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 ${\mathbb{Q}}_p$. 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.

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