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

Chetan Balwe; Amit Hogadi; Anand Sawant

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arXiv · 2205.07527

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

Chetan Balwe·Amit Hogadi·Anand Sawant

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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.

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BibTeX RIS

Chetan Balwe, Amit Hogadi, Anand Sawant. 2022-05-16. Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups. https://arxiv.org/abs/2205.07527

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