UC Berkeley

The theme of this dissertation is the Brauer group of algebraic stacks. Antieau and Meier showed that if $k$ is an algebraically closed field of $\on{char} k \ne 2$, then $\Br(\ms{M}_{1,1,k}) = 0$, where $\ms{M}_{1,1}$ is the moduli stack of elliptic curves. We show that if $\on{char} k = 2$ then $\Br(\ms{M}_{1,1,k}) = \Z/(2)$. In another direction, we compute the cohomological Brauer group of $\G_{m}$-gerbes; this is an analogue of a result of Gabber which computes the cohomological Brauer group of Brauer-Severi schemes. We also discuss two kinds of algebraic stacks $X$ for which not all torsion classes in $\H_{\et}^{2}(X,\G_{m})$ are represented by Azumaya algebras on $X$ (i.e. $\Br \ne \Br'$).