The de Rham–Fargues–Fontaine cohomology

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 ‫ނ‬ 1 -homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves. 1. Introduction 2097 2. Adic étale motives 2100 3. Relative overconvergent varieties and motives 2111 4. The relative overconvergent de Rham cohomology 2116 5. A rigid analytic Fargues-Fontaine construction 2132 6. The de Rham-Fargues-Fontaine cohomology 2142 Acknowledgments 2148 References 2148The authors are partially supported by the Agence Nationale de la Recherche (ANR), project ANR-19-CE40-0015 and ANR-18-CE40-0017.