-connected components of classifying spaces and purity for torsors
Elden Elmanto
Department of Mathematics, Harvard University, 1 Oxford St., Cambridge MA 02138, USAGirish Kulkarni
Mathematical Institute, Heinrich Heine Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, GermanyMatthias Wendt
Fachgruppe Mathematik und Informatik, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany
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