Derived Azumaya algebras and twisted $K$-theory

We construct a relative version of topological $K$-theory of dg categories over an arbitrary quasi-compact, quasi-separated $\mathbb{C}$-scheme $X$. This has as input a $\text{Perf}(X)$-linear stable $\mathrm{\infty }$-category and output a sheaf of spectra on $X(\mathbb{C})$, the space of complex points of $X$. We then characterize the values of this functor on inputs of the form $Mo{d}_A^{\omega }$, for $A$ a derived Azumaya algebra over $X$. In such cases we show that this coincides with the $\alpha$-twisted topological $K$-theory of $X(\mathbb{C})$ 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.