On Chow-weight homology of motivic complexes and its relation to motivic homology

TL;DR

This paper explores the relationship between Chow-weight homology and motivic homology, establishing new links and properties, including vanishing implications and torsion bounds, within the framework of Voevodsky motives and their complexes.

Contribution

It generalizes previous results by relating Chow-weight homology to motivic homology, especially regarding vanishing and torsion properties, and extends the theory to motivic complexes.

Findings

Vanishing of higher motivic homology implies vanishing of Chow-weight homology.

Higher motivic homology groups that are torsion have uniformly bounded exponents.

Relations between motivic homology and Chow groups are established for motives with compact support.

Abstract

We study in detail the so-called Chow-weight homology of Voevodsky motivic complexes and relate it to motivic homology. We generalize earlier results and prove that the vanishing of higher motivic homology groups of a motif $M$ implies similar vanishing for its Chow-weight homology along with effectivity properties of the higher terms of its weight complex $t(M)$ and of higher Deligne weight quotients of its cohomology. Applying this statement to motives with compact support we obtain a similar relation between the vanishing of Chow groups and the cohomology with compact support of varieties. Moreover, we prove that if higher motivic homology groups of a geometric motif or a variety over a universal domain are torsion (in a certain "range") then the exponents of these groups are uniformly bounded. To prove our main results we study Voevodsky slices of motives. Since the slice functors do not respect the compactness of motives, the results of the previous Chow-weight homology paper are not sufficient for our purposes; this is our main reason to extend them to ($w_{Chow}$-bounded below) motivic complexes.