The de Rham–Fargues–Fontaine cohomology,Algebra & Number Theory

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The de Rham–Fargues–Fontaine cohomology
Algebra & Number Theory ( IF 1.3 ) Pub Date : 2023-10-08 , DOI: 10.2140/ant.2023.17.2097
Arthur-César Le Bras , Alberto Vezzani

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 $𝔹^{1}$ -homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves.

更新日期：2023-10-09

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