
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 -construction, we produce a spectrum whose components model the Grothendieck ring of varieties (over a field ) . We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausen's -theory of spaces and to . We end with a conjecture relating and the doubly-iterated -theory of the sphere spectrum.
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