On Chow weight structures without projectivity and resolution of singularities

In this paper certain Chow weight structures on the "big" triangulated motivic categories $DM_R^{eff}\subset DM_R$ are defined in terms of motives of all smooth varieties over the base field. This definition allows studying basic properties of these weight structures without applying resolution of singularities; thus we don't have to assume that the coefficient ring $R$ contains $1/p$ in the case where the characteristic $p$ of the base field is positive. Moreover, in the case where $R$ satisfies the latter assumption our weight structures are "compatible" with Chow weight structures defined in previous papers (in terms of Chow motives). The results of this article yield certain Chow-weight filtration (also) on $p$-adic cohomology of motives and smooth varieties.