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Chow motives without projectivity

Part of: Arithmetic problems. Diophantine geometry $K$-theory in geometry (Co)homology theory Abelian categories

Published online by Cambridge University Press:  04 September 2009

Jörg Wildeshaus
Jörg Wildeshaus*
Affiliation:
LAGA, UMR 7539, Institut Galilée, Université Paris 13, Avenue Jean-Baptiste Clément, F-93430 Villetaneuse, France (email: [email protected])
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Abstract

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In a recent paper, Bondarko [Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), Preprint (2007), 0704.4003] defined the notion of weight structure, and proved that the category DMgm(k) of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander [Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, vol. 143 (Princeton University Press, Princeton, NJ, 2000)], is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive [J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656], we describe a method to construct intrinsically in DMgm(k) a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work [J. Wildeshaus, On the interior motive of certain Shimura varieties: the case of Hilbert–Blumenthal varieties, Preprint (2009), 0906.4239], this method will be applied to Shimura varieties.

Keywords

weight structuresweight filtrationsChow motivesgeometrical motivesmotives for modular formsboundary motiveinterior motive

MSC classification

Secondary: 14F42: Motivic cohomology; motivic homotopy theory 14F20: Étale and other Grothendieck topologies and (co)homologies 14F25: Classical real and complex (co)homology 14F30: $p$-adic cohomology, crystalline cohomology 14G35: Modular and Shimura varieties 18E30: Derived categories, triangulated categories 19E15: Algebraic cycles and motivic cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1196 - 1226
Copyright
Copyright © Foundation Compositio Mathematica 2009

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