Torsors of isotropic reductive groups over Laurent polynomials

Stavrova, Anastasia

Documenta mathematica, Tome 26 (2021), pp. 661-673.

Résumé

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R=k[x_1^{\pm 1},\ldots,x_n^{\pm 1}]$. We prove that $G$ is isotropic over $R$ if and only if it is isotropic over the field of fractions $k(x_1,\ldots,x_n)$ of $R$, and if this is the case, then the natural map $H^1_{\acute{e}t}(R,G)\to H^1_{\acute{e}t}(k(x_1,\ldots,x_n),G)$ 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 $H^1_{Zar}(R,G)=\ast$ for such groups $G$. We also deduce that if $G$ is a reductive group over $R$ of isotropic rank $\ge 2$, then the natural map of non-stable $K_1$-functors $K_1^G(R)\to K_1^G\bigl( k((x_1))\ldots((x_n)) \bigr)$ is injective, and an isomorphism if $G$ is moreover semisimple.

Détail

Classification : 14F20, 20G35, 17B67, 19B28, 11E72
Keywords: isotropic reductive group, loop reductive group, Laurent polynomials, \(G\)-torsor, non-stable \(K_1\)-functor, Whitehead group