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The de Rham–Fargues–Fontaine cohomology

Arthur-César Le Bras and Alberto Vezzani

Vol. 17 (2023), No. 12, 2097–2150
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 𝔹1-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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