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Tuesday, October 17, 2017

**Abstract:** Topological K-theory of dg-categories is a localizing invariant of dg-categories over C taking values in the infinity category of KU-modules. In this talk I describe a relative version of this construction; namely for X a quasi-compact, quasi-separated C-scheme I construct a functor valued in Shv_{Sp}(X(C)), the infinity category of sheaves of spectra on X(C). For inputs of the form Perf(X, A) where A is an Azumaya algebra over X, I characterize the values of this functor in terms of the twisted topological K-theory of X(C). From this I deduce a certain decomposition, for X a finite CW-complex equipped with a bundle of projective spaces P over X, of KU(P) in terms of the twisted topological K-theory of X; this is a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer schemes.