Rational analytic syntomic cohomology

Abstract: We define and study the rational analytic syntomification $X^{\mathrm{Syn}}$ of a partially proper rigid-analytic variety $X$ over $\mathbb{Q}_p$. We establish Poincaré duality and a theory of first Chern classes for the resulting cohomology theory, identify vector bundles on $X^{\mathrm{Syn}}$ with de Rham bundles on the Fargues--Fontaine curve of $X^{\diamondsuit}$ and recover several classical comparison theorems in $p$-adic Hodge theory. We also develop analogues of our results and constructions over $\mathbb{C}_p$.

• The paper introduces a geometric construction of an analytic syntomification X^(Syn) for partially proper rigid-analytic varieties over Q_p and C_p.
• It employs analytic prismatisation, Nygaardification, and gluing techniques to recover syntomic cohomology and establish six-functor compatibility and duality properties.
• Strong comparison theorems and applications to p-adic Hodge theory, Fargues–Fontaine geometry, and motivic cohomology are demonstrated.

Rational Analytic Syntomic Cohomology: Cohomological Syntomification of Rigid-Analytic Varieties

Introduction and Motivation

The development of $p$-adic Hodge theory has led to a multitude of distinct cohomology theories, each probing interrelations between de Rham, étale, and log-crystalline cohomologies for varieties over $p$-adic fields. Syntomic cohomology functions as a universal intermediary encapsulating comparison theorems, and thus its geometric construction and categorical properties are of fundamental importance—particularly over the field of rigid-analytic spaces.

"Rational analytic syntomic cohomology" (2604.15193) introduces an analytic counterpart to syntomification for rigid-analytic varieties over $Q_p$, leveraging the recent framework of Gelfand stacks. This work achieves a geometric construction of the rational analytic syntomification $X^{\mathrm{Syn}}$ attached to any partially proper rigid-analytic variety $X$, thereby providing a space whose quasi-coherent cohomology recovers syntomic cohomology. The main theoretical advances synthesized by this manuscript integrate geometric transmutation principles (in the sense of Bhatt and Drinfeld), six-functor formalisms, and prismatic methodologies, all in the analytic context, and rigorously establish deep comparison results throughout the landscape of $p$-adic Hodge theory.

Construction of Analytic Syntomification

The heart of the paper is the construction of the syntomification $X^{\mathrm{Syn}}$ for a Gelfand stack $X$ over $Q_p$, following a series of intermediate geometric avatars:

• Analytic Prismatisation ($X^\triangle$): The variety $p$0 is first associated to its analytic prismatisation as a Gelfand stack over $p$1. The construction makes essential use of the structure of nilperfectoid rings and the pro-étale and prismatic geometry underlying $p$2-adic cohomology.
• Analytic Nygaardification ($p$3): $p$4 refines $p$5 by incorporating a filtered structure analogous to the Nygaard filtration, via a pullback construction involving both de Rham and Hodge–Tate period maps, imitating the pushout formalism of Gardner–Madapusi for the algebraic Nygaardification.
• Analytic Syntomification ($p$6): One then defines $p$7 as a stack obtained by gluing two copies of $p$8 embedded disjointly in $p$9 (via $Q_p$0, $Q_p$1).

This geometric approach induces a syntomic site equipped with the necessary period structure and functoriality to recover classical syntomic cohomology by taking global sections of the "Breuil–Kisin twist" line bundle on $Q_p$2: $Q_p$3

This construction is defined in a manner compatible with coefficients in arbitrary perfect complexes (analytic $Q_p$4-gauges), thus allowing broad generalization.

Fundamental Properties: Duality, Functoriality, Chern Classes

Poincaré Duality

$Q_p$5 is shown to enjoy a robust form of Poincaré duality: for $Q_p$6, the cohomology functor admits a dualizing sheaf $Q_p$7 and the pushforward functor $Q_p$8 along $Q_p$9 satisfies: $X^{\mathrm{Syn}}$0

The shift by $X^{\mathrm{Syn}}$1 (rather than the expected $X^{\mathrm{Syn}}$2) is explained via the contractible topological dimension inherent to $X^{\mathrm{Syn}}$3.

Six-Functor Compatibility

The syntomification construction is compatible with the full six-functor formalism for quasi-coherent sheaves on Gelfand stacks. For a smooth proper morphism $X^{\mathrm{Syn}}$4 of relative dimension $X^{\mathrm{Syn}}$5, the induced morphism $X^{\mathrm{Syn}}$6 is cohomologically smooth with explicit dualizing sheaf $X^{\mathrm{Syn}}$7, and $X^{\mathrm{Syn}}$8 preserves perfectness.

Chern Classes and Projective Bundle Formula

A system of syntomic Chern classes is defined in the sense of Zavyalov, via a morphism $X^{\mathrm{Syn}}$9, satisfying a syntomic variant of the classical projective bundle formula.

Comparison Theorems in $X$0-adic Hodge Theory

The structure of $X$1 is exploited to derive strong versions and generalizations of the classical comparison isomorphisms. These include:

De Rham Comparison (Scholze's Theorem)

A categorification of the de Rham comparison via filtered vector bundles with integrable connection is given. The associated period maps induce, for proper smooth $X$2 over $X$3 and de Rham local systems $X$4,

$X$5

with compatibility for cohomology with coefficients.

Syntomic–Hyodo–Kato and Syntomic–Pro-étale Comparison

Hyodo–Kato: For any smooth derived Berkovich space $X$6,

$X$7

and syntomic cohomology fits into a pullback diagram with Hyodo–Kato and filtered de Rham cohomologies.

Pro-étale: For any vector bundle $X$8 of compatible Hodge–Tate weights,

$X$9

in low degrees, yielding syntomic–pro-étale compatibility up to truncation at the Hodge–Tate range.

Vector Bundles and Fargues–Fontaine Geometry

The stack of vector bundle analytic $p$0-gauges on $p$1 is described fully in terms of "de Rham" vector bundles on the relative Fargues–Fontaine curve $p$2. For $p$3, this yields an equivalence between $p$4 and the category of $p$5-equivariant de Rham vector bundles on the Fargues–Fontaine curve, tightly connecting this formalism to the heart of the $p$6-adic Langlands program.

Extensions to Analytic Syntomification over $p$7: Geometric Syntomic Cohomology

The construction is generalized to rigid-analytic varieties over $p$8, producing syntomic cohomology classes in the context of the Fargues–Fontaine curve and $p$9-cohomology. A stack-theoretic description is given for $X^{\mathrm{Syn}}$0-cohomology compatible with filtrations, and comparison diagrams are proven analogous to the arithmetic case.

Implications and Prospects

The presented analytic syntomification construction solves the longstanding open problem of formulating a geometric incarnation of syntomic cohomology for rigid-analytic spaces with $X^{\mathrm{Syn}}$1-coefficients, creating a bridge between relative prismatic and Hodge–Tate theories in the analytic setting. The full six-functor formalism and the explicit geometric and moduli interpretations open perspectives for:

• Cohomological computations in $X^{\mathrm{Syn}}$2-adic Hodge theory with arbitrary coefficients;
• Stack-theoretic approaches to $X^{\mathrm{Syn}}$3-adic regulators, motivic cohomology, and $X^{\mathrm{Syn}}$4-theory;
• Categorification and geometric representation theory over the Fargues–Fontaine curve.

Further potential developments include the extension to integral coefficients, possible stack-theoretic generalizations to families or deformation contexts, and applications in the stack-theoretic approach to $X^{\mathrm{Syn}}$5-adic motives and Langlands correspondences.

Conclusion

This paper provides a comprehensive, stack-theoretic construction and analysis of rational analytic syntomic cohomology for rigid-analytic varieties over $X^{\mathrm{Syn}}$6 and $X^{\mathrm{Syn}}$7. The geometric approach not only recapitulates but systematizes comparison theorems, intertwines Fargues–Fontaine and prismatic geometries, and equips the theory with the apparatus needed for deeper categorical, motivic, and arithmetic applications. As such, rational analytic syntomic cohomology, as developed here, serves as a foundational tool for advancing $X^{\mathrm{Syn}}$8-adic Hodge theory and its associated domains.