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

TL;DR

This paper establishes a fundamental $p$-adic comparison theorem linking pro-étale and de Rham cohomologies for smooth rigid analytic spaces over an algebraic closure of a $p$-adic field, using Hyodo-Kato theory.

Contribution

It introduces a new comparison theorem for $p$-adic cohomologies and develops a geometric framework on perfectoid spaces, advancing the understanding of $p$-adic Hodge theory.

Findings

Proves a $p$-adic comparison theorem for smooth rigid analytic varieties.

Constructs a Hyodo-Kato isomorphism connecting crystalline and Hyodo-Kato cohomologies.

Geometrizes cohomology theories as sheaves on perfectoid spaces.

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-\'etale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over $\bf{B}^+_{\rm dR}$). 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-\'etale 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).