Chevalley groups over Laurent polynomial rings

TL;DR

This paper proves injectivity of a natural map between Chevalley group quotients over Laurent polynomial rings and their iterated Laurent series fields, extending known results for special linear groups to more general Chevalley groups.

Contribution

It establishes the injectivity (and in the case of one variable, isomorphism) of a key map for Chevalley groups over Laurent polynomial rings, generalizing prior results for specific groups.

Findings

Injectivity of the map for n ≥ 1 over certain rings.

Isomorphism for n=1 over all rings considered.

Extension of results from special linear groups to general Chevalley groups.

Abstract

Let $G$ be a simply connected Chevalley--Demazure group scheme without $SL_2$-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. We prove that the map $$ G(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])/E(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])\to G\bigl(R((x_1))\ldots((x_n))\bigr)/E\bigl(R((x_1))\ldots((x_n))\bigr) $$ 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.