{
  "id": "1707.01799",
  "version": "v1",
  "published": "2017-07-06T14:04:04.000Z",
  "updated": "2017-07-06T14:04:04.000Z",
  "title": "On topological cyclic homology",
  "authors": [
    "Thomas Nikolaus",
    "Peter Scholze"
  ],
  "comment": "165 pages, 3 appendices",
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "abstract": "Topological cyclic homology is a refinement of Connes' cyclic homology which was introduced by B\\\"okstedt--Hsiang--Madsen in 1993 as an approximation to algebraic $K$-theory. There is a trace map from algebraic $K$-theory to topological cyclic homology, and a theorem of Dundas--Goodwillie--McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing $K$-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum. The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the $\\infty$-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum $X$ with $S^1$-action (in the most naive sense) together with $S^1$-equivariant maps $\\varphi_p: X\\to X^{tC_p}$ for all primes $p$. Here $X^{tC_p}=\\mathrm{cofib}(\\mathrm{Nm}: X_{hC_p}\\to X^{hC_p})$ is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-06T14:04:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19D55",
      "16E40",
      "13D03",
      "55P42",
      "55P43",
      "55P91",
      "55P92"
    ],
    "keywords": [
      "topological cyclic homology",
      "cyclotomic spectrum",
      "genuine equivariant homotopy theory",
      "explicit point-set models",
      "construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 165,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}