The de Rham–Fargues–Fontaine cohomology

Le Bras, Arthur-César; Vezzani, Alberto

doi:10.2140/ant.2023.17.2097

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The de Rham–Fargues–Fontaine cohomology

Arthur-César Le Bras and Alberto Vezzani

Vol. 17 (2023), No. 12, 2097–2150

DOI: 10.2140/ant.2023.17.2097

Abstract

We show how to attach to any rigid analytic variety $V$ over a perfectoid space $P$ a rigid analytic motive over the Fargues–Fontaine curve $𝒳 (P)$ functorially in $V$ and $P$ . We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over $𝒳 (P)$ , and we show that its cohomology groups are vector bundles if $V$ is smooth and proper over $P$ or if $V$ is quasicompact and $P$ is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit $𝔹^{1}$ -homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves.

Keywords

p-adic Hodge theory, Fargues–Fontaine curve, de Rham cohomology, rigid analytic varieties, perfectoid spaces, motives

Mathematical Subject Classification

Primary: 14F30, 14F42, 14G45

Milestones

Received: 13 December 2021

Revised: 5 December 2022

Accepted: 3 January 2023

Published: 8 October 2023

Authors

Arthur-César Le Bras
Institut de Recherche Mathématique Avancée CNRS - Université de Strasbourg Strasbourg France

Alberto Vezzani
Dipartimento di Matematica “F. Enriques” Università degli Studi di Milano Milan Italy

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Vol. 17, No. 12, 2023