On Brauer groups of tame stacks

TL;DR

This paper introduces new tools for computing the Brauer group of tame algebraic stacks by relating it to the Brauer group of their coarse spaces and fiber Picard groups, demonstrated through examples.

Contribution

It develops general methods to compute Brauer groups of tame stacks, simplifying calculations by connecting them to coarse spaces and fiber Picard groups.

Findings

Computed Brauer group of the moduli stack of elliptic curves over certain schemes.

Determined Brauer groups of various stacky curves with generic stabilizers.

Provided tools to relate stack Brauer groups to coarse space Brauer groups.

Abstract

We develop some general tools for computing the Brauer group of a tame algebraic stack $\mathscr X$ by studying the difference between it and the Brauer group of the coarse space $X$ of $\mathscr X$. It is our hope that these tools will be used to simplify future computations of Brauer groups of stacks. Informally, we show that $\operatorname{Br}\mathscr X$ is often built from $\operatorname{Br} X$ and "information about the Picard groups of the fibers of $\mathscr X\to X$''. Along these lines, we compute, for example, the Brauer group of the moduli stack $\mathscr Y(1)_S$ of elliptic curves, over any regular noetherian $\mathbb Z[1/2]$-scheme $S$ as well as the Brauer groups of (many) stacky curves (allowing generic stabilizers) over algebraically closed fields.