We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre–Grothendieck conjecture for isotropic reductive groups (Panin et al., 2015; Panin, 2019) to obtain similar injectivity and A1-invariance theorems for non-stable K1-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 non-trivial normal semisimple R-subgroup of G contains (Gm,R)n. We show that if G has isotropic rank ≥2 and R is a regular domain containing a field, then K1G(R[x])=K1G(R), where K1G(R)=G(R)/E(R) is the corresponding non-stable K1-functor, also called the Whitehead group of G. If R is, moreover, local, then we show that K1G(R)→K1G(K) is injective, where K is the field of fractions of R.
Scopus subject areas
- Isotropic reductive group
- Non-stable K-functor
- Serre–Grothendieck conjecture
- Whitehead group