Torsors of Isotropic Reductive Groups over Laurent Polynomials

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Torsors of Isotropic Reductive Groups over Laurent Polynomials. / Stavrova, Anastasia.

In: Documenta Mathematica, Vol. 26, 2021, p. 661-673.

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

BibTeX

@article{fbdfeb30fb384de1ae1a0e6731e1b386,

title = "Torsors of Isotropic Reductive Groups over Laurent Polynomials",

abstract = "Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R = k [(Formula presented)]. We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x1,…, xn) of R, and if this is the case, then the natural map (Formula presented)(R, G) (Formula presented)(k(x1,..., xn), G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that HZar (R, G) = * for such groups G. We also deduce that if G is a reductive group over R of isotropic rank > 2, then the natural map of non-stable K1-functors (Formula presented) ((x1))...((xn))) is injective, and an isomorphism if G is moreover semisimple.",

keywords = "G-torsor, Isotropic reductive group, Laurent polynomials, loop reductive group, non-stable K1-functor, Whitehead group",

author = "Anastasia Stavrova",

year = "2021",

doi = "10.25537/dm.2021v26.661-673",

language = "English",

volume = "26",

pages = "661--673",

journal = "Documenta Mathematica",

issn = "1431-0635",

publisher = "Deutsche Mathematiker Vereinigung",

}

RIS

TY - JOUR

T1 - Torsors of Isotropic Reductive Groups over Laurent Polynomials

AU - Stavrova, Anastasia

PY - 2021

Y1 - 2021

N2 - Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R = k [(Formula presented)]. We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x1,…, xn) of R, and if this is the case, then the natural map (Formula presented)(R, G) (Formula presented)(k(x1,..., xn), G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that HZar (R, G) = * for such groups G. We also deduce that if G is a reductive group over R of isotropic rank > 2, then the natural map of non-stable K1-functors (Formula presented) ((x1))...((xn))) is injective, and an isomorphism if G is moreover semisimple.

AB - Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R = k [(Formula presented)]. We prove that G is isotropic over R if and only if it is isotropic over the field of fractions k(x1,…, xn) of R, and if this is the case, then the natural map (Formula presented)(R, G) (Formula presented)(k(x1,..., xn), G) has trivial kernel and G is loop reductive. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that HZar (R, G) = * for such groups G. We also deduce that if G is a reductive group over R of isotropic rank > 2, then the natural map of non-stable K1-functors (Formula presented) ((x1))...((xn))) is injective, and an isomorphism if G is moreover semisimple.

KW - G-torsor

KW - Isotropic reductive group

KW - Laurent polynomials

KW - loop reductive group

KW - non-stable K1-functor

KW - Whitehead group

UR - http://www.scopus.com/inward/record.url?scp=85114791869&partnerID=8YFLogxK

U2 - 10.25537/dm.2021v26.661-673

DO - 10.25537/dm.2021v26.661-673

M3 - Article

AN - SCOPUS:85114791869

VL - 26

SP - 661

EP - 673

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

ER -

ID: 86101135