  We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an
example. In some cases we show a derived equivalence which corresponds to a
derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are
Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) -
c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3
surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack
of Azumaya algebras with the same underlying gerbe, if non empty, is a proper
algebraic space.

Algebraic Geometry