On centrality of $K_2$ for Chevalley groups of type $E_l$

TL;DR

This paper proves that for Chevalley groups of type E_l over any ring, the K_2 group lies in the center of the Steinberg group, and establishes a local-global principle for K_2.

Contribution

It demonstrates the centrality of K_2 in Steinberg groups of type E_l and introduces a local-global principle for K_2 in this context.

Findings

K_2( ext{Phi}, R) is contained in the center of St( ext{Phi}, R)

Established a Quillen--Suslin type local-global principle for K_2( ext{Phi}, R)

Extended known results to Chevalley groups of type E_l over arbitrary rings

Abstract

For a root system $\Phi$ of type $E_l$ and arbitrary commutative ring $R$ we show that the group $K_2(\Phi, R)$ is contained in the centre of the Steinberg group $St(\Phi, R)$. In course of the proof we also demonstrate an analogue of Quillen---Suslin local-global principle for $K_2(\Phi, R)$.