On Chow-weight homology of geometric motives

Abstract:

We describe new Chow-weight (co)homology theories on the category $DM^{\mathrm {eff}}_{gm}(k,R)$ of effective geometric Voevodsky motives ($R$ is the coefficient ring). These theories are interesting “modifications” of motivic homology; Chow-weight homology detects whether a motive $M\in ObjDM^{\mathrm {eff}}_{gm}(k,R)$ is $r$-effective (i.e., belongs to the $r$th Tate twist $DM^{\mathrm {eff}}_{gm}(k,R)(r)$ of effective motives), bounds the weights of $M$ (in the sense of the Chow weight structure defined by the first author), and detects the effectivity of “the lower weight pieces” of $M$. Moreover, we calculate the connectivity of $M$ (in the sense of Voevodsky’s homotopy $t$-structure, i.e., we study motivic homology) and prove that the exponents of the higher motivic homology groups (of an “integral” motive) are finite whenever these groups are torsion. We apply the latter statement to the study of higher Chow groups of arbitrary varieties.

These motivic properties of $M$ have plenty of applications. They are closely related to the (co)homology of $M$; in particular, if the Chow groups of a variety $X$ vanish up to dimension $r-1$ then the highest Deligne weight factors of the (singular or étale) cohomology of $X$ with compact support are $r$-effective.

Our results yield vast generalizations of the so-called “decomposition of the diagonal” theorems, and we re-prove and extend some of earlier statements of this sort.