Étale motives

Abstract

We define a theory of etale 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 rigity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion etale motives essentially coincide with the usual complexes of torsion etale 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 etale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion etale 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.

Final version. To appear in Compositio Mathematica

References in corpus (2)

• Orientation theory in arithmetic geometry
• On weights for relative motives with integral coefficients

Cited by in corpus (27)

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• Perfection in motivic homotopy theory
• The intersection motive of the moduli stack of shtukas
• On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive
• Equivariant Hodge theory and noncommutative geometry
• Intermediate extension of Chow motives of Abelian type
• On the rational motivic homotopy category
• Rigidity in etale motivic stable homotopy theory
• On Modules Over Motivic Ring Spectra
• Künneth formulas for motives and additivity of traces
• The motivic Satake equivalence
• Three real Artin-Tate motives
• Rigidity for rigid analytic motives
• On the Nori and Hodge realisations of Voevodsky motives
• Algebraic Cobordism and Étale Cohomology
• Weights and conservativity
• tt-geometry of Tate motives over algebraically closed fields
• Rigidity for equivariant pseudo pretheories
• A note on Tannakian categories and mixed motives
• The Berkovich realization for rigid analytic motives
• Chow motives without projectivity, II
• The singular support of an $\ell$-adic sheaf
• On some finiteness results in real étale cohomology
• On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn)
• Étale degree map and 0-cycles
• Motivic homological stability of configuration spaces
• Trace maps in motivic homotopy and local terms