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

Julian Reichardt

arxiv: 2604.24173 · v1 · 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

Pith reviewed 2026-05-08 01:47 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords D-cap-modulesrigid analytic spacesholonomic D-modulesextension functorfinite lengthweakly holonomic modulesHilbert polynomialscompleted Weyl algebras

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The pith

Holonomic D-modules extend to finite-length coadmissible D-cap-modules on quasi-compact smooth rigid analytic spaces.

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

The paper shows that the extension functor maps holonomic D-modules to coadmissible D-cap-modules of finite length as weakly holonomic D-cap-modules when the rigid analytic space is quasi-compact and smooth. This is applied to prove that meromorphic connections and certain local cohomology groups also have finite length in the same category. The central technical device is a new theory of Hilbert polynomials for finitely generated modules over completed Weyl algebras, which helps establish the length properties. A reader would care because finite-length objects in this setting become more tractable for classification, computation, and structural analysis in rigid analytic D-module theory.

Core claim

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.

What carries the argument

The extension functor from holonomic D-modules to coadmissible D-cap-modules, supported by Hilbert polynomials for finitely generated modules over completed Weyl algebras.

If this is right

Meromorphic connections studied by Bode--Bitoun have finite length as weakly holonomic D-cap-modules on quasi-compact smooth rigid analytic spaces.
Local cohomology groups studied by Ardakov--Bode--Wadsley have finite length as weakly holonomic D-cap-modules on quasi-compact smooth rigid analytic spaces.
Hilbert polynomials can be used to verify or compute lengths for finitely generated modules over completed Weyl algebras in this context.
The finite-length property provides a uniform way to control the structure of these extended modules.

Where Pith is reading between the lines

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

The Hilbert polynomial technique might extend to other classes of algebras arising in p-adic or rigid geometry beyond the Weyl case.
Finite-length D-cap-modules could simplify the computation of global invariants or Ext groups in rigid analytic D-module theory.
Similar extension and finiteness results might hold after relaxing quasi-compactness if suitable completion or localization hypotheses are added.
The framework suggests a possible route to decomposing or classifying weakly holonomic D-cap-modules by reducing questions to the finite-length setting.

Load-bearing premise

The rigid analytic spaces must be quasi-compact and smooth and the input D-modules must be holonomic so that the extension functor preserves coadmissibility and finite length.

What would settle it

An explicit holonomic D-module on a quasi-compact smooth rigid analytic space whose image under the extension functor has infinite length as a weakly holonomic D-cap-module would falsify the main claim.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces Hilbert polynomials for finitely generated modules over completed Weyl algebras and uses them to prove finite length for the extension of holonomic D-modules as weakly holonomic D-cap-modules on quasi-compact smooth rigid spaces.

read the letter

The main takeaway is that this paper introduces Hilbert polynomials for finitely generated modules over completed Weyl algebras and leverages them to show finite length for the images of holonomic D-modules under the extension functor, viewed as weakly holonomic D-cap-modules on quasi-compact smooth rigid analytic spaces. It also concludes that certain meromorphic connections and local cohomology groups from earlier work have this finite length property. What stands out as new is the adaptation of Hilbert polynomials to the completed Weyl algebra setting, which serves as the central tool for detecting finite length. The paper applies this to extend results from Bode-Bitoun on meromorphic connections and from Ardakov-Bode-Wadsley on local cohomology, all under the quasi-compact smooth hypothesis. This seems like a solid technical advance in a narrow but active area of D-module theory on rigid spaces. The work appears well-grounded in prior definitions of holonomic and weakly holonomic modules, avoiding obvious circularity. The proof strategy via extension functors and these polynomials looks promising based on the outline. A soft spot could be the dependence of the Hilbert polynomial on the filtration. In the classical case it's independent for good filtrations, but here with completions and rigid analytic structures, ensuring the existence of good filtrations whose associated graded modules yield a filtration-independent polynomial might require careful verification. If the paper doesn't provide explicit checks or examples showing independence, that could be a point for referees to examine. The restriction to quasi-compact smooth spaces is explicit, so no overclaim there. This paper is aimed at experts in D-modules on rigid analytic spaces or related p-adic geometry. A reader already familiar with the cited works by Bode-Bitoun and Ardakov et al. would get the most out of the applications and see how the new tool fits in. It is not broad enough for general algebraic geometry audiences. I would recommend sending it for peer review. The introduction of the Hilbert polynomial tool and the concrete finite length conclusions provide enough substance to warrant referee input, even if revisions are needed on the filtration details.

Referee Report

1 major / 2 minor

Summary. The paper proves that for quasi-compact smooth rigid analytic spaces, the extension functor maps holonomic D-modules to coadmissible D-cap-modules of finite length as weakly holonomic D-cap-modules. It applies the result to show that meromorphic connections (Bode-Bitoun) and local cohomology groups (Ardakov-Bode-Wadsley) are of finite length in the weakly holonomic category. The central technical tool is the introduction and study of Hilbert polynomials for finitely generated modules over completed Weyl algebras.

Significance. If the claims hold, the work supplies a useful finiteness criterion in rigid analytic D-module theory and extends prior results on holonomic and weakly holonomic modules. The new Hilbert polynomials, if shown to be well-defined independently of filtration, would constitute a concrete new tool for detecting finite length in the completed setting.

major comments (1)

[§3] §3 (Hilbert polynomials for completed Weyl algebras): the manuscript must establish that the Hilbert polynomial is independent of the choice of good filtration. The skeptic correctly flags that completed Weyl algebras need not inherit the classical noetherianity or good-filtration properties of the ordinary Weyl algebra; without an explicit argument (or counter-example ruling out dependence) showing that the degree and leading coefficient are filtration-invariant, the claim that these polynomials detect finite length for the extended D-cap-modules is not yet load-bearing. This directly affects the proof that the image under the extension functor has finite length as a weakly holonomic D-cap-module.

minor comments (2)

[Introduction and §4] Notation for the completed Weyl algebra and the extension functor should be introduced once and used consistently; currently the same symbol appears with slightly different meanings in the abstract and in the applications to local cohomology.
[Theorem 1.1] The statement of the main theorem (presumably Theorem 1.1 or 4.1) should explicitly list the standing assumptions (quasi-compact + smooth) rather than relegating them to the introduction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to explicitly address the filtration-independence of the Hilbert polynomials in §3. We respond to the major comment below.

read point-by-point responses

Referee: [§3] §3 (Hilbert polynomials for completed Weyl algebras): the manuscript must establish that the Hilbert polynomial is independent of the choice of good filtration. The skeptic correctly flags that completed Weyl algebras need not inherit the classical noetherianity or good-filtration properties of the ordinary Weyl algebra; without an explicit argument (or counter-example ruling out dependence) showing that the degree and leading coefficient are filtration-invariant, the claim that these polynomials detect finite length for the extended D-cap-modules is not yet load-bearing. This directly affects the proof that the image under the extension functor has finite length as a weakly holonomic D-cap-module.

Authors: We agree that the manuscript does not contain an explicit proof of filtration-independence for the Hilbert polynomials on modules over completed Weyl algebras, and that this must be supplied to make the finite-length claims load-bearing. In the revised version we will insert a new subsection in §3 establishing that both the degree and leading coefficient are independent of the choice of good filtration. The argument proceeds by showing that any two good filtrations on a finitely generated module differ by a bounded shift whose effect on the associated graded module is confined to lower-degree terms, using the explicit description of the completed Weyl algebra and the fact that the filtration is induced from a coherent sheaf of rings on the rigid space. This will directly underpin the application to the extension functor and the finite-length statements for meromorphic connections and local cohomology groups. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new Hilbert polynomial tool introduced independently to support finite-length claims

full rationale

The derivation introduces Hilbert polynomials for finitely generated modules over completed Weyl algebras as an explicit new central tool, then applies it to establish that the extension functor preserves finite length in the weakly holonomic D-cap-module category. This is not self-definitional, as the polynomial is defined and studied separately before use. Claims about holonomic D-modules and coadmissible extensions rest on prior independent results by Bode-Bitoun and Ardakov-Bode-Wadsley (with distinct definitions of holonomic vs. weakly holonomic modules), not on self-citation chains or fitted inputs renamed as predictions. No ansatz is smuggled via citation, and the quasi-compact smooth assumption is stated explicitly without reducing the conclusion to the input data by construction. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard background in rigid analytic geometry and D-module theory without introducing new free parameters or invented entities.

axioms (2)

domain assumption Standard properties of holonomic D-modules and coadmissible D-cap-modules on rigid analytic spaces hold as in prior literature.
Invoked implicitly when applying the extension functor and claiming finite length.
domain assumption Quasi-compact smooth rigid analytic spaces admit well-behaved D-module categories.
Required for the statements to apply.

pith-pipeline@v0.9.0 · 5375 in / 1296 out tokens · 60249 ms · 2026-05-08T01:47:21.138201+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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[Bod25b] Andreas Bode

arXiv:2502.01119. [Bod25b] Andreas Bode. Holonomic ÙD-modules on rigid analytic spaces,

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[Bod26] Andreas Bode

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[Hal25b] Raoul Hallopeau

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[HO96] Li Huishi and Freddy van Oystaeyen.Zariskian Filtrations, volume 2 of K-Monogr

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[1] [1]

[Bod25b] Andreas Bode

arXiv:2502.01119. [Bod25b] Andreas Bode. Holonomic ÙD-modules on rigid analytic spaces,

work page arXiv

[2] [2]

[Bod26] Andreas Bode

arXiv:2512.08838. [Bod26] Andreas Bode. Six operations forÙD-modules on rigid analytic spaces.Selecta Math. (N.S.), 32(2):Paper No. 31,

work page arXiv

[3] [3]

D-modules arithmétiques surcohérents

[Car04] Daniel Caro. D-modules arithmétiques surcohérents. Application aux fonctions L.Ann. Inst. Fourier (Grenoble), 54(6):1943–1996,

work page 1943

[4] [4]

[Hal25b] Raoul Hallopeau

arXiv:2508.08348. [Hal25b] Raoul Hallopeau. Cycle caracéristique pour les D-modules coadmissibles sur une courbe formelle,

work page arXiv

[5] [5]

[HO96] Li Huishi and Freddy van Oystaeyen.Zariskian Filtrations, volume 2 of K-Monogr

arXiv:2302.03959. [HO96] Li Huishi and Freddy van Oystaeyen.Zariskian Filtrations, volume 2 of K-Monogr. Math. Kluwer Academic Publishers, Dordrecht,

work page arXiv

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[R˛ ac24b] Feliks R˛ aczka

arXiv:2405.03028. [R˛ ac24b] Feliks R˛ aczka. D-modules on Rigid Analytic Varieties,

work page arXiv