On Brauer groups of tame stacks

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