Z[1/p]-motivic resolution of singularities

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