A^1-invariance of non-stable K_1-functors in the equicharacteristic case

TL;DR

This paper proves A^1-invariance and injectivity properties of non-stable K_1-functors for isotropic reductive groups over regular domains containing a field, extending techniques used in the proof of the Serre-Grothendieck conjecture.

Contribution

It establishes A^1-invariance and injectivity theorems for non-stable K_1-functors of isotropic reductive groups in the equicharacteristic case, using techniques from Panin's work.

Findings

K_1^G(R[x])=K_1^G(R) for regular domains with isotropic rank >=2

Injectivity of K_1^G(R) to K_1^G(K) for local rings

Extension of techniques from the Serre-Grothendieck conjecture proof

Abstract

We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre-Grothendieck conjecture for isotropic reductive groups (I. Panin, A. Stavrova, N. Vavilov, 2015; I. Panin, 2019) to obtain similar injectivity and A^1-invariance theorems for non-stable K_1-functors associated to isotropic reductive groups. Namely, let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We show that if G has isotropic rank >=2 and R is a regular domain containing a field, then K_1^G(R[x])=K_1^G(R) for any n>=1, where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. If R is, moreover, local, then we show that K_1^G(R)->K_1^G(K) is injective, where K is the field of fractions of R.