{
  "id": "2309.05039",
  "version": "v1",
  "published": "2023-09-10T14:26:48.000Z",
  "updated": "2023-09-10T14:26:48.000Z",
  "title": "The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory",
  "authors": [
    "Guchuan Li",
    "Ningchuan Zhang"
  ],
  "comment": "45 pages. Comments welcome!",
  "categories": [
    "math.AT"
  ],
  "abstract": "In this paper, we study profinite descent theory for Picard groups in $K(n)$-local homotopy theory through their inverse limit topology. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over $K(n)$-local commutative ring spectrum is equivalent to the limit of its base changes by the tower of generalized Moore spectra of type $n$. As a result, the $K(n)$-local Picard groups are endowed with a natural inverse limit topology. This topology allows us to identify the entire $E_1$ and $E_2$-pages of a descent spectral sequence for Picard spaces of $K(n)$-local profinite Galois extensions. Our main examples are $K(n)$-local Picard groups of homotopy fixed points $E_n^{hG}$ of the Morava $E$-theory $E_n$ for all closed subgroups $G$ of the Morava stabilizer group $\\mathbb{G}_n$. The $G=\\mathbb{G}_n$ case has been studied by Heard and Mor. At height $1$, we compute Picard groups of $E_1^{hG}$ for all closed subgroups $G$ of $\\mathbb{G}_1=\\mathbb{Z}_p^\\times$ at all primes as a Mackey functor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-10T14:26:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C22",
      "55P43",
      "20E18",
      "55N22",
      "55T25"
    ],
    "keywords": [
      "local homotopy theory",
      "local picard groups",
      "generalized moore spectra",
      "natural inverse limit topology",
      "local profinite galois extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}