A Hodge-Tate decomposition with rigid analytic coefficients

TL;DR

This paper extends the Hodge--Tate decomposition to rigid analytic spaces with coefficients in any commutative locally p-divisible rigid group, generalizing previous results and introducing new spectral sequences that degenerate at E2.

Contribution

It introduces a generalized Hodge--Tate decomposition with locally p-divisible coefficients and proves the degeneration of associated spectral sequences at E2.

Findings

Hodge--Tate spectral sequences degenerate at E2.

Generalization of Hodge--Tate decomposition to broader coefficients.

Applications to analytic Brauer groups and p-adic Simpson correspondence.

Abstract

Let $X$ be a smooth proper rigid analytic space over a complete algebraically closed field extension $K$ of $\mathbb{Q}_p$. We establish a Hodge--Tate decomposition for $X$ with $G$-coefficients, where $G$ is any commutative locally $p$-divisible rigid group. This generalizes the Hodge--Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb{G}_a$. For this, we introduce geometric analogs of the Hodge--Tate spectral sequence with general locally $p$-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over $X$ and in the relative setting of smooth proper morphisms $X\rightarrow S$ of smooth rigid spaces. We deduce applications to analytic Brauer groups and the geometric $p$-adic Simpson correspondence.