Derived Azumaya algebras and twisted $K$-theory

TL;DR

This paper develops a relative topological $K$-theory for dg categories over complex schemes, linking derived Azumaya algebras to twisted $K$-theory and extending classical algebraic results to a topological setting.

Contribution

It introduces a sheaf-theoretic topological $K$-theory for dg categories over schemes and characterizes it for derived Azumaya algebras, connecting algebraic and topological $K$-theories.

Findings

Constructs a sheaf of spectra on complex points of schemes.

Shows the $K$-theory for derived Azumaya algebras matches twisted topological $K$-theory.

Provides a topological analogue of Quillen's result on Severi-Brauer varieties.

Abstract

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.