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

By Mikhail V. Bondarko

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
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