$\mathbb{G}_m$-cohomology of $p$-adic Stein spaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910194389221376 |
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| author | Gilles, Sally Junger, Damien |
| author_facet | Gilles, Sally Junger, Damien |
| contents | 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 $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04696 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathbb{G}_m$-cohomology of $p$-adic Stein spaces Gilles, Sally Junger, Damien Number Theory Algebraic Geometry 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 $\mathbb{G}_m/U$-cohomology is obtained using Kummer exact sequences. In particular, our formula applies to the case of Drinfeld upper-half space. |
| title | $\mathbb{G}_m$-cohomology of $p$-adic Stein spaces |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2605.04696 |