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

TL;DR

This paper constructs a Chow weight structure for cdh-motives with integral coefficients over equicharacteristic schemes, extending previous rational coefficient results and enabling new spectral sequences and filtrations for motive (co)homology.

Contribution

It defines a Chow weight structure with integral coefficients for cdh-motives, generalizing prior rational coefficient results and applying to a broad class of base schemes.

Findings

Established Chow weight structures for integral coefficients

Extended properties of weight structures from rational to integral coefficients

Developed spectral sequences and filtrations for motive (co)homology

Abstract

The main goal of this paper is to define a certain Chow weight structure $w_{Chow}$ on the category $DM_c(S)$ 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 $w_{Chow}$ 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 for any (co)homology of motives.