Stacky Abelianization of an Algebraic Group
Masoud Kamgarpour

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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology
