Pith. sign in

REVIEW 2 major objections 1 cited by

On quasi-compact smooth rigid analytic spaces the extension of holonomic D-modules produces coadmissible D-cap-modules of finite length as weakly holonomic modules.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

On quasi-compact smooth rigid analytic spaces the extension functor sends holonomic D-modules to coadmissible D-cap-modules of finite length as weakly holonomic D-cap-modules.

T0 review reviewed 2026-07-15 challenge →

load-bearing objection Abstract-only view of a coherent finite-length result in p-adic D-module theory; body is missing so we cannot check the Hilbert-polynomial tool or the proofs. the 2 major comments →

arxiv 2604.24173 v2 pith:YMHMJO4W submitted 2026-04-27 math.NT math.AG

On D-cap-Modules of Finite Length on Rigid Analytic Spaces

Julian Reichardt This is my paper
classification math.NT math.AG MSC 14F1014G2232C38
keywords D-modulesrigid analytic spacesholonomic modulesD-cap modulesweakly holonomicHilbert polynomialscompleted Weyl algebrasmeromorphic connections
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 proves that when the underlying rigid analytic space is quasi-compact and smooth, the natural extension functor carries holonomic D-modules to coadmissible modules over the sheaf of completed differential operators (D-cap), and that these modules have finite length once they are regarded as weakly holonomic D-cap-modules. The same finiteness is then deduced for the meromorphic connections studied by Bode–Bitoun and for the local-cohomology groups studied by Ardakov–Bode–Wadsley. The argument rests on a new theory of Hilbert polynomials for finitely generated modules over completed Weyl algebras, which supplies the growth control needed to bound lengths. A reader interested in p-adic geometry or non-Archimedean D-module theory cares because finite length is a strong structural property that makes further classification and cohomological calculations feasible.

Core claim

For every quasi-compact smooth rigid analytic space the extension functor sends holonomic D-modules to coadmissible D-cap-modules that are of finite length when viewed as weakly holonomic D-cap-modules; the same finite-length statement therefore holds for the meromorphic connections of Bode–Bitoun and the local cohomology modules of Ardakov–Bode–Wadsley.

What carries the argument

Hilbert polynomials attached to finitely generated modules over completed Weyl algebras; they measure asymptotic growth and thereby force the modules that arise by extension to have finite length as weakly holonomic D-cap-modules.

Load-bearing premise

The rigid analytic space must be both quasi-compact and smooth; the finite-length claim is stated only under these two geometric hypotheses.

What would settle it

An explicit holonomic D-module on a quasi-compact smooth rigid space whose extension is coadmissible yet has infinite length as a weakly holonomic D-cap-module, or a computation showing that the associated Hilbert polynomial fails to be eventually constant.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Meromorphic connections of Bode–Bitoun become finite-length weakly holonomic D-cap-modules on quasi-compact smooth rigid spaces.
  • Local cohomology groups of Ardakov–Bode–Wadsley likewise acquire finite length in the weakly holonomic category.
  • Any holonomic D-module extends to a coadmissible D-cap-module that admits a finite composition series of weakly holonomic subquotients.
  • The Hilbert-polynomial technique supplies a uniform length bound once the space is fixed and quasi-compact.

Where Pith is reading between the lines

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

  • Composition series of finite length open the possibility of inductive arguments and classification results for weakly holonomic D-cap-modules that were previously unavailable.
  • The same Hilbert-polynomial control may adapt, after suitable modification of the completed Weyl algebra, to mildly singular or non-quasi-compact rigid spaces.
  • Finite-length statements of this type typically imply that the modules are coherent and that their characteristic cycles are well-defined algebraic cycles.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The abstract claims that, for quasi-compact smooth rigid analytic spaces, the extension functor sends holonomic D-modules to coadmissible D-cap-modules that are of finite length as weakly holonomic D-cap-modules, and that this implies finite length for the meromorphic connections of Bode–Bitoun and the local cohomology groups of Ardakov–Bode–Wadsley; the central technical tool is the introduction of Hilbert polynomials for finitely generated modules over completed Weyl algebras. The supplied full manuscript body, however, is an unrelated statistics paper (divergence-based model weighting and averaging of probabilistic predictions) whose title, abstract, theorems, experiments, and arXiv identifier do not match the claimed math.NT work.

Significance. If the claimed finite-length results for coadmissible and weakly holonomic D-cap-modules hold under the stated quasi-compact smooth hypotheses, they would be a useful contribution to non-commutative geometry and rigid-analytic D-module theory, clarifying the behaviour of extension, meromorphic connections, and local cohomology. Because the body of the claimed paper is absent and an unrelated manuscript has been supplied in its place, no assessment of the actual mathematical contribution, novelty of the Hilbert-polynomial construction, or correctness of the arguments is possible.

major comments (2)
  1. The full manuscript text provided under paper_id 2604.24173 is not the paper described by the title and abstract. It is instead the complete text of an unrelated statistics preprint (arXiv:2604.24172) on divergence-based model averaging. Consequently there are no statements, proofs, constructions of Hilbert polynomials over completed Weyl algebras, or geometric arguments that can be checked against the claims in the abstract. Evaluation of the central finite-length assertions is impossible on the supplied document.
  2. Even the abstract’s load-bearing geometric hypotheses (“quasi-compact smooth rigid analytic spaces”) and the applications to Bode–Bitoun meromorphic connections and Ardakov–Bode–Wadsley local cohomology cannot be verified, because the body that would contain the definitions of the extension functor, the weakly holonomic category, and the Hilbert-polynomial control is missing. The mismatch is not a presentation issue; it prevents any technical review of the claimed results.

Circularity Check

0 steps flagged

No significant circularity; finite-length claim is a standard existence statement under explicit geometric hypotheses, with no definitional tautology or fitted-parameter prediction visible from available text.

full rationale

Only the abstract of the claimed math.NT paper (arXiv:2604.24173) is present; the supplied full manuscript body is an unrelated statistics paper. From the abstract alone the central claim is a coherent finiteness/existence result: the extension functor takes holonomic D-modules to coadmissible D-cap-modules of finite length as weakly holonomic modules on quasi-compact smooth rigid analytic spaces, with Hilbert polynomials over completed Weyl algebras as the main technical tool, and with applications to meromorphic connections and local cohomology. Nothing in the abstract reduces the conclusion to a definition, a fitted free parameter, or a self-citation uniqueness theorem. The geometric hypotheses (quasi-compact + smooth) are stated explicitly as load-bearing rather than smuggled. Consequently no circular step of any of the six enumerated kinds can be exhibited by quotation. Score 0 is the honest finding given the available text.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 1 invented entities

Abstract-only review. The load-bearing geometric hypotheses (quasi-compact smooth rigid analytic spaces) and the background theory of coadmissible/weakly holonomic D-cap-modules are taken from the existing literature (Ardakov–Wadsley et al.). The paper invents a Hilbert-polynomial formalism for completed Weyl algebras; no free numerical parameters appear. All other axioms are standard commutative/non-commutative algebra or rigid geometry.

axioms (3)
  • domain assumption The ambient space is a quasi-compact smooth rigid analytic space over a complete non-archimedean field.
    Stated explicitly as the setting of every main claim in the abstract; without it the finite-length conclusion is not asserted.
  • domain assumption Existence and basic properties of the extension functor from D-modules to coadmissible D-cap-modules, and of the categories of holonomic / weakly holonomic / coadmissible modules.
    Taken as background from the Ardakov–Wadsley / Bode school; the paper builds on rather than re-derives them.
  • standard math Standard facts about finitely generated modules over (completed) Weyl algebras and the existence of Hilbert polynomials in that setting once introduced.
    Classical commutative-algebra techniques adapted to the completed non-commutative case.
invented entities (1)
  • Hilbert polynomials for finitely generated modules over completed Weyl algebras no independent evidence
    purpose: Central technical tool used to control growth and deduce finite length of the extended modules.
    Explicitly introduced in the abstract as new; no independent external evidence is supplied in the abstract itself.

reviewed 2026-07-15 · how reviews work

0 comments
Cite this review

Pith. "Pith review of On D-cap-Modules of Finite Length on Rigid Analytic Spaces." pith.science (2026). https://pith.science/paper/YMHMJO4W

@misc{pith2026260424173,
  author       = {Pith},
  title        = {Pith review of: On D-cap-Modules of Finite Length on Rigid Analytic Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YMHMJO4W}},
  note         = {Machine review of arXiv:2604.24173}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We show that for quasi-compact smooth rigid analytic spaces, the extension functor sends holonomic D-modules to coadmissible D-cap-modules which are of finite length as weakly holonomic D-cap-modules. Using this, we show that the meromorphic connections considered by Bode--Bitoun and the local cohomology groups considered by Ardakov--Bode--Wadsley are of finite length as weakly holonomic D-cap-modules for quasi-compact smooth rigid analytic spaces. As a central tool, we introduce and study Hilbert polynomials for finitely generated modules over completed Weyl algebras.

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

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

  1. A Mackey criterion for locally analytic representations

    math.NT 2026-07 accept novelty 6.5

    Compact induction of a uniformly simple admissible locally analytic representation from an open compact-mod-centre subgroup is topologically irreducible when there are no nontrivial intertwiners with its conjugates.

Reference graph

Works this paper leans on

5 extracted references · cited by 1 Pith paper

  1. [1]

    [Yes] � ���������������� ������ ��� ��������� ��� ��������� ����� ����������� (b) An analysis of the properties and complexity (time, space, sample size) of any algorithm

    For all models and algorithms presented, check if you include: (a) A clear description of the mathematical set- ting, assumptions, algorithm, and/or model. [Yes] � ���������������� ������ ��� ��������� ��� ��������� ����� ����������� (b) An analysis of the properties and complexity (time, space, sample size) of any algorithm. [Not Applicable] The algorith...

  2. [2]

    [Yes] (b) Complete proofs of all theoretical results

    For any theoretical claim, check if you include: (a) Statements of the full set of assumptions of all theoretical results. [Yes] (b) Complete proofs of all theoretical results. [Yes] This is provided in the appendix. (c) Clear explanations of any assumptions. [Yes] In some cases, more complete explanations are provided in the appendix

  3. [3]

    [Yes] (b) All the training details (e.g., data splits, hy- perparameters, how they were chosen)

    For all figures and tables that present empirical results, check if you include: (a) The code, data, and instructions needed to reproduce the main experimental results (ei- ther in the supplemental material or as a URL). [Yes] (b) All the training details (e.g., data splits, hy- perparameters, how they were chosen). [Yes] Some of this information has been...

  4. [4]

    [Yes] (b) The license information of the assets, if ap- plicable

    If you are using existing assets (e.g., code, data, models) or curating/releasing new assets, check if you include: (a) Citations of the creator If your work uses ex- isting assets. [Yes] (b) The license information of the assets, if ap- plicable. [Yes] (c) New assets either in the supplemental mate- rial or as a URL, if applicable. [Not Applica- ble] (d)...

  5. [5]

    generated quantities

    If you used crowdsourcing or conducted research with human subjects, check if you include: (a) The full text of instructions given to partici- pants and screenshots. [Not Applicable] (b) Descriptions of potential participant risks, with links to Institutional Review Board (IRB) approvals if applicable. [Not Appli- cable] (c) The estimated hourly wage paid...

    2019

This paper was first reviewed by grok-4.5 on July 15, 2026.