Weights for relative motives; relation with mixed complexes of sheaves

摘要

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