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A fully faithful p-adic Riemann-Hilbert functor for coadmissible D-cap-modules

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Pith's one-line read The paper proves that replacing the positive overconvergent de Rham period ring by the almost de Rham ring makes the solution functor fully faithful on all C-complexes, so every coadmissible p-adic D-module is canonically reconstructed…

desk verdict A plausible and significant extension of Scholze's theorem, but the load-bearing solid-to-ind-Banach bridge needs a closer look. read the letter →

arxiv 2506.12601 v2 pith:LCRIZUTL submitted 2025-06-14 math.AG math.NT

Finn Wiersig This is my paper · ORCID
classification math.AGmath.NT MSC 14F3014G2214F10
keywords p-adicRiemann-HilbertcorrespondencecoadmissibleD-cap-modulesC-complexesoverconvergentalmostdeRhamperiodringpro-étalesitecontinuousGaloiscohomologyrigidanalyticgeometrysolidvectorspaces
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 establishes a p-adic analogue of the Riemann-Hilbert correspondence for coadmissible modules over the sheaf of infinite-order differential operators on smooth rigid-analytic varieties. The solution functor that sends a complex of modules to its solutions valued in the positive overconvergent de Rham period sheaf is not fully faithful on the whole triangulated category of C-complexes, and the paper identifies the obstruction in the first cohomology of the period structure sheaf. Adding a formal logarithm of the period element $t$, so that solutions are valued in the overconvergent almost de Rham period sheaf $B^\dagger_{\mathrm{pdR}}$, removes the obstruction. The main theorem states that this modified solution functor is a fully faithful embedding of triangulated categories, and that every C-complex is canonically reconstructed from the solutions it produces. If correct, every coadmissible module is determined by its $B^\dagger_{\mathrm{pdR}}$-valued solutions.

What carries the argument

The load-bearing object is the overconvergent almost de Rham period ring $B^\dagger_{\mathrm{pdR}}$, obtained from the positive overconvergent de Rham period ring $B^{\dagger,+}_{\mathrm{dR}}$ by inverting $t$ and adjoining a formal variable $\log t$; the corresponding structure sheaf $\mathcal{O}_{B^\dagger_{\mathrm{pdR}}}$ is the bimodule that makes the reconstruction work. The reconstruction functor $\mathrm{Rec}(F^\bullet) = R\nu_* \mathrm{RHom}_{B^{\dagger,+}_{\mathrm{dR}}}(F^\bullet, \mathcal{O}_{B^\dagger_{\mathrm{pdR}}})$ is the mechanism: Theorem J identifies $R\nu_*\mathcal{O}_{B^\dagger_{\mathrm{pdR}}} \simeq \mathcal{O}$, and Theorem L promotes that to $M^\bullet \simeq \mathrm{Rec}(\mathrm{Sol}(M^\bullet))$ for every C-complex. The proof of the key identity runs through the continuous Galois cohomology of $B^\dagger_{\mathrm{pdR}}$ and its relatives, computed with eta-operator techniques and the cohomology of cyclotomic twists.

What would settle it

Compute $H^1(X,\mathcal{O}_{B^\dagger_{\mathrm{pdR}}})$ for an affinoid $X$ with \'etale coordinates: Theorem J predicts it is zero, and a single nonzero class would make $\mathcal{O}\simeq R\nu_*\mathcal{O}_{B^\dagger_{\mathrm{pdR}}}$ fail, breaking the reconstruction theorem and the fullness of the modified solution functor.

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Extended reading notes

Core claim

The central claim is Theorem C: the functor $\mathrm{D}^C(\widehat{\mathcal{D}})^{\mathrm{op}} \to \mathrm{D}(B^\dagger_{\mathrm{pdR}})$ sending $M^\bullet \mapsto \mathrm{Sol}(M^\bullet) \otimes^{\mathbb{L}}_{B^{\dagger,+}_{\mathrm{dR}}} B^\dagger_{\mathrm{pdR}}$ is a fully faithful embedding of triangulated categories. Equivalently, via the reconstruction functor $\mathrm{Rec}(F^\bullet) = R\nu_* \mathrm{RHom}_{B^{\dagger,+}_{\mathrm{dR}}}(F^\bullet, \mathcal{O}_{B^\dagger_{\mathrm{pdR}}})$, every C-complex $M^\bullet$ satisfies $M^\bullet \simeq \mathrm{Rec}(\mathrm{Sol}(M^\bullet))$. The proof reduces this to the computation that the derived pushforward of the overconvergent almost de Rham period structure sheaf is the structure sheaf, $\mathcal{O} \simeq R\nu_* \mathcal{O}_{B^\dagger_{\mathrm{pdR}}}$, and to explicit continuous Galois cohomology calculations for the new period rings. A covariant version for the de Rham functor after duality is also fully faithful.

Load-bearing premise

The proof needs each cohomology group involved to have a countable basis of bounded subsets; without that, exactness in the solid world does not automatically give the strict exactness in the ind-Banach world that the reconstruction theorem requires.

Editorial extensions

If this is right

  • Every coadmissible $\widehat{\mathcal{D}}$-module, viewed as a C-complex concentrated in degree zero, is canonically isomorphic to $\mathrm{Rec}(\mathrm{Sol}(M))$, so its $B^\dagger_{\mathrm{pdR}}$-valued solutions determine the module completely.
  • On vector bundles with integrable connection, the modified solution functor agrees with the pro-\'etale horizontal sections functor, so the theorem contains and extends that fully faithful embedding.
  • The result is a p-adic counterpart of the Archimedean reconstruction theorem: the natural map $\widehat{\mathcal{D}} \to R\nu_* \mathrm{RHom}_{B^\dagger_{\mathrm{pdR}}}(\mathcal{O}_{B^\dagger_{\mathrm{pdR}}}, \mathcal{O}_{B^\dagger_{\mathrm{pdR}}})$ is an isomorphism.
  • The covariant de Rham functor, obtained by applying the duality functor and shifting by the dimension, is also fully faithful after the same base change.
  • The essential image of the fully faithful functor has an explicit local description as complexes built from the period sheaves and closed under direct summands, giving a concrete target category for the correspondence.

Reading between the lines

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

  • One natural next step, not taken in the paper, is to find a full subcategory of C-complexes on which the original $B^{\dagger,+}_{\mathrm{dR}}$-valued solution functor is already fully faithful; the paper expects such a category to play the role of holonomic D-modules.
  • The explicit description of the obstruction suggests a sharpness test for any proposed enlargement of the period ring: full faithfulness should hold exactly when $R^1\nu_*\mathcal{O}_{B}$ vanishes for the chosen period structure sheaf; the paper shows that inverting $t$ alone leaves $R^1\simeq\mathcal{O}$, and only adjoining $\log t$ kills it.
  • The Galois cohomology computations for the new period rings are likely reusable in locally analytic representation theory, where coadmissible $\widehat{\mathcal{D}}$-modules arise naturally; a reconstruction statement there would give a new bridge between solutions and representations.
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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

0 major / 4 minor

Summary. The paper develops a p-adic Riemann-Hilbert correspondence for Ardakov-Wadsley's coadmissible D-cap-modules and, more generally, for Bode's C-complexes. It introduces new period sheaves (the overconvergent de Rham period sheaf and the overconvergent almost de Rham period structure sheaves), computes their continuous Galois cohomology and derived pushforwards, constructs a reconstruction functor, and proves that the solution functor, after tensoring with the overconvergent almost de Rham period sheaf, is a fully faithful embedding of triangulated categories (Theorem C / Theorem 7.15). A covariant version for the de Rham functor is also obtained (Theorem F).

Significance. If correct, this is a substantial advance: it provides an explicit, fully faithful solution functor on the full triangulated category of C-complexes, hence on all coadmissible D-cap-modules, rather than only on vector bundles with integrable connection. The proof is long and structured, with direct computations for the newly introduced period rings; the paper explicitly identifies the cohomological obstruction to fullness (the non-vanishing of H^1 of the positive overconvergent de Rham structure sheaf) and gives explicit Galois-cohomology computations. The author is also honest about the delicate passage from solid to ind-Banach categories (Remark 1.10). No machine-checked formalization is supplied, but I did not find an internal contradiction in the visible text.

minor comments (4)
  1. [§5.3.9 (proof of Theorem 5.7)] The final application of Proposition 2.65 is extremely compressed, and given that Remark 1.10 warns that this passage is subtle, I recommend expanding it as follows: apply Proposition 2.65 to each triple K^{i-1} to K^i to K^{i+1}, and note that the countable-basis hypothesis holds termwise because C^j_cts(G, B†,+_dR) is the filtered colimit of Hom_cts(G^j, B_q^+) with injective transition maps, while C^j_cts(G, C) is a Banach space. This would remove any residual doubt about the solid-to-ind-Banach upgrade.
  2. [Theorem D statement] In Theorem D, 'The cohomology H^{-dim X}(Sol(E))' should presumably read H^{-dim X}(dR(E)); as written it refers to the wrong functor.
  3. [Throughout] There are numerous typographical errors ('cohmology', 'immedeate', 'principle ideal', 'the the', 'strict exactnes') that should be corrected in the final version.
  4. [§5.3.9 (proof of Theorem 5.7)] The cross-reference 'Corollary 5.27' in the proof of Theorem 5.7 appears to be a misnumbering; the intended reference is presumably Lemma 5.27 or Corollary 5.28. Please correct the cross-reference.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reconstruction and full-faithfulness theorems are powered by direct Galois-cohomology computations, and self-citations only supply input definitions or technical lemmas.

full rationale

The derivation chain is not circular in any exhibited sense. The central result Theorem C is obtained from the reconstruction theorem Theorem L, which is proved from Proposition 7.43 and Theorem J (the isomorphism O ≅ Rν_* OB†_pdR). Theorem J in turn relies on the Galois-cohomology computation Theorem K, which is proved by explicit computation, ultimately reducing to external results of Barthel–Schlank–Stapleton–Weinstein on Galois cohomology of Tate twists. The key cohomology statements Theorems G, I, and K are proved directly from the definitions of the period rings; they are not assumed from the desired full faithfulness. Self-citations to [55] introduce the solution functor, the period sheaf B†,+_dR, and structural facts about period rings, but they do not assert or presuppose the full-faithfulness conclusion. The bridge Proposition 2.65, flagged in Remark 1.10 as subtle, is a potential correctness risk, not a circular step: the paper checks exactness in the solid category and then upgrades via a stated criterion, and any failure would be a proof gap rather than a reduction of the conclusion to its inputs. No equation or definition was found to be equivalent by construction to the output theorem, and no fitted parameter is renamed as a prediction. Therefore the paper earns a low circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The central theorem depends on a large stack of background results (Scholze's pro-etale site, Ardakov-Wadsley's coadmissible modules, Bode's six operations, BSSW's Galois cohomology) and on newly introduced period sheaves whose key cohomological properties are proved internally. No data fitting or free parameters are involved. The ledger is therefore dominated by domain assumptions and background theorems rather than ad hoc postulates.

assumptions (5)
  • domain assumption Standing setup: k is a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field; X is a smooth rigid-analytic k-variety.
    Restricts the scope of all theorems; cited in §1.6 and used throughout.
  • standard math Barthel-Schlank-Stapleton-Weinstein's Galois cohomology of Tate twists O_C(n) ([12, Theorems 4.0.4 and 4.0.5]) is correct.
    These theorems underlie the proofs of Theorems I and K, as stated in Remark 1.12; the paper flags that 'local field' is used nonstandardly there.
  • standard math Scholze's horizontal sections functor is fully faithful ([48, Theorem 7.6]).
    Used to prove Theorems A and D for O-modules with integrable connection (§1.1).
  • standard math Bode's six-functor formalism for C-complexes, including duality with D^2 ≃ id, is available ([22]).
    The proof of Theorem M uses D^2 ≃ id to pass from Sol to dR and from Theorem B/E to Theorem C/F (§1.2.6).
  • domain assumption The ind-Banach to solid formal bridge (Proposition 2.65) is valid in every cohomological degree used.
    Proposition 2.65 is proved in the paper but its hypothesis is described as 'surprisingly subtle'; it is load-bearing for Theorem 5.7 and its analogues.
invented entities (3)
  • B†dR, the overconvergent de Rham period sheaf
    purpose: Inverting Fontaine's t locally on the pro-etale site to remove the H^1 obstruction seen with B†+dR.
    Introduced in this paper (§4.1.3); its cohomology is computed internally via Theorems H and I, with no independent external confirmation.
  • B†pdR and OB†pdR, the overconvergent almost de Rham period (structure) sheaves
    purpose: Adjoining a formal variable log t so that Rν_* OB†pdR ≅ O, which makes the reconstruction functor fully faithful.
    Introduced in §4.1.4 and §4.3; the key property is Theorem J/K, proved inside the paper rather than checked externally.
  • B†+dR, the positive overconvergent de Rham period ring/sheaf
    purpose: Target of the original solution functor; already introduced in companion paper [55].
    Not an independent benchmark; its properties are used as input and were established by the same author in [55].

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Cite this review

Pith. "Pith review of A fully faithful p-adic Riemann-Hilbert functor for coadmissible D-cap-modules." pith.science (2026). https://pith.science/paper/LCRIZUTL

@misc{pith2026250612601,
  author       = {Pith},
  title        = {Pith review of: A fully faithful p-adic Riemann-Hilbert functor for coadmissible D-cap-modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LCRIZUTL}},
  note         = {Machine review of arXiv:2506.12601}
}
read the original abstract

This article establishes a Riemann-Hilbert correspondence in rigid-analytic geometry. We construct an explicit solution functor and prove that it is fully faithful on Ardakov-Wadsley's coadmissible D-cap-modules. For vector bundles with flat connection, our functor is canonically identified with Scholze's horizontal sections functor.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The p-adic Cauchy Theorem and Overconvergent Period Sheaves

    math.NT 2026-06 unverdicted novelty 6.0 of 10

    The horizontal sections functor using the overconvergent de Rham period structure sheaf agrees with Scholze's using OBdR on smooth rigid-analytic varieties, identifying it with the de Rham functor for D-cap-modules.

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