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Étale motives

Part of: $K$-theory in geometry Cycles and subschemes (Co)homology theory

Published online by Cambridge University Press:  24 September 2015

Denis-Charles Cisinski  and
Frédéric Déglise
Denis-Charles Cisinski
Affiliation:
Université Paul Sabatier, Institut de Mathématiques de Toulouse, Institut Universitaire de France, 118, route de Narbonne, 31062 Toulouse Cedex 9, France email denis-charles.cisinski@math.univ-toulouse.fr
Frédéric Déglise
Affiliation:
E.N.S. Lyon, UMPA, 46, allée d’Italie, 69364 Lyon Cedex 07, France email frederic.deglise@ens-lyon.fr
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Abstract

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We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic $K$-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason’s insights, this also provides a conceptual and convenient construction of the $\ell$-adic realization of motives, as the homotopy $\ell$-completion functor.

Keywords

étale motivesh-topologyGrothendieck six functorsrigidity theoreml-adic realizationl-adic completion

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies 14F42: Motivic cohomology; motivic homotopy theory 14F05: Sheaves, derived categories of sheaves and related constructions 19E15: Algebraic cycles and motivic cohomology 14C25: Algebraic cycles
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 3 , March 2016 , pp. 556 - 666
Copyright
© The Authors 2015 

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