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

We establish a relative version of the abstract “affine representability” theorem in A1–homotopy theory from part I of this paper. We then prove some A1–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 A1–homotopy theory.

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motivic homotopy theory, principal bundles
Mathematical Subject Classification 2010
Primary: 14F42, 14L10, 20G15, 55R15
Received: 13 July 2016
Revised: 25 April 2017
Accepted: 24 May 2017
Published: 16 January 2018
Proposed: Haynes R Miller
Seconded: Mark Behrens, Stefan Schwede
Aravind Asok
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Marc Hoyois
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Matthias Wendt
Fakultät Für Mathematik
Universität Duisburg-Essen