Sur la cohomologie \'etale de la courbe de Fargues-Fontaine

Abstract

In this article the \'etale cohomology of constructible torsion sheaves on the \'etale site of the algebraic resp. adic Fargues-Fontaine curve is analyzed. In the $\ell\neq p$-torsion case, two conjectures of Fargues are verified: vanishing in degrees greater than two and the comparison between the \'etale cohomology of the adic and the algebraic curve. In the $p$-torsion case, under a certain assumption, the vanishing of the \'etale cohomology in degrees greater than two of those Zariski-constructible sheaves on the adic curve that come via pullback from the algebraic curve is proven.