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).
|Titolo:||The de Rham-Fargues-Fontaine cohomology|
VEZZANI, ALBERTO (Corresponding)
|Parole Chiave:||Mathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; Mathematics - K-Theory and Homology; Mathematics - Number Theory|
|Settore Scientifico Disciplinare:||Settore MAT/02 - Algebra|
Settore MAT/03 - Geometria
|Data di pubblicazione:||2021-05-27|
|Appare nelle tipologie:||24 - Pre-print|