Rational $p$-adic Hodge theory for rigid-analytic varieties

TL;DR

This paper develops a new cohomology theory for rigid-analytic varieties over _p, establishing comparison theorems with existing rational p-adic cohomologies and proving a conjecture of Le Bras.

Contribution

It introduces a cohomology framework that works without properness or smoothness, connecting various p-adic cohomologies via the Fargues-Fontaine curve and condensed formalism.

Findings

Proves a conjecture of Le Bras.

Establishes comparison theorems relating different p-adic cohomologies.

Extends results of Colmez-Niziol on p-adic cohomology.

Abstract

We study a cohomology theory for rigid-analytic varieties over $\mathbb{C}_p$, without properness or smoothness assumptions, taking values in filtered quasi-coherent complexes over the Fargues-Fontaine curve, which compares to other rational $p$-adic cohomology theories for rigid-analytic varieties $-$ namely, the rational $p$-adic pro-\'etale cohomology, the Hyodo-Kato cohomology, and the infinitesimal cohomology over the positive de Rham period ring. In particular, this proves a conjecture of Le Bras. Such comparison results are made possible thanks to the systematic use of the condensed and solid formalisms developed by Clausen-Scholze. As applications, we deduce some general comparison theorems that describe the rational $p$-adic pro-\'etale cohomology in terms of de Rham data, thereby recovering and extending results of Colmez-Niziol.