On the cohomology of 𝑝-adic analytic spaces, I: The basic comparison theorem

The purpose of this paper is to prove a basic p p -adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure C C of a p p -adic field: p p -adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over B dR + {\mathbf B}^+_{\operatorname {dR} } ). The key computation is the passage from absolute crystalline cohomology to Hyodo–Kato cohomology and the construction of the related Hyodo–Kato isomorphism. We also “geometrize” our comparison theorem by turning p p -adic pro-étale and syntomic cohomologies into sheaves on the category P e r f C {\mathrm {Perf}}_C of perfectoid spaces over C C and the period morphisms into maps between such sheaves (this geometrization will be crucial in our study of the C s t C_{\mathrm {st}} -conjecture in the sequel to this paper and in the formulation of duality for geometric p p -adic pro-étale cohomology).