Centrality of ${\mathrm{K}}_2$ for Chevalley groups: a pro-group approach

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 $\ge 3$. 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 $\Phi$.

References in corpus (1)

• Centrality of ${\mathrm{K}}_2$-functor revisited

Cited by in corpus (1)

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