{
  "id": "2405.07654",
  "version": "v1",
  "published": "2024-05-13T11:30:48.000Z",
  "updated": "2024-05-13T11:30:48.000Z",
  "title": "A note on étale $(\\varphi,Γ)$-modules in families",
  "authors": [
    "Marvin Schneider"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Lambda$ 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 $(\\Lambda,F)$-crystals on $\\mathfrak{X}$ is equivalent to $\\Lambda$-\\'etale local systems on the generic adic fiber of $\\mathfrak{X}$ and that the cohomology of $(\\Lambda,F)$-crystals recovers the pro-\\'etale 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 \\'etale $(\\varphi,\\Gamma)$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-13T11:30:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite residue field",
      "adic formal scheme",
      "reprove dees classical result",
      "generic adic fiber",
      "etale local systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}