## 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 (Panin et al., 2015; 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 non-trivial normal semisimple R-subgroup of G contains (G_{m,R})^{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), 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.

Original language | English |
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Journal | Indagationes Mathematicae |

DOIs | |

State | E-pub ahead of print - 20 Aug 2021 |

## Scopus subject areas

- Mathematics(all)

## Keywords

- Isotropic reductive group
- Non-stable K-functor
- Serre–Grothendieck conjecture
- Whitehead group