Detecting motivic equivalences with motivic homology

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