On the cohomology of p-adic analytic spaces, I: The basic comparison theorem

2021/04/27 by Pierre Colmez, Colmez, Pierre, Wiesława Nizioł +1
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2104.13448

openalex publication_date 2021/04/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to prove a basic p-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure C of a p-adic field: p-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over \bfB⁺_{\rm dR}). The key computation is the passage from absolute crystalline cohomology to Hyodo-Kato cohomology and the construction of the related Hyodo-Kato isomorphism. We also "geometrize" our comparison theorem by turning p-adic pro-étale and syntomic cohomologies into sheaves on the category \rm Perf_C of perfectoid spaces over C (this geometrization will be crucial in our proof of the C_{\rm st}-conjecture in the sequel to this paper).

Related

• On p-adic comparison theorems for rigid analytic varieties, I
• On the cohomology of p-adic analytic spaces, II: The C_{\rm st}-conjecture
• Compactly supported p-adic pro-étale cohomology of analytic varieties
• Duality for p-adic geometric pro-étale cohomology
• A comparison theorem for l-adic cohomology