The annihilator of the Lefschetz motive

TL;DR

This paper constructs a spectrum to analyze the Grothendieck ring of varieties, revealing new insights into classes annihilated by the affine line and providing alternative proofs of known structural results.

Contribution

It introduces a spectrum-based approach to study the Grothendieck ring, offering new proofs and geometric interpretations of classes related to the affine line.

Findings

Classes in the kernel of multiplication by [A^1] can be represented as differences of varieties with specific properties.

The spectrum reveals that certain classes are not piecewise isomorphic despite having equal products with A^1.

New proofs of Larsen--Lunts' results on the structure of K_0[Var_k]/([A^1]) are provided.

Abstract

In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0 K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_0[\mathcal{V}_k]$ and show that classes in the kernel of multiplication by $[\mathbb{A}^1]$ can always be represented as $[X]-[Y]$ where $X$ and $Y$ are varieties such that $[X] \neq [Y]$, $X\times \mathbb{A}^1$ and $Y\times \mathbb{A}^1$ are not piecewise isomorphic, but $[X\times \mathbb{A}^1] =[Y\times \mathbb{A}^1]$ in $K_0[\mathcal{V}_k]$. Along the way we present new proofs of the result of Larsen--Lunts on the structure on $K_0[\mathcal{V}_k]/([\mathbb{A}^1])$.