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

The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over ${\mathbf B}^+_{\operatorname {dR} }$). 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 $p$-adic pro-étale and syntomic cohomologies into sheaves on the category ${\mathrm {Perf}}_C$ of perfectoid spaces over $C$ and the period morphisms into maps between such sheaves (this geometrization will be crucial in our study of the $C_{\mathrm {st}}$-conjecture in the sequel to this paper and in the formulation of duality for geometric $p$-adic pro-étale cohomology).