G-bundles on the absolute Fargues-Fontaine curve

TL;DR

This paper establishes an equivalence between vector bundles on the absolute Fargues-Fontaine curve and isocrystals, extending to G-bundles and sections over classifying stacks for profinite groups.

Contribution

It proves a canonical equivalence between vector bundles on the absolute Fargues-Fontaine curve and isocrystals, and extends this to G-bundles and sections over classifying stacks.

Findings

Category of vector bundles on the absolute Fargues-Fontaine curve is equivalent to isocrystals.

Results extend to G-bundles for reductive groups.

Includes sections over classifying stacks for locally profinite groups.

Abstract

We prove that the category of ``vector bundles on the absolute Fargues--Fontaine curve'' (more precisely the category of sections over some discrete algebraically closed field of the $v$-stack $\mathrm{Bun}_\mathrm{FF}$ of vector bundles on the Fargues--Fontaine curve) is canonically equivalent to the category of isocrystals. We deduce a similar result for ``$G$-bundles on the absolute Fargues--Fontaine curve'' for some reductive group $G$ as well as for sections of $\mathrm{Bun}_\mathrm{FF}$ over classifying stacks for locally profinite groups.