On the $\mathbb{A}^1$-invariance of $\mathrm{K}_2$ of Chevalley groups of simply-laced type

Abstract

In this paper we study the $\mathbb{A}^1$-invariance of the unstable functor $\mathrm{K}_2(\Phi, R)$ in the case when $\Phi$ is an irreducible root system of type $\mathsf{ADE}$ containing $\mathsf{A}_4$ and not of type $\mathsf{E}_8$. We show that in the geometric case, i. e. when $R$ is a regular ring containing a field $k$ one has $\mathrm{K}_2(\Phi, R[t]) = \mathrm{K}_2(\Phi, R)$, which allows one to interpret the unstable $\mathrm{K}_2$ groups as $\mathbb{A}^1$-fundamental groups of Chevalley--Demazure group schemes in the $\mathbb{A}^1$-homotopy category. We also prove a variant of "early stability" theorem which allows one to find a generating set of $\mathrm{K}_2(\Phi, A[X_1, \ldots X_n])$ in the case when $A$ is a Dedekind domain.