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## Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do

### Abstract

The goal of this paper is to prove: if certain 'standard' conjectures on motives over algebraically closed fields hold, then over any 'reasonable' \$S\$ there exists a motivic \$t\$-structure for the category of Voevodsky's \$S\$-motives (as constructed by Cisinski and Deglise). If \$S\$ is 'very reasonable' (for example, of finite type over a field) then the heart of this \$t\$-structure (the category of mixed motivic sheaves over \$S\$) is endowed with a weight filtration with semi-simple factors. We also prove a certain 'motivic decomposition theorem' (assuming the conjectures mentioned) and characterize semi-simple motivic sheaves over \$S\$ in terms of those over its residue fields. Our main tool is the theory of weight structures. We actually prove somewhat more than the existence of a weight filtration for mixed motivic sheaves: we prove that the motivic \$t\$-structure is transversal to the Chow weight structure for \$S\$-motives (that was introduced previously and independently by D. Hebert and the author; weight structures and their transversality with t-structures were also defined by the author in recent papers). We also deduce several properties of mixed motivic sheaves from this fact. Our reasoning relies on the degeneration of Chow-weight spectral sequences for 'perverse 'etale homology' (that we prove unconditionally); this statement also yields the existence of the Chow-weight filtration for such (co)homology that is strictly restricted by ('motivic') morphisms.Comment: a few minor corrections mad

Topics: Mathematics - Algebraic Geometry, Mathematics - K-Theory and Homology, 14C15 (Primary) 14F42 (Secondary), 19E15, 14F43, 18G40, 14C25, 14F20, 18E30, 13D15
Year: 2014
DOI identifier: 10.1112/S0010437X14007763
OAI identifier: oai:arXiv.org:1105.0420