A note on étale $(φ,Γ)$-modules in families
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929341475061760 |
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| author | Schneider, Marvin |
| author_facet | Schneider, Marvin |
| contents | Let $Λ$ be a complete noetherian local ring with finite residue field of characteristic $p$ and $K/\mathbb{Q}_p$ a $p$-adic field. We show that, by deformation of the structure sheaf on the (transversal) prismatic site of a bounded $p$-adic formal scheme $\mathfrak{X}$, the category of prismatic $(Λ,F)$-crystals on $\mathfrak{X}$ is equivalent to $Λ$-étale local systems on the generic adic fiber of $\mathfrak{X}$ and that the cohomology of $(Λ,F)$-crystals recovers the pro-étale cohomology of the corresponding local systems. The proof follows the strategy used in \cite{bhatt2023prismatic} and \cite{marks2023prismatic}. From this we construct an isomorphism between Iwasawa cohomology of a $p$-adic Lie extension of $K$ and prismatic cohomology. Following \cite{wu2021galois}, we then reprove Dee's classical result \cite{article} on the equivalence between families of Galois representations and étale $(φ,Γ)$-modules. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07654 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on étale $(φ,Γ)$-modules in families Schneider, Marvin Number Theory Let $Λ$ be a complete noetherian local ring with finite residue field of characteristic $p$ and $K/\mathbb{Q}_p$ a $p$-adic field. We show that, by deformation of the structure sheaf on the (transversal) prismatic site of a bounded $p$-adic formal scheme $\mathfrak{X}$, the category of prismatic $(Λ,F)$-crystals on $\mathfrak{X}$ is equivalent to $Λ$-étale local systems on the generic adic fiber of $\mathfrak{X}$ and that the cohomology of $(Λ,F)$-crystals recovers the pro-étale cohomology of the corresponding local systems. The proof follows the strategy used in \cite{bhatt2023prismatic} and \cite{marks2023prismatic}. From this we construct an isomorphism between Iwasawa cohomology of a $p$-adic Lie extension of $K$ and prismatic cohomology. Following \cite{wu2021galois}, we then reprove Dee's classical result \cite{article} on the equivalence between families of Galois representations and étale $(φ,Γ)$-modules. |
| title | A note on étale $(φ,Γ)$-modules in families |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.07654 |