Solid Theory: Locally Analytic Representations

• Solid Theory of Locally Analytic Representations is a framework that uses condensed mathematics to view p-adic Lie group actions as solid modules with robust homological properties.
• It leverages Fréchet–Stein algebras and coadmissible modules to construct derived categories and resolve analytic vectors within an abelian setting.
• Extensions to mixed characteristic and semilinear cases establish strong links to p-adic Hodge theory, eigenvarieties, and the p-adic Langlands program.

The solid theory of locally analytic representations investigates $p$-adic Lie group representations on vector spaces equipped with analytic structures, situated in the robust framework of condensed mathematics and solid functional analysis. Solid representations are objects in the abelian category of solid (condensed) modules, which replaces the quasi-abelian world of topological vector spaces and enables precise homological and categorical techniques. Recent developments have extended the solid theory to encompass mixed characteristic coefficients, semilinear modules, and deeper connections to $p$-adic Hodge theory, eigenvarieties, and the $p$-adic Langlands program.

1. Solid Representations and Condensed Mathematics

Solid representations are $p$-adic Lie group representations considered within the abelian category of solid vector spaces, as formulated in the condensed mathematics framework of Clausen and Scholze. Every complete locally convex $K$-vector space (Banach, Fréchet, LB, LF, etc.) can be embedded as a solid module. This allows the systematic application of homological algebra and derived category methods, circumventing pathologies of the traditional topological setting. Solid locally analytic representations are those where every vector's orbit map is locally analytic, but defined and manipulated within the abelian solid category rather than the quasi-abelian category of locally convex spaces (Jacinto et al., 2021, Jacinto et al., 2023).

The solid category enables the construction of derived functors (such as the derived functor of analytic vectors), extensions, resolutions, and the comparison of cohomology theories, crucial for conceptual clarity and technical power in $p$-adic representation theory.

2. Fréchet–Stein Algebras and Coadmissible Modules

A core structural feature in the solid theory is the use of Fréchet–Stein algebras. For a $p$-adic Lie group $G$ over $L$ and a complete non-archimedean extension $K/L$, the locally analytic distribution algebra is assembled as a projective limit of Banach algebras: $D^{la}(G,K) = \varprojlim_h D^h(G,K)$ where each $D^h(G,K)$ is noetherian, Auslander, has global dimension $\leq \dim_L(G)$, and the transition maps are flat and dense. Finite $D^h(G,K)$-modules (coadmissible modules) are thus Banach spaces with finite projective resolutions; coadmissible $D^{la}(G,K)$-modules correspond anti-equivalently to admissible solid locally analytic representations (Jacinto et al., 2023).

This Fréchet–Stein property is essential for defining the derived category of quasi-coherent sheaves of solid representations and underpins the "noncommutative geometry" perspective for $G$-representations.

3. Extension to Mixed Characteristic and Semilinear Cases

The recent extension of solid locally analytic theory to mixed characteristic coefficients significantly broadens applicability. New coefficient rings include $\mathbf{F}_p((X))$ and certain analytic function rings such as $\mathbf{Z}_p[[X]]\langle p/X\rangle [1/X]$, together with semilinear group actions. This required the generalization of analytic distribution algebras, completion techniques, and functional analytic arguments to these more exotic (nonarchimedean analytic) base rings (Porat, 15 Oct 2025).

Consequences of this extension include potential advances in:

• Mixed characteristic $p$-adic Hodge theory
• The construction and paper of eigenvarieties over arbitrary coefficient fields
• Generalizations of the $p$-adic Langlands correspondence beyond characteristic zero coefficients

4. Cohomological Theory and Comparison Results

Solid functional analysis permits robust homological tools for locally analytic representations. Central results include the generalization of Lazard's isomorphisms: for a solid $G$-representation $V$ (or suitable complex $C$),

$$R_{K[G]}(K, C) \cong R_{D^{h^+}(G, K)}(K, C) \cong (R_{U(\mathfrak{g})}(K, C))^G$$

i.e., group cohomology, distribution algebra cohomology, and Lie algebra cohomology coincide for derived locally analytic objects (Jacinto et al., 2021, Jacinto et al., 2023). Comparison theorems between continuous cohomology and cohomology of analytic vectors are greatly strengthened due to the abelian nature of the solid category.

The Lazard–Serre resolution and its solid analytic generalization (in the spirit of Kohlhaase) are applied to Banach module contexts, including twisted distribution algebras and Banach pairs, yielding derived comparison results and facilitating spectral sequence arguments.

5. Derived Functors, Resolutions, and Duality

The solid environment enables derived functors of analytic vectors, explicit projective resolutions (e.g., via Chevalley–Eilenberg complexes), and wall complex constructions for solid representations. For admissible solid locally analytic representations $V$, the category admits explicit resolutions by analytic coefficient systems, and extension groups $\operatorname{Ext}_G^n(V, W)$ can be calculated using derived functor machinery (Agrawal et al., 8 Sep 2024).

Duality statements are natural in this context. For (weakly D-proper) morphisms of solid D-stacks (e.g., arising from families of $(\varphi,\Gamma)$-modules on relative Fargues–Fontaine curves), a Poincaré duality is proven: $$\operatorname{Hom}_{\mathbf{Q}_p}\left(\Gamma(X, M), \mathbf{Q}_p\right) \cong \Gamma\left(X, \operatorname{Hom}\left(M, f^! \mathbf{1}_S\right)\right)$$ This duality provides new proofs and generalizations of finiteness and duality properties for cohomology in solid analytic contexts (Mikami, 2 Apr 2025).

6. Relations to $p$-adic Hodge Theory and the Langlands Program

Solid locally analytic theory, especially in mixed characteristic and for semilinear representations, underpins links to $p$-adic Hodge theory and the $p$-adic Langlands program. Foundational works by Berger, Colmez, and collaborators elucidate the correspondence between locally analytic representations, overconvergent $(\varphi, \Gamma)$-modules, and $p$-adic differential equations, information that is synthesized in the solid setting (Porat, 15 Oct 2025).

The analytic stack and 6-functor formalism, as developed by Clausen–Scholze and Heyer–Mann, is employed to formulate duality and finiteness theorems for families of $(\varphi, \Gamma)$-modules, extending the classic results of Kedlaya–Pottharst–Xiao and providing a categorical context for generalizations relating to eigenvarieties and the Langlands correspondence (Mikami, 2 Apr 2025, Porat, 15 Oct 2025).

7. Bibliography and Interconnections

The solid theory of locally analytic representations synthesizes foundational tools:

• Analytic distributions and admissible representations (Schneider–Teitelbaum)
• Extensions of the Lazard–Serre resolution (Lazard, Kohlhaase)
• Condensed mathematics and solid module theory (Clausen–Scholze, Mann)
• Analytic and (pro-)Banach techniques in mixed characteristic (Lourenço, Stacks Project)
• $p$-adic Hodge modules and eigenvariety techniques (Kedlaya, Colmez, Emerton, Hansen–Newton, Coleman–Mazur, Bellovin)

This integration provides a comprehensive framework, with robust homological devices, non-commutative geometry perspectives, and analytic techniques tailored for applications in arithmetic geometry, automorphic forms, and $p$-adic representation theory.

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Summary Table: Fundamental Structures in Solid Locally Analytic Representations

Structure	Description	Main Reference
Solid Representation	Object in abelian category of condensed modules (solid vector spaces)	(Jacinto et al., 2021)
Fréchet–Stein Algebra	Inverse limit of noetherian Banach algebras with flat transition maps	(Jacinto et al., 2023)
Cohomological Comparison	$$R_{K[G]} \cong R_{D^{la}(G, K)} \cong (R_{U(\mathfrak{g})})^G$$	(Jacinto et al., 2021)
Resolution by Wall Complex	Analytic and geometric (Lie algebraic) resolutions combined for Ext-groups	(Agrawal et al., 8 Sep 2024)
6-Functor Formalism and Duality	Poincaré duality for solid D-stacks and $(\varphi,\Gamma)$-modules families	(Mikami, 2 Apr 2025)
Mixed Characteristic/Nonarchimedean Base	Generalized to $\mathbf{F}_p((X))$, $\mathbf{Z}_p[[X]]\langle p/X\rangle$	(Porat, 15 Oct 2025)

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The solid theory of locally analytic representations delivers a comprehensive, categorical, and analytic apparatus for the paper of $p$-adic Lie group representations, particularly in settings requiring robust homological control, dualities, and general base coefficients relevant both in arithmetic geometry and higher representation theory.