On equivariant vector bundles on the Fargues--Fontaine curve over a finite extension

Abstract

Let $K/E/\mathbb{Q}_p$ be a tower of finite extensions with $E$ Galois. We relate the category of $G_K$-equivariant vector bundles on the Fargues--Fontaine curve with coefficients in $E$ with $E$-$G_K$-$B$-pairs and describe crystalline and de Rham objects in explicit terms. When $E$ is a proper extension, we give a new description of the category in terms of compatible tuples of $\mathbf{B}_e$-modules, which allows us to compute Galois cohomology in terms of an explicit \v{C}ech complex which can serve as a replacement of the fundamental exact sequence.