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

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

2011-05-02

arxiv.org/abs/1105.0420v1

Mathematics

Algebraic Geometry

Recent results of Cisinski, Deglise, Hebert, and the author make it possible to prove the following result: if certain 'standa

Scientific paper

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 an equicharacteristic scheme, 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.

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