Z[1/p]-motivic resolution of singularities

TL;DR

This paper proves that Chow motives with Z[1/p]-coefficients generate the effective geometric Voevodsky motives category over a perfect field of characteristic p, enabling a Chow weight structure and related tools for cohomology theories.

Contribution

It establishes the generation of DM^{eff}_{gm}[1/p] by Chow motives and constructs a Chow weight structure, extending previous results to positive characteristic fields.

Findings

Existence of a Chow weight structure on DM^{eff}_{gm}[1/p]

Construction of a conservative weight complex functor

Development of Chow-weight spectral sequences and filtrations

Abstract

The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with $Z[1/p]$-coefficients over a perfect field $k$ of characteristic $p$ generate the category $DM^{eff}_{gm}[1/p]$ (of effective geometric Voevodsky's motives with $Z[1/p]$-coefficients). It follows that $DM^{eff}_{gm}[1/p]$ could be endowed with a Chow weight structure $w_{Chow}$ whose heart is $Chow^{eff}[1/p]$ (weight structures were introduced in a preceding paper, where the existence of $w_{Chow}$ for $DM^{eff}_{gm}Q$ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor $DM^{eff}_{gm}[1/p]\to K^b(Chow^{eff}[1/p])$ (which induces an isomorphism on $K_0$-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on $DM^{eff}_{gm}[1/p]$. We also define a certain Chow t-structure for $DM_{-}^{eff}[1/p]$ and relate it with unramified cohomology. To this end we study birational motives and birational homotopy invariant sheaves with transfers.