Transfers on $\mathbb A^1$-connected components of quasi-split groups and the norm principle

TL;DR

This paper proves that the sheaf of a1^1-connected components of quasi-split groups over perfect fields is strictly a1^1-invariant with transfers, leading to the validation of the norm principle for such groups.

Contribution

It establishes the strict a1^1-invariance with transfers of the sheaf of a1^1-connected components for quasi-split groups over perfect fields, confirming the norm principle.

Findings

Sheaf of a1^1-connected components is strictly a1^1-invariant with transfers.

Norm principle holds for all quasi-split groups over perfect fields.

Provides new tools for understanding algebraic groups in a1^1-homotopy theory.

Abstract

We show that the sheaf of $\mathbb A^1$-connected components of a quasi-split group over a perfect field is a strictly $\mathbb A^1$-invariant sheaf with (Voevodsky) transfers. As a consequence, we show that the norm principle holds for any quasi-split group over a perfect field.