On Chow weight structures without projectivity and resolution of singularities

Abstract:

In this paper certain Chow weight structures on the “big” triangulated motivic categories $\mathrm {DM}^{\mathrm {eff}}_{R}{}\subset \mathrm {DM}_{R}$ are defined in terms of motives of all smooth varieties over the ground field. This definition allows the study of basic properties of these weight structures without applying resolution of singularities; thus, it is possible to lift the assumption that the coefficient ring $R$ contains $1/p$ in the case where the characteristic $p$ of the ground field is positive. Moreover, in the case where $R$ does satisfy the last assumption, our weight structures are “compatible” with the weight structures that were defined in previous papers in terms of Chow motives; it follows that a motivic complex has nonnegative weights if and only if its positive Nisnevich hypercohomology vanishes. The results of this article yield certain Chow-weight filtration (also) on $p$-adic cohomology of motives and smooth varieties.