Centrality of $\mathrm K_2$ for Chevalley groups: a pro-group approach
Andrei Lavrenov, Sergey Sinchuk, Egor Voronetsky
TL;DR
This work delivers a uniform proof of the centrality of the Chevalley $K_2$-functor $K_2(\Phi, R)$ for irreducible root systems with rank at least $3$, by employing a pro-group localization framework and elementary root-elimination techniques. Central to the approach is the Steinberg pro-group $\mathrm{St}^{(\infty)}(\Phi, R)$ and its action by localization on Chevalley pro-groups, enabling a construction of a crossed module on $\mathrm{St}(\Phi, R) \to \mathrm{G}_{\mathrm{sc}}(\Phi, R)$, which in turn demonstrates the centrality of $K_2(\Phi, R)$. The paper also extends these ideas to exceptional types, notably $\mathsf{F}_4$, and aligns Steinberg and Quillen unstable $K_2$ via exact sequences, with implications for normality and homology computations. Overall, the pro-group method provides a streamlined, localization-based pathway to centrality across a broad class of root systems, including for some that resisted earlier approaches. The results yield a robust connection between Steinberg presentations and central extensions within a unified framework that is applicable to a wide range of Chevalley groups.
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 $\mathrm K_2(Φ,\, R)$ for any root system $Φ$ 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 $\mathrm{St}(Φ, R) \to \mathrm{G}_\mathrm{sc}(Φ, R)$, which has not been known previouly for exceptional $Φ$.