$\mathbb{G}_m$-cohomology of $p$-adic Stein spaces

TL;DR

This paper calculates the étale -cohomology of certain p-adic Stein spaces, employing filtrations and p-adic Hodge theory, with applications to Drinfeld upper-half spaces.

Contribution

It provides explicit formulas for -cohomology of p-adic Stein spaces, including Drinfeld upper-half space, using novel methods from p-adic Hodge theory.

Findings

Computed -cohomology for p-adic Stein spaces.

Applied p-adic Hodge theory and Kummer sequences in the computation.

Derived formulas applicable to Drinfeld upper-half space.

Abstract

We compute the \'etale $\mathbb{G}_m$-cohomology of some $p$-adic rigid analytic Stein spaces. The computation is done by considering the filtration induced by the subgroup of principal units $U=1+ \mathfrak{m} \mathcal{O}^+$ of $\mathbb{G}_m$. We then determine the $U$-cohomology via methods from $p$-adic Hodge theory (passage to the pro-\'etale site, comparison theorems with $p$-adic cohomologies), while the $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.