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

We compute the étale ${\mathbb{G}}_m$-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-étale site, comparison theorems with $p$-adic cohomologies), while the ${\mathbb{G}}_m\mathrm{/}U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space.