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

TL;DR

This paper explores Morita theory for twisted sheaves on $$-gerbes of line bundles, establishing criteria for Azumaya algebra equivalence and providing a counterexample to Cldraru's Conjecture in the context of Deligne--Mumford stacks.

Contribution

It offers explicit conditions for Morita equivalence of Azumaya algebras on $$-gerbes and demonstrates that Cldraru's Conjecture does not hold universally for Deligne--Mumford stacks.

Findings

Derived explicit Morita equivalence conditions for Azumaya algebras.

Provided a counterexample to Cldraru's Conjecture.

Enhanced understanding of twisted sheaves on $$-gerbes.

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.