Stacky Abelianization of an Algebraic Group

TL;DR

This paper proves a conjecture about the existence of a special etale cover of the commutator subgroup of a connected algebraic group, which simplifies the structure of central extensions and relates to universal Picard stacks.

Contribution

It establishes the existence of a connected etale group cover of the commutator subgroup that characterizes splitting of central extensions and lifts the commutator map, confirming Drinfeld's conjecture.

Findings

Existence of a connected etale cover H of [G,G]

H characterizes splitting of central extensions of G

The quotient stack G/H is the universal Deligne-Mumford Picard stack

Abstract

Let G be a connected algebraic group and let [G,G] be its commutator subgroup. We prove a conjecture of Drinfeld about the existence of a connected etale group cover H of [G,G], characterized by the following properties: every central extension of G, by a finite etale group scheme, splits over H, and the commutator map of G lifts to H. We prove, moreover, that the quotient stack of G by the natural action of H is the universal Deligne-Mumford Picard stack to which G maps.