A1-invariance of non-stable K1-functors in the equicharacteristic case

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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 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.

Original languageEnglish
JournalIndagationes Mathematicae
DOIs
StateE-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

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