Morita theory of twisted sheaves on $\mu_{n}$-gerbes of line bundles

Abstract

We study Morita theory of twisted sheaves on $\mu_{n}$-gerbes of line bundles $\mathscr{X}$. In this context, we find explicit equivalent conditions for when two Azumaya algebras on $\mathscr{X}$ are Morita equivalent. Additionally, we provide an example showing that C\u{a}ld\u{a}raru's Conjecture does not hold for Deligne--Mumford stacks in general.