  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.

Algebraic Geometry;K-Theory and Homology