Cohomology of $p$-adic Stein spaces

TL;DR

This paper computes the $p$-adic étale and pro-étale cohomologies of Drinfeld half-spaces across all dimensions, introducing a new comparison theorem for $p$-adic Stein spaces' cohomology.

Contribution

It presents a novel comparison theorem for $p$-adic pro-étale cohomology of $p$-adic Stein spaces, enabling explicit cohomology calculations for Drinfeld half-spaces.

Findings

Computed $p$-adic étale cohomology of Drinfeld half-spaces.

Established a new comparison theorem for $p$-adic pro-étale cohomology.

Extended cohomology calculations to all dimensions of Drinfeld half-spaces.

Abstract

We compute the $p$-adic \'etale and the pro-\'etale cohomologies of the Drinfeld half-space of any dimension. The main input is a new comparison theorem for the $p$-adic pro-\'etale cohomology of $p$-adic Stein spaces.