Geodesic
$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors
Documenta mathematica, Tome 27 (2022), pp. 2657-2689 Cet article a éte moissonné depuis la source EMS Press

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In this paper, we study the Nisnevich sheafification Heˊt1​(G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G-torsors, for a reductive group G. We show that if G-torsors on affine lines are extended, then Heˊt1​(G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G-torsors. We also identify the sheaf Heˊt1​(G) with the sheaf of A1-connected components of the classifying space Beˊt​G. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A1-connected components in terms of unramified G-torsors over function fields whenever Nisnevich-local purity holds for G-torsors.
DOI : 10.4171/dm/x38
Classification : 14F42, 14L15, 19E15
Mots-clés : classifying spaces, torsors, motivic homotopy theory 
@article{10_4171_dm_x38,
     author = {Girish Kulkarni and Matthias Wendt and Elden Elmanto},
     title = {$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors},
     journal = {Documenta mathematica},
     pages = {2657--2689},
     year = {2022},
     volume = {27},
     doi = {10.4171/dm/x38},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/x38/}
}
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Girish Kulkarni; Matthias Wendt; Elden Elmanto. $\mathbb{A}^1$-connected components of classifying spaces and purity for torsors. Documenta mathematica, Tome 27 (2022), pp. 2657-2689. doi: 10.4171/dm/x38

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