Motivically Functorial Coniveau Spectral Sequences; Direct Summands of Cohomology of Function Fields

Abstract

The goal of this paper is to prove that coniveau spectral sequences are motivically functorial for all cohomology theories that could be factorized through motives. To this end the motif of a smooth variety over a countable field $k$ is decomposed (in the sense of Postnikov towers) into twisted (co)motives of its points; this is generalized to arbitrary Voevodsky's motives. In order to study the functoriality of this construction, we use the formalism of weight structures (introduced in the previous paper). We also develop this formalism (for general triangulated categories) further, and relate it with a new notion of a nice duality (pairing) of (two distinct) triangulated categories; this piece of homological algebra could be interesting for itself. We construct a certain Gersten weight structure for a triangulated category of comotives that contains $DM^{eff}_{gm}$ as well as (co)motives of function fields over $k$. It turns out that the corresponding weight spectral sequences generalize the classical coniveau ones (to cohomology of arbitrary motives). When a cohomological functor is represented by a $Y\in Obj DM^{eff}_-$, the corresponding coniveau spectral sequences can be expressed in terms of the (homotopy) $t$-truncations of $Y$; this extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the comotif of a smooth connected semi-local scheme is a direct summand of the comotif of its generic point; comotives of function fields contain twisted comotives of their residue fields (for all geometric valuations). Hence similar results hold for any cohomology of (semi-local) schemes mentioned.