Weights for relative motives; relation with mixed complexes of sheaves

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.

a few minor corrections made

References in corpus (5)

• Triangulated categories of mixed motives
• Structures de poids à la Bondarko sur les motifs de Beilinson
• Homotopy categories and idempotent completeness, weight structures and weight complex functors
• On weights for relative motives with integral coefficients
• On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

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• K-theory Soergel Bimodules
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• Motivic Intersection Complex of Certain Shimura varieties