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

Wieslawa Niziol (1) , Pierre Colmez

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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-\'etale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over $\bf{B}^+_{\rm dR}$). 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-\'etale and syntomic cohomologies into sheaves on the category ${\rm Perf}_C$ of perfectoid spaces over $C$ (this geometrization will be crucial in our proof of the $C_{\rm st}$-conjecture in the sequel to this paper).

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

Wieslawa Niziol, Pierre Colmez. On the cohomology of $p$-adic analytic spaces, I: The basic comparison theorem. 2021. ⟨hal-03431807⟩