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

TL;DR

This paper investigates the étale cohomology of constructible torsion sheaves on the Fargues-Fontaine curve, confirming conjectures in the $ ext{l} eq p$ case and establishing vanishing results in the $p$-torsion case under certain assumptions.

Contribution

It verifies two conjectures of Fargues for the $ ext{l} eq p$ case and proves vanishing of étale cohomology in degrees greater than two for certain sheaves in the $p$-torsion case.

Findings

Verification of Fargues' conjectures in the $ ext{l} eq p$ case.

Proof of vanishing of étale cohomology in degrees > 2 for specific sheaves.

Comparison results between étale cohomology of algebraic and adic Fargues-Fontaine curves.

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.