Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups

TL;DR

This paper proves that the sheaf of $A^1$-connected components of reductive algebraic groups over perfect fields is strongly $A^1$-invariant, leading to new insights into torsors, $A^1$-fiber sequences, and $R$-equivalence classes.

Contribution

It establishes strong $A^1$-invariance for sheaves of $A^1$-connected components of reductive algebraic groups, removing the need for perfectness assumptions in related results.

Findings

Sheaf of $A^1$-connected components is strongly $A^1$-invariant.

Torsors under such groups induce $A^1$-fiber sequences.

Sections of $A^1$-connected components match $R$-equivalence classes for certain groups.

Abstract

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