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 $\mathcal{𝒳}(P)$ 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 $\mathcal{𝒳}(P)$, 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 ${\mathbb{𝔹}}^1$-homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves.