Torsors of isotropic reductive groups over Laurent polynomials
Documenta mathematica, Tome 26 (2021), pp. 661-673
Cet article a éte moissonné depuis la source EMS Press
Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x1±1,...,xn±1]. 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 Heˊt1(R,G)→Heˊt1(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 HZar1(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 K1G(R)→K1G(k((x1))...((xn))) is injective, and an isomorphism if G is moreover semisimple.
Classification :
11E72, 14F20, 17B67, 19B28, 20G35
Mots-clés : isotropic reductive group, loop reductive group, Laurent polynomials, G-torsor, non-stable K1-functor, Whitehead group
Mots-clés : isotropic reductive group, loop reductive group, Laurent polynomials, G-torsor, non-stable K1-functor, Whitehead group
@article{10_4171_dm_825,
author = {Anastasia Stavrova},
title = {Torsors of isotropic reductive groups over {Laurent} polynomials},
journal = {Documenta mathematica},
pages = {661--673},
year = {2021},
volume = {26},
doi = {10.4171/dm/825},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/825/}
}
Anastasia Stavrova. Torsors of isotropic reductive groups over Laurent polynomials. Documenta mathematica, Tome 26 (2021), pp. 661-673. doi: 10.4171/dm/825
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