The Brauer-Manin obstruction on algebraic stacks

TL;DR

This paper extends the theory of Brauer-Manin obstructions to algebraic stacks over number fields, establishing foundational concepts, properties, and equivalences with other obstructions.

Contribution

It generalizes classical Brauer-Manin obstruction theory from varieties to algebraic stacks, including definitions, properties, and relations with other cohomological obstructions.

Findings

Brauer groups of certain stacks are torsion.

Brauer-Manin obstruction coincides with other cohomological obstructions under mild conditions.

Properties like descent and product preservation hold for these stacks.

Abstract

For algebraic stacks over number fields, we define their Brauer-Manin sets, Brauer-Manin pairings, and extend the descent theory of Colliot-Th\'el\`ene and Sansuc. By extending Sansuc's exact sequence, we show the torsionness of Brauer groups of stacks that are locally quotients of varieties by linear groups. With mild assumptions, for stacks that are locally quotients or Deligne-Mumford, we show that the Brauer-Manin obstruction coincides with some other cohomological obstructions such as obstructions given by torsors under connected groups or abelian gerbes. For Brauer-Manin sets of these stacks, we show the properties such as descent along a torsor, product preservation are still correct. These results extend classical theories of those on varieties.