Weights for relative motives; relation with mixed complexes of sheaves

TL;DR

This paper develops the Chow weight structure for Beilinson motives over schemes, exploring its properties, functoriality, and connections to motivic cohomology and mixed complexes of sheaves.

Contribution

It introduces the Chow weight structure for motives over schemes, analyzes its functoriality, and relates it to motivic cohomology and mixed sheaf complexes.

Findings

Chow weight structure defined for Beilinson motives over schemes.

Functoriality properties similar to weights in mixed sheaves.

Spectral sequences and filtrations linked to motivic cohomology.

Abstract

The main goal of this paper is to define the so-called Chow weight structure for the category of Beilinson motives over any 'reasonable' base scheme $S$ (this is the version of Voevodsky's motives over $S$ defined by Cisinski and Deglise). We also study the functoriality properties of the Chow weight structure (they are very similar to the well-known functoriality of weights for mixed complexes of sheaves). As shown in a preceding paper, the Chow weight structure automatically yields an exact conservative weight complex functor (with values in $K^b(Chow(S))$). Here $Chow(S)$ is the heart of the Chow weight structure; it is 'generated' by motives of regular schemes that are projective over $S$. Besides, Grothendiek's group of $S$-motives is isomorphic to $K_0(Chow(S))$; we also define a certain 'motivic Euler characteristic' for $S$-schemes. We obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation to Beilinson's 'integral part' of motivic cohomology and to weights of mixed complexes of sheaves. For the study of the latter we introduce a new formalism of relative weight structures.