deforms log de Rham cohomology (Theorem 5.13), establishes a log de Rham–étale comparison (Theorem 5.19), and states the main degeneration theorem (Theorem 5.20): for proper log smooth X of dimension d, the Hodge–log de Rham spectral sequence degenerates at E_1, the Hodge–Tate spectral sequence degenerates at E_2, and H^i_{B_dR^+}(X) is finite free over B_dR^+.

Significance. If the main results are fully established, this is a substantial extension of Scholze's p-adic Hodge theory for proper smooth rigid analytic varieties to the logarithmic setting, with potential applications to semistable comparison theorems and to the cohomology of open varieties via compactification. The paper's structure is commendable: it reduces many statements to established results of Scholze, Bosco, Diao–Lan–Liu–Zhu, Katz, and Kiehl, and the use of condensed mathematics is consistent. The construction of logarithmic B_dR^+-cohomology via Lη_f is a natural and promising framework. No free parameters or fitted constants enter the arguments; all claims are meant to be derived from external proven results. However, the central degeneration theorem depends on two assertions whose proofs are not supplied in adequate detail, in particular the logarithmic Hodge–Tate isomorphism of Lemma 5.16 and the existence of log smooth B_dR^+/t^2-lifts in Proposition 5.21(3). These gaps are load-bearing for the main theorem and need to be completed before the central claims can be regarded as established.

major comments (2)
  1. [Section 5.2, Lemma 5.16] The proof of Lemma 5.16 is a single sentence, 'Similar to [Sch13b, Lemma 3.24]', but this lemma is load-bearing: Proposition 5.15 uses it to identify R^iλ_*Ô^+_X with Ω^{i,log}_X(−i), Proposition 5.25 uses that identification to split Rν_*Ô^+_X, and Theorem 5.20 uses the splitting to identify the E_2-page of the Hodge–Tate spectral sequence and to count dimensions. The log-specific content is not a formal consequence of the smooth case: one must prove that the boundary map M^gp_X → R^1λ_*Z_p(1) induced by the Kummer sequence extends O_X-linearly to all of Ω^{1,log}_X and is an isomorphism, and that its exterior powers give the identifications for all i ≥ 2. As written, the proof omits these steps; if the log Hodge–Tate map is only an injection with nontrivial cokernel, the splitting in Proposition 5.25 fails and the dimension count in Theorem 5.20 does not go through. A complete proof of Lemma 5.16 is required.
  2. [Section 5.4.1, Proposition 5.21(3)] Proposition 5.21(3) asserts that every proper log smooth rigid analytic variety over C admits a log smooth B_dR^+/t^2-lift, and this is used in Proposition 5.25 and Theorem 5.20. The proof is a sketch in several respects. First, it identifies the log structure of X with the compactifying log structure induced by the trivial locus U = X^tr; this requires justification for arbitrary log smooth X over C, not only those étale locally modeled on toric charts. Second, the deformation-theoretic argument using a versal deformation of the closed immersion D → X needs to be made precise: the compatibility of the versal deformation with the log structure and the existence of a formally smooth classifying map are asserted rather than proved. Third, the claim that B_dR^+/t^2 → C has a 'natural F̄-structure' is not immediate: B_dR^+ has a natural W(k)-structure, but an embedding of an algebraic closure F̄ of F into B_dR^+ compatible with θ is not automatic and must be checked explicitly. Since this lifting is a key input to the splitting of Rν_*Ô^+_X and to the proof of Theorem 5.20, a complete proof is needed.
minor comments (6)
  1. [Introduction, opening line] The phrase 'a complete discrete valuation ting' should read 'ring'.
  2. [Theorem 5.20] The notation H^i_{B_dR^+}(X/B_dR^+) is nonstandard and appears to conflict with the notation H^i_{B_dR^+}(X) used elsewhere; please make consistent.
  3. [Proof of Theorem 5.20] The displayed formula 'dX i=1 dim_C H^j(X, Ω^{i,log}_X)' should be a sum ∑_{i=1}^d, and the variable appears to be missing in several places.
  4. [Proposition 5.25] The map is constructed as a splitting of Rν_*Ô^+_X, but the proof invokes Rλ_*Ô^+_X; the relationship between the two morphisms of sites should be made explicit.
  5. [Corollary 1.3] The phrase 'a (non filtered!) natural quasi-isomorphisms' should be singular: 'a natural quasi-isomorphism'.
  6. [Section 5.4.1, Proposition 5.21(3)] There is a typo 'beteween' in the proof; also the notation X is reused for both the rigid variety and the formal scheme, which is confusing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the one-line proof of Lemma 5.16 is an incompleteness gap, not a circular input.

full rationale

The derivation chain is independent: Section 4 computes Kummer pro-étale cohomology of BdR from the log Poincaré lemma, the local structure of OBdR,log, and prior results of Scholze, Bosco, and DLLZ23a, none of which assume the paper's target statements. Section 5 defines logarithmic B+dR-cohomology via Lηt following Bosco, and Theorem 5.12, Theorem 5.13, and Theorem 5.19 are proved from those independent inputs plus the paper's own Proposition 4.8. The central degeneration theorem 5.20 uses the splitting of Rν*Ô_{X_prokét} obtained in Proposition 5.25, whose quasi-isomorphism claim relies on Proposition 5.15, which in turn invokes Lemma 5.16. Lemma 5.16 is indeed proved only by the sentence 'Similar to [Sch13b, Lemma 3.24]', and the log-specific verification of OX-linearity of the boundary map and its exterior powers is not supplied. That makes Lemma 5.16 a load-bearing proof gap and a correctness risk, but it is not circularity: [Sch13b] is an external reference that does not assume the log Hodge–Tate isomorphism, and the paper does not define Ω^{i,log}_X in terms of the cohomology it predicts. There are no fitted parameters renamed as predictions, no uniqueness theorem imported from the authors' own prior work, no ansatz smuggled in via self-citation, and no target result used in its own proof. The self-citations [Sha25a] and [Sha25b] are announced future work and do not carry the argument. A missing proof of a cited lemma should be reported as an incompleteness or correctness concern, not as circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities are present; the listed axioms are the prior results and structural assumptions on which the theorems depend.

assumptions (7)
  • domain assumption The condensed mathematics formalism of Clausen and Scholze, with a fixed uncountable strong limit cardinal κ, is a valid foundation for the cohomology theories used.
    Invoked in the notation section; all cohomology groups are valued in condensed abelian groups and set-theoretic issues are avoided by restricting to κ-small spaces.
  • domain assumption Log smooth rigid analytic varieties have locally free logarithmic differentials Ω^{1,log}_X ([DLLZ23b, Lemma 3.3.15]).
    Needed to define the log de Rham complex and to tensor period sheaves with logarithmic differential forms.
  • domain assumption Kiehl's comparison theorem for log de Rham cohomology with meromorphic poles, Theorem 2.8 from [Kie67], holds in the condensed setting.
    Used in Corollary 1.3 and Proposition 5.17 to identify log de Rham cohomology with cohomology of the open locus.
  • domain assumption The Poincaré lemma for the log period sheaf OB_dR,log, Proposition 3.11 from [DLLZ23a, Corollary 2.4.2], is valid.
    This exact sequence is the core input for Proposition 4.8 and for Theorem 4.6.
  • ad hoc to paper The logarithmic Hodge-Tate isomorphism of Lemma 5.16, R^i λ_* Ô_X ≃ Ω^{i,log}_X(-i), holds; the paper gives only 'Similar to [Sch13b, Lemma 3.24]' as proof.
    This is a new log statement on which Proposition 5.15 and Theorem 5.20 depend. It is not derived in the text and is the main unproved input.
  • ad hoc to paper Every proper log smooth rigid analytic X over C admits a log smooth B_dR^+/t²-lift, Proposition 5.21(3).
    The proof is a sketch using deformation theory and formal smoothness; the lift is required for the splitting in Proposition 5.25.
  • standard math Rotman's theorem [Rot60, Theorem 3] that certain torsion-free modules over a complete DVR are free.
    Used in Theorem 5.20 to conclude that H^i_{B_dR^+}(X) is finite free over B_dR^+.

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Pith. "Pith review of On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$." pith.science (2026). https://pith.science/paper/GWMTVP4I

@misc{pith2026250116916,
  author       = {Pith},
  title        = {Pith review of: On the Kummer pro-\'etale cohomology of $\mathbb B_\operatornamedR$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWMTVP4I}},
  note         = {Machine review of arXiv:2501.16916}
}
abstract

We investigate $p$-adic cohomologies of log rigid analytic varieties over a $p$-adic field. For a log rigid analytic variety $X$ defined over a discretely valued field, we compute the Kummer pro-\'etale cohomology of $\mathbb{B}_{\mathrm{dR}}^+$ and $\mathbb{B}_{\mathrm{dR}}$. When $X$ is defined over $\mathbb{C}_p$, we introduce a logarithmic ${B}_{\mathrm{dR}}^+$-cohomology theory, serving as a deformation of log de Rham cohomology. Additionally, we establish the log de Rham-\'etale comparison in this setting and prove the degeneration of both the Hodge-Tate and Hodge-log de Rham spectral sequences when $X$ is proper and log smooth.

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