An Explicit Derived Equivalence of Azumaya Algebras on K3 Surfaces via Koszul Duality

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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 c₂(A) - c₂(A'). In particular this implies that if A is an Azumaya algebra on a K3 surface and 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.