The relative Hodge-Tate spectral sequence for rigid analytic spaces

2024/02/01 by Ben Heuer, Heuer, Ben
Mathematics · #14F30 #14G22 #14G45 #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematical Analysis and Transform Methods #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.00842

openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• A Hodge--Tate decomposition with rigid analytic coefficients
• A Note on Hodge-Tate Spectral Sequences
• The relative Hodge-Tate spectral sequence -- an overview
• La suite spectrale de Hodge-Tate
• p-adic Hodge theory for rigid-analytic varieties