Coadmissible Modules

• Coadmissible modules are topologically robust modules over Fréchet–Stein algebras, defined as inverse limits of finitely generated modules over Noetherian Banach algebras.
• They exhibit strong abelian and homological properties such as Auslander regularity, finite global dimension, and compatibility with Grothendieck’s six-functor formalism.
• Their framework underpins analytic applications in nonarchimedean geometry, enabling Kashiwara’s equivalence, analytic localization, and p-adic Riemann–Hilbert correspondences.

A coadmissible module is a topologically robust module over certain infinite-dimensional Fréchet–Stein algebras that arise in nonarchimedean geometry, representation theory, and $p$-adic D-module theory. The concept organizes analytic families of modules on rigid analytic spaces, completed enveloping algebras, and distribution algebras, providing a pivotal generalization of the coherent module paradigm to the analytic and $p$-adic context. The category of coadmissible modules exhibits abelian, finiteness, and duality properties essential for both algebraic and analytic applications.

1. Fréchet–Stein Algebras and the Basic Notion of Coadmissibility

A $K$-algebra $A$ (with $K$ a complete discretely valued nonarchimedean field) is called Fréchet–Stein if $A \cong \varprojlim_{n} A_n$ for a sequence of Noetherian Banach $K$-algebras $A_n$ with flat transition maps $A_{n+1} \to A_n$ and dense image (Lyubinin, 2014). The prototypical examples include:

• The sheaf $\widehat D_X$ of completed infinite-order differential operators on a smooth rigid analytic $p$0-variety $p$1: $p$2, where $p$3 is a completed enveloping algebra associated to a Lie–Rinehart pair built from a Lie lattice in the tangent sheaf (Ardakov et al., 2015, Bode, 12 May 2025).
• The $p$4-adic Arens–Michael envelope $p$5 of a reductive Lie algebra $p$6 (Schmidt, 2010).
• The locally analytic distribution algebra $p$7 of a compact $p$8-adic Lie group $p$9 (Zábrádi, 2010).

A left $K$0-module $K$1 is coadmissible if $K$2 with each $K$3 a finitely generated $K$4-module and $K$5 an isomorphism. This structure endows $K$6 with a canonical Fréchet topology (Lyubinin, 2014, Schmidt, 2010).

The category of coadmissible modules (for a fixed Fréchet–Stein structure) is abelian, stable under kernels, cokernels, and extensions, and contains all finitely presented $K$7-modules (Lyubinin, 2014, Schmidt, 2010). On rigid spaces, coadmissible sheaves are locally inverse limits of coherent modules over the Banach pieces.

2. Homological Regularity and the Auslander Condition

A central structural result asserts that for any smooth rigid analytic variety $K$8, there exist affinoid covers $K$9 such that each Banach algebra $A$0 in the Fréchet–Stein presentation $A$1 is Auslander regular with finite global dimension: $A$2, where $A$3 (Bode, 12 May 2025). The Auslander regularity implies:

• For any finitely generated $A$4-modules $A$5, one has $A$6 for $A$7.
• Coadmissible $A$8-modules inherit finite projective resolutions and dimension-theoretic finiteness: e.g., Bernstein's inequality $A$9 for the grade $K$0.

This regularity grounds the homological algebra of the coadmissible category and underpins duality and six-functor formalisms (Bode, 12 May 2025, Bode, 2021).

3. Fundamental Constructions and Functoriality

The geometric and categorical framework for coadmissible modules includes:

Six Operations and Adjointness

The category of coadmissible $K$1-modules supports all classical Grothendieck six functors in the context of complete bornological sheaves (Bode, 2021, Bode, 12 May 2025):

• Direct and proper pushforwards ($K$2),
• Extraordinary and usual inverse images ($K$3),
• Tensor products and (derived) internal Hom,
• Duality functor $K$4.

Projection formula and adjunction theorems have been established for these functors: $K$5

$K$6

under standard hypotheses (Bode, 12 May 2025).

Kashiwara's Equivalence for Closed Embeddings

If $K$7 is the closed embedding of a smooth subvariety, the functors $K$8 (pushforward) and $K$9 (pullback to support on $A \cong \varprojlim_{n} A_n$0) yield quasi-inverse equivalences between coadmissible modules on $A \cong \varprojlim_{n} A_n$1 and those on $A \cong \varprojlim_{n} A_n$2 supported on $A \cong \varprojlim_{n} A_n$3 (Ardakov et al., 2015, Bode, 2021). This is an analytic version of the algebraic Kashiwara equivalence, crucial for microlocal analysis and the construction of simple coadmissible modules.

4. Key Examples and Applications

Coadmissible modules encapsulate the correct analytic counterparts of algebraic D-modules, as evidenced in several contexts:

• Equivariant and Induced Modules: The category admits induction and restriction functors compatible with $A \cong \varprojlim_{n} A_n$4-adic group actions, and forms a bridge to the theory of admissible locally analytic representations. The geometric induction functor and its properties of support and (ir)reducibility are decisive in the classification of equivariant D-modules (Ardakov et al., 13 Jan 2025, Ardakov, 2020).
• Global Sections and Beilinson–Bernstein Localisation: On rigid analytic flag varieties, global sections of coadmissible D-modules correspond to admissible locally analytic group representations, and the global sections functor preserves coadmissibility (Bode, 2018, Schmidt, 2010).
• Meromorphic Connections and Extension Criteria: For meromorphic connections on rigid spaces, coadmissibility is governed by the roots of $A \cong \varprojlim_{n} A_n$5-functions: positivity of type ensures extension to a coadmissible D-module, while certain Liouville exponents obstruct coadmissibility (Bitoun et al., 2018).
• Hilbert Polynomials and Finite-Length Criteria: The length and multiplicity formalism for modules over completed Weyl algebras provides tools for establishing finite-length results for analytic analogues of holonomic D-modules (Reichardt, 27 Apr 2026).

5. Structural Properties, Duality, and Dimension Theory

The homological framework inherited from the Auslander regularity yields:

• Dimension Function: For $A \cong \varprojlim_{n} A_n$6 in the coadmissible category over a Fréchet–Stein, Auslander–Gorenstein algebra, one defines a dimension $A \cong \varprojlim_{n} A_n$7 which sharply discriminates between various classes: e.g., weakly holonomic if $A \cong \varprojlim_{n} A_n$8 (Schmidt et al., 2020, Ardakov et al., 2019).
• Duality: The derived duality $A \cong \varprojlim_{n} A_n$9 is involutive on the subcategory of minimally dimensioned (weakly holonomic) coadmissible modules, with $K$0, ensuring biduality parallels to the algebraic case (Schmidt et al., 2020).
• Kashiwara's Equivalence and Simple Modules: The pushforward $K$1 of the structure sheaf of a closed subvariety $K$2 yields simple coadmissible D-modules on $K$3 supported on $K$4, with simple objects thus parametrized by subvarieties (Ardakov et al., 2015).

Table: Fundamental Features of Coadmissible $K$5-Modules

Feature	Description	Reference
Topological type	Inverse limit of finite Banach modules (Fréchet–Stein)	(Lyubinin, 2014)
Homological regularity	Auslander regularity, finite global dimension at each level	(Bode, 12 May 2025)
Functoriality	Stable under six operations, Kashiwara equivalence, duality	(Bode, 2021)
Length/Muliplicity	Controlled by Hilbert polynomial computation	(Reichardt, 27 Apr 2026)
Applications	$K$6-adic Beilinson–Bernstein, analytic representation theory	(Bode, 2018)

6. Connections to Representation Theory and Further Developments

Coadmissible modules over completed enveloping algebras and distribution algebras serve as analytic analogues of finite-dimensional representations:

• Representation-Theoretic Correspondences: There are anti-equivalences between the category of coadmissible arithmetic D-modules on formal models of flag varieties and that of admissible locally analytic group representations (Huyghe et al., 2015). For compact $K$7-adic Lie groups, projective coadmissible modules are finitely generated, but the category does not have enough projectives (Zábrádi, 2010).
• Riemann–Hilbert Correspondence: Recent work establishes a full "solution-to-module" and de Rham correspondence for coadmissible D-modules, extending the Riemann–Hilbert dictionary to the $K$8-adic infinite-order setting (Wiersig, 14 Jun 2025).
• Dimension and Finiteness: For modules of algebraic origin (extensions/meromorphic connections/local cohomology of connections), coadmissibility ensures finite length and well-behaved stratification, analogous to the holonomic theory over the complex numbers (Reichardt, 27 Apr 2026, Hallopeau, 2022).

7. Significance, Pathologies, and Outlook

Coadmissible module theory for completed rings of differential operators forms a bridge between $K$9-adic rigid analytic geometry, nonarchimedean functional analysis, and $A_n$0-adic representation theory. The Auslander regularity of Banach algebra components suffices to permit a robust microlocal and homological theory. Despite their favorable properties, coadmissible modules can display pathologies not seen in the algebraic setting: infinite length weakly holonomic modules, infinite-dimensional fibers, or failures of extension for non-classical meromorphic connections (Ardakov et al., 2019, Bitoun et al., 2018). Nevertheless, the framework is indispensable for structuring six-functor formalisms, compatibility with representation theory, analytic analogues of geometric localization, and the analytic Riemann–Hilbert correspondence (Bode, 2021, Wiersig, 14 Jun 2025).