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The relative Hodge-Tate spectral sequence for rigid analytic spaces

Ben Heuer

Abstract

We construct a relative Hodge-Tate spectral sequence for any smooth proper morphism of rigid analytic spaces over a perfectoid field extension of $\mathbb Q_p$. To this end, we generalise Scholze's strategy in the absolute case by using smoothoid adic spaces. As our main additional ingredient, we prove a perfectoid version of Grothendieck's "cohomology and base-change". We also use this to prove local constancy of Hodge numbers in the rigid analytic setting, and deduce that the relative Hodge-Tate spectral sequence degenerates.

The relative Hodge-Tate spectral sequence for rigid analytic spaces

Abstract

We construct a relative Hodge-Tate spectral sequence for any smooth proper morphism of rigid analytic spaces over a perfectoid field extension of . To this end, we generalise Scholze's strategy in the absolute case by using smoothoid adic spaces. As our main additional ingredient, we prove a perfectoid version of Grothendieck's "cohomology and base-change". We also use this to prove local constancy of Hodge numbers in the rigid analytic setting, and deduce that the relative Hodge-Tate spectral sequence degenerates.
Paper Structure (15 sections, 32 theorems, 68 equations)

This paper contains 15 sections, 32 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The relative Hodge--Tate sequence
  3. Previous results on the relative Hodge--Tate spectral sequence
  4. Degeneration
  5. Splittings
  6. Cohomology and base-change
  7. Setup and Recollections
  8. Cohomology and base change in rigid geometry
  9. Base change in Kiehl's Theorem
  10. Grauert's Theorem in rigid analytic geometry
  11. Cohomology and base change
  12. A limit version of the Primitive Comparison Theorem
  13. The relative $p$-adic Hodge--Tate sequence
  14. Splittings in the case of curve fibrations
  15. Commuting Cohomology and Completion

Key Result

Theorem 1.2

Let $K$ be a perfectoid field over $\mathbb{Q}_p$. Let $f: X\to S$ be a smooth proper morphism of reduced rigid spaces over $K$. Let $S_{{\operatorname{pro\acute{e}t}}}$ be the pro-étale site with its completed structure sheaf $\widehat{\mathcal{O}}_S$. Let $\nu: S_{{\operatorname{pro\acute{e}t}}}\

Theorems & Definitions (76)

  • Theorem 1.2: \ref{['t:relHTSS']}
  • Theorem 1.4: \ref{['p:Deligne-constant-fibre-dim']}
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4: BMS
  • Proposition 2.5: heuer-sheafified-paCS
  • Definition 2.6
  • Proposition 3.1: StacksProject
  • ...and 66 more