Universal birational invariants and ${\mathbb{A}}^1$-homology

Abstract

Let $k$ be a field admitting a resolution of singularities. In this paper, we prove that the functor of zeroth ${\mathbb{A}}^1$-homology ${\mathbf{H}}_0^{{\mathbb{A}}^1}$ is universal as a functorial birational invariant of smooth proper $k$-varieties taking values in a category enriched by abelian groups. For a smooth proper $k$-variety $X$, we also prove that the dimension of ${\mathbf{H}}_0^{{\mathbb{A}}^1}(X;\mathbb{Q})(\mathrm{Spec}k)$ coincides with the number of $R$-equivalence classes of $X(k)$. We deduce these results as consequences of the structure theorem that for a smooth proper $k$-variety $X$, the sheaf ${\mathbf{H}}_0^{{\mathbb{A}}^1}(X)$ is the free abelian presheaf generated by the birational ${\mathbb{A}}^1$-connected components ${\pi }_0^{b{\mathbb{A}}^1}(X)$ of Asok-Morel.