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

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A¹-invariance of non-stable K₁-functors in the equicharacteristic case. / Stavrova, Anastasia.

In: Indagationes Mathematicae, 20.08.2021.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{ea95b0abe5d94d2f9851df6a31b22d14,

title = "A1-invariance of non-stable K1-functors in the equicharacteristic case",

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.",

keywords = "Isotropic reductive group, Non-stable K-functor, Serre–Grothendieck conjecture, Whitehead group, Non-stable K1-functor",

author = "Anastasia Stavrova",

note = "Publisher Copyright: {\textcopyright} 2021 Royal Dutch Mathematical Society (KWG)",

year = "2021",

month = aug,

day = "20",

doi = "10.1016/j.indag.2021.08.002",

language = "English",

journal = "Indagationes Mathematicae",

issn = "0019-3577",

publisher = "Elsevier",

}

RIS

TY - JOUR

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

AU - Stavrova, Anastasia

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 -

ID: 86101087