Chevalley groups over Laurent polynomial rings

Abstract

Let $G$ be a simply connected Chevalley--Demazure group scheme without $S{L}_2$-factors. For any unital commutative ring $R$, we denote by $E(R)$ the standard elementary subgroup of $G(R)$, that is, the subgroup generated by the elementary root unipotent elements. We prove that the map $$G(R[x_1^{\pm 1},\dots ,x_n^{\pm 1}])/E(R[x_1^{\pm 1},\dots ,x_n^{\pm 1}])\to G(R((x_1))\dots ((x_n)))/E(R((x_1))\dots ((x_n)))$$ is injective for any $n\ge 1$, if $R$ is either a Dedekind domain or a Noetherian ring that is geometrically regular over a Dedekind domain with perfect residue fields. For $n=1$ this map is also an isomorphism. As a consequence, we show that if $D$ is a PID such that $S{L}_2(D)=E_2(D)$ (e.g. $D=\mathbb{Z}$), then $G(D[x_1^{\pm 1},\dots ,x_n^{\pm 1}])=E(D[x_1^{\pm 1},\dots ,x_n^{\pm 1}])$. This extends earlier results for special linear and symplectic groups due to A. A. Suslin and V. I. Kopeiko.