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 \(\infty\)-category and output a sheaf of spectra on \(X(\mathbb{C})\), the space of complex...

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Main Author: Moulinos, Tasos
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description 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.
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title Derived Azumaya algebras and twisted \(K\)-theory
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