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A Hodge-Tate decomposition with rigid analytic coefficients

Lucas Gerth

TL;DR

The authors extend the Hodge--Tate framework to proper smooth rigid spaces $X$ with coefficients in locally $p$-divisible groups $G$, establishing a $G$-coefficient HT decomposition and proving a geometric HT spectral sequence degenerates at $E_2$. They develop diamantine higher direct images to organize moduli of $G$-torsors, proving a degeneracy for the geometric spectral sequence and deriving both the standard HT decomposition and a relative version over a base $S$. The results connect to analytic Brauer groups and provide a geometric $p$-adic Simpson correspondence, including explicit treatments for abeloid varieties and curves via fundamental-group descriptions. The work blends good/adic space theory, smoothoid geometry, and tilde-limit approximation to relate étale and $v$-topologies, yielding tools with broad impact on non-abelian $p$-adic Hodge theory and relative cohomology.

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.

A Hodge-Tate decomposition with rigid analytic coefficients

TL;DR

The authors extend the Hodge--Tate framework to proper smooth rigid spaces with coefficients in locally -divisible groups , establishing a -coefficient HT decomposition and proving a geometric HT spectral sequence degenerates at . They develop diamantine higher direct images to organize moduli of -torsors, proving a degeneracy for the geometric spectral sequence and deriving both the standard HT decomposition and a relative version over a base . The results connect to analytic Brauer groups and provide a geometric -adic Simpson correspondence, including explicit treatments for abeloid varieties and curves via fundamental-group descriptions. The work blends good/adic space theory, smoothoid geometry, and tilde-limit approximation to relate étale and -topologies, yielding tools with broad impact on non-abelian -adic Hodge theory and relative cohomology.

Abstract

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

Paper Structure

This paper contains 19 sections, 44 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. The Hodge--Tate decomposition
  3. The geometric Hodge--Tate spectral sequence
  4. Application to non-abelian $p$-adic Hodge theory
  5. Preliminaries
  6. Good adic spaces
  7. Smoothoid spaces
  8. Zariski-constructible sheaves
  9. Locally $p$-divisible groups
  10. Hodge--Tate spectral sequences
  11. The group theoretic spectral sequence
  12. Rigid approximation
  13. The diamantine higher direct images
  14. The geometric spectral sequence
  15. Degeneration
  16. ...and 4 more sections

Key Result

Theorem 1.1

(Theorem Theorem: The classical sseq splits) Let $X$ be a proper smooth rigid space over $K$ and let $G$ be a locally $p$-divisible rigid group. Then a choice of a $\mathop{\mathrm{\mathrm{B_{dR}^+}}}\nolimits+/\xi^2$-lift $\mathbb{X}$ of $X$ and the datum of an exponential $\mathop{\mathrm{Exp}}\no

Theorems & Definitions (129)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • ...and 119 more