{
  "id": "1206.0247",
  "version": "v2",
  "published": "2012-06-01T16:57:28.000Z",
  "updated": "2013-10-05T19:02:23.000Z",
  "title": "On the algebraic K-theory of truncated polynomial algebras in several variables",
  "authors": [
    "Vigleik Angeltveit",
    "Teena Gerhardt",
    "Michael A. Hill",
    "Ayelet Lindenstrauss"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "abstract": "We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on N^n. If the characteristic of k does not divide any of the a_i we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k=Z. To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write TC(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)) as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand. Updated: This is a substantial revision. We corrected several errors in the description of the Witt vectors on a truncation set on N^n and modified the key proofs accordingly. We also replaces several topological statement with purely algebraic ones. Most arguments have been reworked and streamlined.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-05T19:02:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "19D55",
      "55Q91"
    ],
    "keywords": [
      "truncated polynomial algebra",
      "algebraic k-theory",
      "big witt vectors",
      "cyclotomic trace map",
      "k-groups modulo torsion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.0247A"
    }
  }
}