Brauer groups of tame stacky curves and $μ_r$-gerbes over them

Abstract: We fit the Brauer group of a $\mu_r$-gerbe over a (possibly arbitrarily singular) stacky curve into an exact sequence and give characterizations for when it is short exact and a condition for when it splits. We also give a precise formula for the Brauer group of a $\mu_r$-gerbe in the case that the base curve is smooth. We end by showing that $\operatorname{Br}=\operatorname{Br&#39;}=\operatorname{H}<sup>2$ for separated tame one-dimensional algebraic stacks of finite type over a field.