Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach

TL;DR

This paper proves the centrality of K_2 for Chevalley groups over any ring using elementary pro-group techniques, completing the case for all root systems of rank at least 3 and introducing a new crossed module construction.

Contribution

It establishes the centrality of K_2 for all root systems of rank ≥ 3, including exceptional types, using elementary localization and pro-group methods, and constructs a new crossed module.

Findings

Proves centrality of K_2 for F_4 over any ring.

Completes the proof for all root systems of rank ≥ 3.

Constructs a new crossed module for exceptional root systems.

Abstract

We prove the centrality of $\mathrm{K}_2 (\mathsf{F}_4, \,R)$ for an arbitrary commutative ring $R$. This completes the proof of the centrality of $\mathrm K_2(\Phi,\, R)$ for any root system $\Phi$ of rank $\geq 3$. Our proof uses only elementary localization techniques reformulated in terms of pro-groups. Another new result of the paper is the construction of a crossed module on the canonical homomorphism $\mathrm{St}(\Phi, R) \to \mathrm{G}_\mathrm{sc}(\Phi, R)$, which has not been known previouly for exceptional $\Phi$.