The K-Theory Spectrum of Varieties

TL;DR

This paper constructs a spectrum $K(Var_{/k})$ that models the Grothendieck ring of varieties over a field and lifts motivic measures to spectrum-level maps, proposing a conjecture linking it to iterated $K$-theory of the sphere spectrum.

Contribution

It introduces a spectrum-level construction for the Grothendieck ring of varieties and extends motivic measures to spectrum maps, connecting algebraic geometry and $K$-theory.

Findings

Constructed the spectrum $K(Var_{/k})$ modeling the Grothendieck ring.

Lifted motivic measures to spectrum-level maps into $A(*)$ and $K( extbf{Q})$.

Proposed a conjecture relating $K(Var_{/k})$ to iterated $K$-theory of the sphere spectrum.

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.