Affine representability results in 𝔸1–homotopy theory, II : Principal bundles and homogeneous spaces

Asok, Aravind¹ ; Hoyois, Marc² ; Wendt, Matthias³

¹ Department of Mathematics, University of Southern California, Los Angeles, CA, United States
² Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, United States
³ Fakultät Für Mathematik, Universität Duisburg-Essen, Essen, Germany

Geometry & topology, Tome 22 (2018) no. 2, pp. 1181-1225.

Résumé

We establish a relative version of the abstract “affine representability” theorem in ${\mathbb{A}}^1$–homotopy theory from part I of this paper. We then prove some ${\mathbb{A}}^1$–invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass–Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in ${\mathbb{A}}^1$–homotopy theory.

Détail

DOI : 10.2140/gt.2018.22.1181

Classification : 14F42, 14L10, 20G15, 55R15
Keywords: motivic homotopy theory, principal bundles

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