The relative Hodge–Tate spectral sequence for rigid analytic spaces

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.