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

TL;DR

This paper investigates the $A^1$-invariance of the unstable $K_2$ functor for certain root systems, establishing invariance over regular rings and connecting it to $A^1$-fundamental groups in algebraic geometry.

Contribution

It proves $A^1$-invariance of $K_2$ for root systems of type ADE containing $A_4$, and relates these groups to $A^1$-fundamental groups in the $A^1$-homotopy category.

Findings

$K_2( ext{Phi}, R[t]) = K_2( ext{Phi}, R)$ for regular rings containing a field.

Interpretation of unstable $K_2$ groups as $A^1$-fundamental groups.

A variant of early stability theorem for Dedekind domains.

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.