A1 -connectedness in reductive algebraic groups

Abstract

Using sheaves of $ \mathbb{A}^1$-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be $ \mathbb{A}^1$-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of $ \mathbb{A}^1$-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize $ \mathbb{A}^1$-connected reductive algebraic groups over a field of characteristic 0.