On Chow weight structures for $cdh$-motives with integral coefficients

Abstract:

The paper is aimed at defining a certain Chow weight structure $w_{\mathrm {Chow}}$ on the category $\mathcal {DM}_c(S)$ of (constructible) $cdh$-motives over an equicharacteristic scheme $S$. In contrast to the previous papers of D. Hébert and the first author on weights for relative motives (with rational coefficients), this goal is achieved for motives with integral coefficients (if $\mathrm {char}\thinspace S=0$; if $\mathrm {char}\thinspace S=p>0$, then motives with ${\mathbb {Z}}[\frac {1}{p}]$-coefficients are considered). It is proved that the properties of the Chow weight structures that were previously established for ${\mathbb {Q}}$-linear motives can be carried over to this “integral” context (and some of them are generalized using certain new methods). Mostly, the version of $w_{\mathrm {Chow}}$ defined via “gluing from strata” is studied; this makes it possible to define Chow weight structures for a wide class of base schemes.

As a consequence, certain (Chow)-weight spectral sequences and filtrations are obtained on any (co)homology of motives.