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 $\infty$-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 $Mod_{ A}^{ \omega}$, 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.
Categories: math.AT math.AG math.KT