{
  "id": "1503.00317",
  "version": "v1",
  "published": "2015-03-01T17:36:13.000Z",
  "updated": "2015-03-01T17:36:13.000Z",
  "title": "Analogy between the cyclotomic trace map $K \\rightarrow TC$ and the Grothendieck trace formula via noncommutative geometry",
  "authors": [
    "Ilias Amrani"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.AT",
    "math.AG",
    "math.KT",
    "math.NT"
  ],
  "abstract": "In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \\infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of noncommuative schemes over $\\mathbf{F}_{p}$. Our analogy suggests the existence of a categorification of the $l$-adic cohomology trace formula in the noncommutative setting for $l\\neq p$. Finally, we write down the corresponding dictionary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-01T17:36:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19E08",
      "14F30",
      "14A22",
      "55N15",
      "11M38"
    ],
    "keywords": [
      "cyclotomic trace map",
      "grothendieck trace formula",
      "crystalline grothendieck-lefschetz trace formula",
      "noncommutative geometry",
      "adic cohomology trace formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}