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

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 $Obj{d}_{\le m}D{M}_{-,R}^{eff}\cap ObjD{M}_{-,R}^{eff}(i)=Obj{d}_{\le m-i}D{M}_{-,R}^{eff}(i)$ (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.

A few minor corrections made. A shorter version of this paper (without section 3) will probably appear in "Homology, Homotopy and Applications" under the name "Intersecting the dimension and slice filtrations for motives"

References in corpus (4)

• On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors
• On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra
• On relative $K$-motives, weights for them, and negative $K$-groups
• On constructing weight structures and extending them to idempotent extensions

Cited by in corpus (3)

• On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra
• On relative $K$-motives, weights for them, and negative $K$-groups
• On constructing weight structures and extending them to idempotent extensions