Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields

TL;DR

This paper introduces a triangulated analogue of coniveau spectral sequences for motives, decomposing them into twisted motives of points, and demonstrates their functoriality and computational aspects, extending classical results to a broader motivic context.

Contribution

It constructs a 'Gersten' weight structure for a triangulated category of comotives, enabling coniveau spectral sequences for motives and establishing their functoriality and computability.

Findings

Cohomology of a smooth semi-local scheme is a direct summand of its generic fibre cohomology.

Cohomology of function fields contains twisted cohomology of residue fields.

Develops a new theory of 'nice' pairings of triangulated categories.

Abstract

We construct a 'triangulated analogue' of coniveau spectral sequences: the motif of a variety over a countable field is 'decomposed' (in the sense of Postnikov towers) into the twisted (co)motives of its points; this is generalized to arbitrary Voevodsky's motives. To this end we construct a 'Gersten' weight structure for a certain triangulated category of 'comotives': the latter is defined to contain comotives for all projective limits of smooth varieties; the definition of a weight structure was introduced in a preceding paper. The corresponding weight spectral sequences are essentially coniveau one; they are $DM^{eff}_{gm}$-functorial (starting from $E_2$) and can be computed in terms of the homotopy $t$-structure for the category $DM^-_{eff}$ (similarly to the case of smooth varieties). This extends to motives the seminal coniveau spectral sequence computations of Bloch and Ogus. We also obtain that the cohomology of a smooth semi-local scheme is a direct summand of the cohomology of its generic fibre; cohomology of function fields contain twisted cohomology of their residue fields (for all geometric valuations). We also develop further the general theory of weight structures for triangulated categories (independently from the 'motivic' part of the paper). Besides, we develop a certain theory of 'nice' pairings of triangulated categories; this subject seems to be new.