TY - JOUR

T1 - Weights for Relative Motives: Relation with Mixed Complexes of Sheaves

AU - Bondarko, M.V.

PY - 2014

Y1 - 2014

N2 - 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 $S$. We also obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation with Beilinson's 'integral part' of motivic cohomology and with weights of complexes of mixed sheaves. For the study of the latter we also introduce a new formalism of relative weight structures.

AB - 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 $S$. We also obtain (Chow)-weight spectral sequences and filtrations for any cohomology of motives; we discuss their relation with Beilinson's 'integral part' of motivic cohomology and with weights of complexes of mixed sheaves. For the study of the latter we also introduce a new formalism of relative weight structures.

KW - weight structure

KW - Voevodsky motives

KW - triangulated category

KW - mixed complex of sheaves

U2 - 10.1093/imrn/rnt088

DO - 10.1093/imrn/rnt088

M3 - Article

SP - 4715

EP - 4767

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 17

ER -