Detecting motivic equivalences with motivic homology

TL;DR

This paper shows that isomorphisms in Voevodsky's triangulated category of motives can be detected by motivic homology groups after base changes to all separable finitely generated field extensions, linking motivic homology to motivic equivalences.

Contribution

It proves that motivic homology groups detect isomorphisms in the category of motives over fields with invertible exponential characteristic, extending previous conservativity results.

Findings

Motivic homology groups detect isomorphisms in the triangulated category of motives.

Base changes to all separable finitely generated field extensions are sufficient for detection.

Results apply to certain spaces in the pointed motivic homotopy category.

Abstract

Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are detected by motivic homology groups of base changes to all separable finitely generated field extensions of $k$. It then follows from previous conservativity results that these motivic homology groups detect isomorphisms between certain spaces in the pointed motivic homotopy category $\mathcal{H}(k)_*$.