{
  "id": "1501.02436",
  "version": "v1",
  "published": "2015-01-11T10:25:03.000Z",
  "updated": "2015-01-11T10:25:03.000Z",
  "title": "Quotients of MGL, their slices and their geometric parts",
  "authors": [
    "Marc Levine",
    "Girja Shanker Tripathi"
  ],
  "comment": "30 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $x_1, x_2,\\ldots$ be a system of homogeneous polynomial generators for the Lazard ring $\\mathbb{L}^*=MU^{2*}$ and let $MGL_S$ denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme $S$.Take $S$ essentially smooth over a field $k$. Relying on Hopkins-Morel-Hoyois isomorphism of the 0th slice $s_0MGL_S$ for Voevodsky's slice tower with $MGL_S/(x_1, x_2,\\ldots)$ (after inverting the characteristic of $k$), Spitzweck computes the remaining slices of $MGL_S$ as $s_nMGL_S=\\Sigma^n_TH\\mathbb{Z}\\otimes \\mathbb{L}^{-n}$ (again, after inverting the characteristic of $k$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_S/(\\{x_i:i\\in I\\})$ for $I$ an arbitrary subset of $\\mathbb{N}$, as well as the mod $p$ version $MGL_S/(\\{p, x_i:i\\in I\\})$ and localizations with respect to a system of homogeneous elements in $\\mathbb{Z}[\\{x_j:j\\not\\in I\\}]$. In case $S=\\text{Spec}\\, k$, $k$ a field of characteristic zero, we apply this to show that for $\\mathcal{E}$ a localization of a quotient of $MGL$ as above, there is a natural isomorphism for the theory with support \\[ \\Omega_*(X)\\otimes_{\\mathbb{L}^{-*}}\\mathcal{E}^{-2*,-*}(k)\\to \\mathcal{E}^{2m-2*, m-*}_X(M) \\] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=$dim${}_kM$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-11T10:25:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C25",
      "19E15",
      "19E08",
      "14F42",
      "55P42"
    ],
    "keywords": [
      "geometric parts",
      "denote voevodskys algebraic cobordism spectrum",
      "voevodskys slice tower",
      "characteristic",
      "motivic stable homotopy category"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150102436L"
    }
  }
}