Transfers for non-stable K_1-functors of classical type

TL;DR

This paper proves that the sheafification of the non-stable K_1-functor for certain classical algebraic groups is A^1-invariant and admits transfers, leading to birational invariance and a rigidity theorem.

Contribution

It establishes A^1-invariance and transfer properties for the Whitehead K_1-functor of classical groups, extending understanding of their algebraic and geometric behavior.

Findings

Sheafification of K_1^G is A^1-invariant for specified groups.

K_1^G is birationally invariant in characteristic zero.

Rigidity theorem for A^1-invariant torsion presheaves with transfers.

Abstract

Let k be a field. Let G be an absolutely almost simple simply connected k-group of type A_l, l>=2, or D_l, l>=4, containing a 2-dimensional split torus. If G is of type D_l, assume moreover that char k is different from 2. We show that the Nisnevich sheafification of the non-stable K_1-functor K_1^G, also called the Whitehead group of G, on the category of smooth k-schemes is A^1-invariant, and has oriented weak transfers for affine varieties in the sense of Panin-Yagunov-Ross. If k has characteristic 0, this implies that the Nisnevich sheafification of K_1^G is birationally invariant. We also prove a rigidity theorem for \A1-invariant torsion presheaves with oriented weak transfers over infinite fields. As a corollary, we conclude that K_1^G(R)=K_1^G(k) whenever R is a Henselian regular local ring with a coefficient field k.