# 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 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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03444470
Contributor : Arthur-César Le Bras Connect in order to contact the contributor
Submitted on : Friday, November 26, 2021 - 11:52:52 AM
Last modification on : Wednesday, December 1, 2021 - 1:33:19 PM

### File

2105.13028.pdf
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### Identifiers

• HAL Id : hal-03444470, version 1
• ARXIV : 2105.13028

### Citation

Arthur-César Le Bras, Alberto Vezzani. The de Rham-Fargues-Fontaine cohomology. 2021. ⟨hal-03444470⟩

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