$G$-gerbes on perfectoid spaces

TL;DR

This paper proves an equivalence between categories of $G$-gerbes on perfectoid spaces under the v-topology and étale topology, extending known results about $G$-torsors.

Contribution

It establishes a 2-categorical equivalence of $G$-gerbes on perfectoid spaces between the v-topology and étale topology, generalizing previous torsor results.

Findings

Equivalence of categories of $G$-gerbes on perfectoid spaces.

Extension of known torsor equivalence to gerbes.

Supports the use of v-topology in perfectoid geometry.

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 $\nu: X_v \to X_{\mathrm{\acute{e}t}}$. It is known from work of Heuer that the direct image functor $\nu_*$ 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.