{
  "id": "2309.11463",
  "version": "v1",
  "published": "2023-09-20T17:00:13.000Z",
  "updated": "2023-09-20T17:00:13.000Z",
  "title": "Algebraic K-theory of real topological K-theory",
  "authors": [
    "Gabriel Angelini-Knoll",
    "Christian Ausoni",
    "John Rognes"
  ],
  "comment": "48 pages. Comments welcome",
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "abstract": "We calculate the A(1)-homotopy of the topological cyclic homology of the connective real topological K-theory spectrum ko, and show that it is a finitely generated and free F_2[v_2^{32}]-module of even rank between 390 and 444, on explicit generators in stems -1 \\le *\\le 198. This is achieved by using syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical rings to E_\\infty rings. In our case there are nontrivial differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology, unlike in the case of complex K-theory at odd primes that was studied by Hahn-Raksit-Wilson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-20T17:00:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19D50",
      "19D55",
      "55Q51",
      "55P43",
      "14F30",
      "19E20",
      "13D03",
      "55N15",
      "55Q10",
      "55T25"
    ],
    "keywords": [
      "algebraic k-theory",
      "topological cyclic homology",
      "real topological k-theory spectrum ko",
      "syntomic cohomology",
      "connective real topological k-theory spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}