Cartier duality for gerbes of vector bundles

TL;DR

This paper establishes a Cartier duality for gerbes of vector bundles, linking algebraic and analytic structures, and applies it to relate categories of sheaves on the Hodge-Tate stack and Simpson gerbe.

Contribution

It introduces a duality framework for gerbes of vector bundles and connects categories of sheaves in mixed characteristic geometry.

Findings

Proves Cartier duality as an anti-equivalence of Hopf algebras.

Shows equivalence between solid quasi-coherent sheaves on the Hodge-Tate stack and weight 1 sheaves on the Simpson gerbe.

Abstract

We prove a Cartier duality for gerbes of algebraic and analytic vector bundles as an anti-equivalence of Hopf algebras in the category of kernels of analytic stacks. As an application, we prove that the category of solid quasi-coherent sheaves on the Hodge-Tate stack of a smooth rigid variety over an algebraically closed field $C$ of mixed characteristic $(0,p)$ is equivalent to the category of weight $1$ sheaves on Bhatt-Zhang's Simpson gerbe.