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A commutative $\mathbb P ^1$-spectrum representing motivic cohomology over Dedekind domains

A commutative $\mathbb P ^1$-spectrum representing motivic cohomology over Dedekind domains

Markus SPITZWECK
A commutative $\mathbb P ^1$-spectrum representing motivic cohomology over Dedekind domains
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  • Année : 2018
  • Tome : 157
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 14F42, 14C25, 19E20, 19E99, 14F05, 14F20, 14F30, 55P43.
  • Nb. de pages : 114
  • ISBN : 978-2-85629-890-9
  • ISSN : 0249-633-X

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over general base schemes which represents Levine's motivic cohomology, defined via Bloch's cycle complexes, over smooth schemes over Dedekind domains. Our method is by gluing $p$-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated étale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of motives with integral coefficients. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications are a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.

We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over general base schemes which represents Levine's motivic cohomology, defined via Bloch's cycle complexes, over smooth schemes over Dedekind domains. Our method is by gluing $p$-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated étale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of motives with integral coefficients. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications are a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.

Motivic Eilenberg-MacLane spectrum, algebraic cycles, motivic six functor formalism.
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