On Chow weight structures without projectivity and resolution of singularities

TL;DR

This paper introduces new Chow weight structures on motivic categories that do not rely on resolution of singularities, enabling analysis over broader coefficient rings and providing a Chow-weight filtration on p-adic cohomology.

Contribution

It defines Chow weight structures on motivic categories without assuming resolution of singularities, broadening applicability and compatibility with previous structures.

Findings

Weight structures are defined without resolution of singularities.

Compatibility established with previous Chow weight structures.

Chow-weight filtration on p-adic cohomology is obtained.

Abstract

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.