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arXiv · 2605.04696

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

Sally Gilles·Damien Junger
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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.

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Sally Gilles, Damien Junger. 2026-05-06. $\mathbb{G}_m$-cohomology of $p$-adic Stein spaces. https://arxiv.org/abs/2605.04696

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