The K-Theory Spectrum of Varieties

Abstract

Using a construction closely related to Waldhausen's $S_{\bullet }$-construction, we produce a spectrum $K({\mathbf{Var}}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0({\mathbf{Var}}_{/k})$. We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausen's $K$-theory of spaces $A(\ast )$ and to $K(\mathbf{Q})$. We end with a conjecture relating $K({\mathbf{Var}}_{/k})$ and the doubly-iterated $K$-theory of the sphere spectrum.

arXiv commentImproved exposition. Changed some nomenclature. Removed Q-construction for future paper