Locally isotropic Steinberg groups I. Centrality of the $\mathrm K_2$-functor

TL;DR

This paper introduces a new construction of Steinberg groups for locally isotropic reductive groups over rings, proving the centrality of the K_2-functor and establishing their structure as crossed modules.

Contribution

It proposes a novel functorial construction of Steinberg groups as group objects in a completion of presheaves category, demonstrating the centrality of K_2 and their crossed module structure.

Findings

The Steinberg group functor is a crossed module over G with central K_2.

Construction works as a group object in a certain presheaf completion.

When G is globally isotropic, the functor is an ordinary group-valued 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 $G$.