I will describe a way to show that spaces of PGLn-bundles (Azumaya algebras) on an algebraic surface are irreducible. The key is to find a nice compactification and exploit an inductive structure provided by the boundary. The existence of a nice compactification is related to a version of the Skolem-Noether theorem for algebra objects of the derived category, while the properties of the compactified space are best understood in terms of a finite covering by a moduli stack of vector bundles on a stacky version of the original surface. The irreducibility of these spaces has concrete consequences in arithmetic.