Strong A¹‐invariance of A¹‐connected components of reductive algebraic groups

Abstract

We show that the sheaf of A¹-connected components of a reductive algebraic group over a perfect field is strongly A¹-invariant. As a consequence, torsors under such groups give rise to A ¹-fiber sequences. We also show that sections of A ¹-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.