Derived Azumaya algebras and twisted K-theory

We construct a relative version of topological K-theory, in the sense of [11], over an arbitrary quasi-compact, quasi separated C-scheme X. This has as input a Perf(X)-linear stable infinity-category and output a sheaf of spectra on X(C), the space of complex points of X. We then characterize the values of this functor on inputs of the form Perf(X, A), for A a derived Azumaya algebra over X. In such cases we show that this coincides with the alpha-twisted topological K-theory of X(C) for some appropriately defined twist of K-theory. We use this to provide a topological analogue of a classical result of Quillen's on the algebraic K-theory of Severi-Brauer varieties. (C) 2019 Elsevier Inc. All rights reserved.