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

Part of: Homological algebra Abelian categories $K$-theory in geometry Commutative algebra: Homological methods Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  09 January 2015

Mikhail V. Bondarko
Mikhail V. Bondarko*
Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Bibliotechnaya Pl. 2, 198904, St. Petersburg, Russia email [email protected]
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Abstract

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The goal of this paper is to prove that if certain ‘standard’ conjectures on motives over algebraically closed fields hold, then over any ‘reasonable’ scheme $S$ there exists a motivic$t$-structure for the category $\text{DM}_{c}(S)$ of relative Voevodsky’s motives (to be more precise, for the Beilinson motives described by Cisinski and Deglise). If $S$ is 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 semisimple factors. We also prove a certain ‘motivic decomposition theorem’ (assuming the conjectures mentioned) and characterize semisimple 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 $\text{DM}_{c}(S)$ (that was introduced previously by Hébert and the author). 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 étale homology’ (which 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.

Keywords

motivesmixed motivic sheavesperverse sheavesétale homologytriangulated categoriesweight structuresweight filtrationdecomposition theorem

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives
Secondary: 14F42: Motivic cohomology; motivic homotopy theory 19E15: Algebraic cycles and motivic cohomology 14F43: Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 18G40: Spectral sequences, hypercohomology 14C25: Algebraic cycles 14F20: Étale and other Grothendieck topologies and (co)homologies 18E30: Derived categories, triangulated categories 13D15: Grothendieck groups, $K$-theory
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 5 , May 2015 , pp. 917 - 956
Copyright
© The Author 2015 

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