On Chow weight structures for cdh-motives with integral coeffcients

Abstract

The main goal of this paper is to define a certain {\it Chow weight structure} $w_{\operatorname{Chow}}$ on the category $\mathcal{DM}_c(S)$ of (constructible) $cdh$-motives over an equicharacteristic scheme $S$. In contrast to the previous papers of D.~H\'ebert and the first author on weights for relative motives (with rational coefficients), we can achieve our goal for motives with integral coefficients (if $\operatorname{char} S=0$; if $\operatorname{char} S=p>0$ then we consider motives with ${\mathbb{Z}}[\frac{1}{p}]$-coefficients). We prove 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 we generalize some of them using certain new methods). In this paper we mostly study the version of $w_{\operatorname{Chow}}$ defined via "gluing from strata"; this enables us to define Chow weight structures for a wide class of base schemes.

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