TY - JOUR

T1 - A1-invariance of non-stable K1-functors in the equicharacteristic case

AU - Stavrova, Anastasia

N1 - Publisher Copyright: © 2021 Royal Dutch Mathematical Society (KWG)

PY - 2021/8/20

Y1 - 2021/8/20

N2 - 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.

AB - 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.

KW - Isotropic reductive group

KW - Non-stable K-functor

KW - Serre–Grothendieck conjecture

KW - Whitehead group

KW - Non-stable K1-functor

UR - http://www.scopus.com/inward/record.url?scp=85114942401&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/93ad79c2-0650-37d5-9d64-43424f381c16/

U2 - 10.1016/j.indag.2021.08.002

DO - 10.1016/j.indag.2021.08.002

M3 - Article

AN - SCOPUS:85114942401

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

ER -