Morita theory of twisted sheaves on ${\mu }_n$-gerbes of line bundles

We study Morita theory of twisted sheaves on ${\mu }_n$-gerbes of line bundles $\mathcal{X}$. In this context, we find explicit equivalent conditions for when two Azumaya algebras on $\mathcal{X}$ are Morita equivalent. Additionally, we provide an example showing that Căldăraru’s Conjecture does not hold for Deligne–Mumford stacks in general.