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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.

Weights for Relative Motives: Relation with Mixed Complexes of Sheaves