The annihilator of the Lefschetz motive

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 $[X]\ne [Y]$, $X\times {\mathbb{A}}^1$ 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 $K_0[{\mathcal{V}}_k]/([{\mathbb{A}}^1])$.

20 pages

References in corpus (2)

• Topologies and structures of the Cremona groups
• Sofic profile and computability of Cremona groups

Cited by in corpus (1)

• Irrationality of motivic zeta functions