-connected components of classifying spaces and purity for torsors

  • Elden Elmanto

    Department of Mathematics, Harvard University, 1 Oxford St., Cambridge MA 02138, USA
  • Girish Kulkarni

    Mathematical Institute, Heinrich Heine Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
  • Matthias Wendt

    Fachgruppe Mathematik und Informatik, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany
$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors cover
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Abstract

In this paper, we study the Nisnevich sheafification of the presheaf associating to a smooth scheme the set of isomorphism classes of -torsors, for a reductive group . We show that if -torsors on affine lines are extended, then is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for -torsors. We also identify the sheaf with the sheaf of -connected components of the classifying space . This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of -connected components in terms of unramified -torsors over function fields whenever Nisnevich-local purity holds for -torsors.

Cite this article

Elden Elmanto, Girish Kulkarni, Matthias Wendt, -connected components of classifying spaces and purity for torsors. Doc. Math. 27 (2022), pp. 2657–2689

DOI 10.4171/DM/X38