Covering groups and their integral models

TL;DR

This paper extends Brylinski and Deligne's classification of central extensions of reductive groups by K2 from fields to more general base schemes, including DVRs and Dedekind domains, with some results conditional on Gersten's conjecture.

Contribution

It develops the theory of integral models of central extensions of reductive groups, generalizing previous classifications from fields to broader base schemes.

Findings

Classified integral models over DVRs with finite residue fields.

Obtained results for Dedekind domains, often conditional on Gersten's conjecture.

Extended the invariants used for classification to more general schemes.

Abstract

Given a reductive group $\boldsymbol{\mathrm{G}}$ over a base scheme $S$, Brylinski and Deligne studied the central extensions of a reductive group $\boldsymbol{\mathrm{G}}$ by $\boldsymbol{\mathrm{K}}_2$, viewing both as sheaves of groups on the big Zariski site over $S$. Their work classified these extensions by three invariants, for $S$ the spectrum of a field. We expand upon their work to study "integral models" of such central extensions, obtaining similar results for $S$ the spectrum of a sufficiently nice ring, e.g., a DVR with finite residue field or a DVR containing a field. Milder results are obtained for $S$ the spectrum of a Dedekind domain, often conditional on Gersten's conjecture.