Locally isotropic Steinberg groups I. Centrality of the ${\mathrm{K}}_2$-functor

Abstract

We begin to study Steinberg groups associated with a locally isotropic reductive group $G$ over a arbitrary ring. We propose a construction of such a Steinberg group functor as a group object in a certain completion of the category of presheaves. We also show that it is a crossed module over $G$ in a unique way, in particular, that the ${\mathrm{K}}_2$-functor is central. If $G$ is globally isotropic in a suitable sense, then the Steinberg group functor exists as an ordinary group-valued functor and all such abstract Steinberg groups are crossed modules over the groups of points of $G$.

Minor fixes, especially new lemma 6(5) needed to fix a gap in the proof of proposition 1