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

摘要

The goal of this paper is to prove: if certain 'standard' conjectures onmotives 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 'veryreasonable' (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 witha weight filtration with semi-simple factors. We also prove a certain 'motivicdecomposition theorem' (assuming the conjectures mentioned) and characterizesemi-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 somewhatmore than the existence of a weight filtration for mixed motivic sheaves: weprove that the motivic $t$-structure is transversal to the Chow weightstructure for $S$-motives (that was introduced previously and independently byD. Hebert and the author; weight structures and their transversality witht-structures were also defined by the author in recent papers). We also deduceseveral properties of mixed motivic sheaves from this fact. Our reasoningrelies on the degeneration of Chow-weight spectral sequences for 'perverse'etale homology' (that we prove unconditionally); this statement also yieldsthe existence of the Chow-weight filtration for such (co)homology that isstrictly restricted by ('motivic') morphisms.