{
  "id": "2003.06892",
  "version": "v1",
  "published": "2020-03-15T18:52:47.000Z",
  "updated": "2020-03-15T18:52:47.000Z",
  "title": "A motivic Segal theorem for open pairs of smooth schemes over an infinite perfect field",
  "authors": [
    "Aleksei Tsybyshev"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "V. Voevodsyky laid the groundwork of delooping motivic spaces in order to provide a new, more computation-friendly, construction of the stable motivic category $SH(k)$, G. Garkusha and I. Panin made that project a reality, while collaborating with A. Ananievsky, A. Neshitov and A. Druzhinin. In particular, G. Garkusha and I. Panin proved that for an infinite perfect field $k$ and any $k$-smooth scheme $X$ the canonical morphism of motivic spaces $C_*Fr(X)\\to \\Omega^{\\infty}_{\\mathbb{P}^1} \\Sigma^{\\infty}_{\\mathbb{P}^1} (X_+)$ is Nisnevich-locally a group-completion. In the present work, a generalisation of that theorem to the case of smooth open pairs $(X,U),$ where $X$ is a $k$-smooth scheme, $U$ is its open subscheme intersecting each component of $X$ in a nonempty subscheme. We claim that in this case the motivic space $C_*Fr((X,U))$ is Nisnevich-locally connected, and the motivic space morphism $C_*Fr((X,U))\\to \\Omega^{\\infty}_{\\mathbb{P}^1} \\Sigma^{\\infty}_{\\mathbb{P}^1} (X/U)$ is Nisnevich-locally a weak equivalence. Moreover, we show that if the codimension of $S=X-U$ in each component of $X$ is greater than $r \\geq 0,$ the simplicial sheaf $C_*Fr((X,U))$ is locally $r$-connected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-15T18:52:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F42"
    ],
    "keywords": [
      "infinite perfect field",
      "smooth scheme",
      "motivic segal theorem",
      "smooth open pairs",
      "motivic space morphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}