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-& eacute;tale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over B + 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-& eacute;tale and syntomic cohomologies into sheaves on the category 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 s t-conjecture in the sequel to this paper and in the formulation of duality for geometric p-adic pro-& eacute;tale cohomology).