Derived Azumaya algebras and twisted $K$theory
Abstract
We construct a relative version of topological $K$theory of dg categories over an arbitrary quasicompact, quasiseparated $\mathbb{C}$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 SeveriBrauer varieties.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.05810
 Bibcode:
 2017arXiv171005810M
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 19D55;
 18G55;
 19E08;
 14F42;
 14F20;
 14F22;
 55N15
 EPrint:
 Final version, to appear in Advances in Mathematics