The annihilator of the Lefschetz motive

Abstract

In this article, we study a spectrum $K\left({\mathcal{V}}_k\right)$ such that ${\pi }_{0}K\left({\mathcal{V}}_k\right)$ 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 $K_0\left[{\mathcal{V}}_k\right]$ and to show that classes in the kernel of multiplication by $\left[{\mathbb{A}}^1\right]$ can always be represented as $\left[X\right]-\left[Y\right]$, where $\left[X\right]\ne \left[Y\right]$, $X\times {\mathbb{A}}^1$, and $Y\times {\mathbb{A}}^1$ are not piecewise-isomorphic, but $[X\times {\mathbb{A}}^1]=[Y\times {\mathbb{A}}^1]$ in $K_0\left[{\mathcal{V}}_k\right]$. Along the way, we present a new proof of the result of Larsen–Lunts on the structure on $K_0\left[{\mathcal{V}}_k\right]/\left(\right[{\mathbb{A}}^1\left]\right)$.