On Brauer groups of tame stacks

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.