Intersecting the dimension filtration with the slice one for (relative) motivic categories

TL;DR

This paper demonstrates that the intersection of the dimension filtration levels with the slice one in motivic categories is minimal, and applies this to relate a conjecture of Ayoub to an orthogonality assumption, also extending results to relative motivic categories.

Contribution

It proves a minimal intersection property for the dimension filtration and slice one in motivic categories, and generalizes this to relative motivic categories under certain axioms.

Findings

Intersection of filtration levels is as small as possible

Equivalence of Ayoub's conjecture with an orthogonality assumption

Generalization to relative motivic categories with new properties

Abstract

In this paper we prove that the intersections of the levels of the dimension filtration on Voevodsky's motivic complexes over a field $k$ with the levels of the slice one are "as small as possible", i.e., that $Obj d_{\le m}DM^{eff}_{-,R} \cap Obj DM^{eff}_{-,R} (i)=Obj d_{\le m-i} DM^{eff}_{-,R} (i)$ (for $m,i\ge 0$ and $R$ being any coefficient ring in which the exponential characteristic of $k$ invertible). This statement is applied to prove that a conjecture of J. Ayoub is equivalent to a certain orthogonality assumption. We also establish a vast generalization of our intersection result to relative motivic categories (that are required to fulfil a certain list of "axioms"). In the process we prove several new properties of relative motives and of the so-called Chow weight structures for them.