Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories

2025/09/20 by Kaixing Cao, Cao, Kaixing
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.16726

openalex publication_date 2025/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, for adic étale motives over ℂ_p, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal ∞-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding G_\breveℚ_p-equivariant solid quasi-coherent sheaves on the Fargues-Fontaine curve; in this equivariant context, the comparison isomorphism becomes canonical. As an application, we show that the de Rham-Fargues-Fontaine cohomology of any smooth quasi-compact rigid analytic variety over ℂ_p admits a finite slope-increasing filtration.

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