Chevalley groups over Laurent polynomial rings

A. Stavrova

Zapiski Nauchnykh Seminarov POMI, Algebra and number theory. Part 7, Tome 538 (2024), pp. 152-159

Résumé

Let $G$ be a simply connected Chevalley–Demazure group scheme without $\mathrm{SL}_2$-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. Set $K_1^G(R)=G(R)/E(R)$. We prove that the natural map $$ K_1^G(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])\to K_1^G\bigl(R((x_1))\ldots((x_n))\bigr) $$ 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 $SL_2(D)=E_2(D)$ (e. g. $D=\mathbb{Z}$), then $$ G(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])=E(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}]). $$ This extends earlier results for special linear and symplectic groups due to A. A. Suslin and V. I. Kopeiko.