The de Rham-Fargues-Fontaine cohomology

TL;DR

This paper constructs a new cohomology theory for rigid analytic varieties over perfectoid spaces, generalizing Scholze's conjecture by linking de Rham cohomology with vector bundles on the Fargues-Fontaine curve.

Contribution

It introduces a functorial construction of motives over the Fargues-Fontaine curve and proves that their cohomology groups are vector bundles in key cases, extending previous conjectures.

Findings

Cohomology groups are vector bundles for smooth proper varieties

Construction of motives over the Fargues-Fontaine curve

Generalization and proof of Scholze's conjecture

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 quasi-coherent 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 quasi-compact 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 quasi-coherent sheaves.