Weights for Relative Motives: Relation with Mixed Complexes of Sheaves

The main goal of this paper is to define the Chow weight structure w
Chow
for the category DM
c
(S) of (constructible) Beilinson motives over any excellent separated finite-dimensional base scheme S (this is the version of Voevodsky’s motives over S described by Cisinski and Deglise). We also study the functoriality properties of w
Chow
(they are very similar to those for weights of mixed complexes of sheaves, as described in Beilinson et al. [2, section 5]). As shown in a preceding paper, (the existence of) w
Chow
automatically yields a certain exact conservative weight complex functor DM
c
(S)→K
^b
(Chow(S)). Here, Chow(S) is the heart of w
Chow
; it is “generated” by motives of regular schemes that are projective over S. We also obtain that K
0
(DM
c
(S))≅K
0
(Chow(S)) (and define a certain “motivic Euler characteristic” for S-schemes). Besides, we obtain (Chow) weight spectral sequences and filtrations for any (co)homology of motives; we discuss their relation with Beilinson’s “integral part” of motivic cohomology and with weights of mixed complexes of sheaves. For the study of the latter, we also introduce a new formalism of relative weight structures.