{
  "id": "1710.06409",
  "version": "v1",
  "published": "2017-10-17T17:38:49.000Z",
  "updated": "2017-10-17T17:38:49.000Z",
  "title": "The geometry of the cyclotomic trace",
  "authors": [
    "David Ayala",
    "Aaron Mazel-Gee",
    "Nick Rozenblyum"
  ],
  "categories": [
    "math.AT",
    "math.AG",
    "math.KT"
  ],
  "abstract": "We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X) \\to TC(X)$ from algebraic K-theory to topological cyclic homology for any scheme $X$. This construction rests on a new identification of the cyclotomic structure on $THH(C)$, which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization of more primitive data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-17T17:38:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological cyclic homology",
      "precise algebro-geometric interpretation",
      "cyclotomic trace map",
      "cyclotomic trace likewise arises",
      "algebraic k-theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}