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Affine representability results in $$\mathbb A^1$$-homotopy theory. III: Finite fields and complements. (English) Zbl 07262981
Summary: We give a streamlined proof of $$\mathbb{A}^1$$-representability for G-torsors under “isotropic” reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in $$\mathbb{A}^1$$-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.
##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14L10 Group varieties 55R15 Classification of fiber spaces or bundles in algebraic topology 20G15 Linear algebraic groups over arbitrary fields
##### Keywords:
torsors; algebraic groups; motivic homotopy theory
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