An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

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′A′ are Morita equivalent Azumaya algebras of degree r then 2r   divides c2(A)−c2(A′)c2(A)−c2(A′). In particular this implies that if A   is an Azumaya algebra on a K3 surface and c2(A)c2(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.