$G$-gerbes on perfectoid spaces

TL;DR

The paper addresses non-abelian gerbes over perfectoid spaces by comparing $G$-gerbes on the $v$-site and the étale site via the pushforward $\nu_*$. Building on Heuer's torsor equivalence, it develops a centre-based descent framework for rigid groups, establishing that $Z$, the centre of $G$, governs the obstruction and inner-outer automorphism data so that $\nu_*$ yields an equivalence of 2-categories $\mathrm{Gerb}(X_v; G_v) \to \mathrm{Gerb}(X_{\acute{e}t}; G_{\acute{e}t})$. The key steps include constructing centers as Zariski-closed rigid subgroups, proving their compatibility with both the $v$- and étale topologies, and applying Giraud-type descent criteria together with vanishing cohomology hypotheses to relate $\mathrm{Gerb}^E(E';G)$ to $\mathrm{Gerb}(E; f_*G)$. Consequently, for perfectoid spaces $X$ and rigid groups $G$, the two 2-categories of $G$-gerbes on the $v$- and étale sites are equivalent, with the centre data ensuring compatibility across topologies and with further pro-étale and quasi-pro-étale contexts in appropriate cases.

Abstract

Let $K$ be a complete non-archimedean field over $\mathbb{Q}_p$, $G$ be a rigid group over $K$, and $X$ be a perfectoid space over $K$. We consider the natural morphism of sites $ν: X_v \to X_{\mathrm{\acute{e}t}}$. It is known from work of Heuer that the direct image functor $ν_*$ induces an equivalence of the categories of $G$-torsors. In this article, we show that there is an equivalence of 2-categories of $G$-gerbes on these two topologies.

Theorems & Definitions (36)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Remark 1.4
• Definition 2.1
• Lemma 2.2: Far19
• Theorem 2.3: Heu22
• Theorem 2.4: Far19
• Definition 2.5
• Proposition 2.6