Comparison of motives with rational coefficients

Abstract

The theory of rational motives admits several models, including those of Morel, Beilinson, Ayoub, and Voevodsky. An open question has been the equivalence of Voevodsky's Nisnevich-based $\mathrm{DM}(S, \mathbb{Q})$ with the others, which was only known over excellent and geometrically unibranch base schemes. In this paper, we prove that $\mathrm{DM}(S, \mathbb{Q})$ is equivalent to Morel/Beilinson/Ayoub's rational motives over any quasi-excellent base scheme $S$. Our main technical result is a stable motivic equivalence between the plus part of free $\mathbb{Q}$-linear spectrum $\mathbb{Q}[\mathbb{S}]$ and the motivic rational Eilenberg MacLane spectrum $\mathbf{H}\mathbb{Q}$. This equivalence is established whenever Ayoub's motives $\mathrm{DA}(S, \mathbb{Q})$ satisfies h-descent. As a byproduct, we partially confirm Voevodsky's conjecture that the formation of motivic rational Eilenberg Maclane spectrum $\mathbf{H}\mathbb{Q}$ is stable under base change between any quasi-excellent scheme.