The K-Theory Spectrum of Varieties (1505.03136v2)

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.