Comparison of motives with rational coefficients

TL;DR

This paper proves the equivalence of various models of rational motives over any quasi-excellent base scheme, extending previous results and confirming a conjecture about the stability of the motivic rational Eilenberg MacLane spectrum.

Contribution

It establishes the equivalence of Voevodsky's DM with Morel, Beilinson, and Ayoub's rational motives over all quasi-excellent schemes, and proves a key motivic spectrum equivalence.

Findings

Equivalence of DM and other rational motives over any quasi-excellent base scheme.

Stable motivic equivalence between the plus part of the free spectrum and the rational Eilenberg MacLane spectrum.

Partial confirmation of Voevodsky's conjecture on base change stability of the motivic rational Eilenberg MacLane spectrum.

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.