Volume 22, issue 2 (2018)

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Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces

Aravind Asok, Marc Hoyois and Matthias Wendt

Geometry & Topology 22 (2018) 1181–1225
Abstract

We establish a relative version of the abstract “affine representability” theorem in ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–homotopy theory from part I of this paper. We then prove some ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–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 ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–homotopy theory.

Keywords
motivic homotopy theory, principal bundles
Mathematical Subject Classification 2010
Primary: 14F42, 14L10, 20G15, 55R15