$G$-gerbes on perfectoid spaces

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 $\hat{I}½:X_v\to X_{\acute{\mathrm{e}}\mathrm{t}}$. It is known from work of Heuer that the direct image functor $\hat{I}{½}_{\ast }$ 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.

Refined proofs for Prop. 2.9 and Lemma 2.14. Added Remarks on non-perfectoid cases