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. - (2021 May 27).

The de Rham-Fargues-Fontaine cohomology

A. Vezzani
2021-05-27

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.
Mathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; Mathematics - K-Theory and Homology; Mathematics - Number Theory
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
https://arxiv.org/abs/2105.13028
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/861661
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