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

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.