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 quasi-coherent 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 quasi-compact and P is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit B1-homotopies, the motivic proper base change and the formalism of solid quasi-coherent sheaves.
The de Rham-Fargues-Fontaine cohomology / A. Le Bras, A. Vezzani. - In: ALGEBRA & NUMBER THEORY. - ISSN 1937-0652. - 17:12(2023 Oct 08), pp. 2097-2150. [10.2140/ant.2023.17.2097]
The de Rham-Fargues-Fontaine cohomology
A. VezzaniUltimo
2023
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 quasi-coherent 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 quasi-compact and P is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit B1-homotopies, the motivic proper base change and the formalism of solid quasi-coherent sheaves.File | Dimensione | Formato | |
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