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

TL;DR

This work develops a comprehensive framework linking $G_K$-equivariant vector bundles on the Fargues--Fontaine curve $X_E$ with $E$-$B$-pairs, extending the Fargues--Fontaine equivalence to the case $E\neq \mathbb{Q}_p$ and clarifying crystalline and de Rham realizations. It introduces $\mathbf{B}_e$-tuples and a co-covering approach to encode bundles via explicit rings $B_{e,T}$, enabling a concrete description of the category $\operatorname{Bun}_{X_E}$ and its $G_K$-equivariant version, as well as a practical computation of Galois cohomology through a Čech complex. The paper also relates crystalline objects to multivariable filtered $\varphi$-modules and explains how crystallinity can be detected from $D_S(\mathcal{E})$ and the ring $B_S$, connecting to $D_{cris}$ and $\operatorname{MF}^{\varphi}$-modules. In addition, a bridge to Berger’s multivariable $(\varphi,\Gamma)$-theory is established in the unramified setting, translating between $\varphi_q$-modules with filtrations and multivariable $(\varphi_q,\Gamma_E^{LT})$-modules, and identifying the geometric objects $\mathcal{V}(D)$ with Berger’s constructions, thus enriching the toolkit for p-adic Hodge theory on the Fargues--Fontaine curve.

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 Čech complex which can serve as a replacement of the fundamental exact sequence.

Theorems & Definitions (66)

• Theorem 1
• Theorem 2
• Definition 1.1
• Definition 1.2
• Proposition 1.3
• proof
• Theorem 1.4
• proof
• Proposition 1.5
• proof