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 $R=k[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$. We prove that $G$ is isotropic over $R$ if and only if it is isotropic over the field of fractions $k(x_1,\dots ,x_n)$ of $R$, and if this is the case, then the natural map $H_{\acute{e}t}^1(R,G)\to H_{\acute{e}t}^1(k(x_1,\dots ,x_n),G)$ 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 $H_{Zar}^1(R,G)=\ast$ 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 $K_1^G(R)\to K_1^G(k((x_1))\dots ((x_n)))$ is injective, and an isomorphism if $G$ is moreover semisimple.