Strong \mathbb A¹-invariance of \mathbb A¹-connected components of reductive algebraic groups

2022/05/16 by Chetan Balwe, Balwe, Chetan, Amit Hogadi +3
Mathematics · #14F42 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2205.07527

openalex publication_date 2022/05/16 · openalex created_date 2022/05/22 · openalex updated_date 2026/07/28

Abstract

We show that the sheaf of \mathbb A¹-connected components of a reductive algebraic group over a perfect field is strongly \mathbb A¹-invariant. As a consequence, torsors under such groups give rise to \mathbb A¹-fiber sequences. We also show that sections of \mathbb A¹-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their R-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

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