A commutative P1-spectrum representing motivic cohomology over dedekind domains

"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."-- Page 4 of printed paper wrapper 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. [4e de couverture] Chapter 1. Introduction Acknowledgements Chapter 2. Preliminaries and Notation Chapter 3. Motivic complexes I Chapter 4. The construction 4.1. The p-parts 4.1.1. Finite coefficients 4.1.2. The p-completed parts 4.2. The completed part 4.3. The rational parts 4.4. The definition Chapter 5. Motivic Complexes II 5.1. A strictification 5.2. Properties of the motivic complexes 5.2.1. Comparison to flat maps 5.2.2. Some localization triangles 5.2.3. The étale cycle class map 5.3. The naive Gm-spectrum Chapter 6. Motivic complexes over a field Chapter 7. Comparisons 7.1. The exceptional inverse image of M 7.2. Pullback to the generic point 7.3. Weight 1 motivic complexes 7.4. Rational spectra 7.5. The isomorphism between MZ and M Chapter 8. Base change Chapter 9. The motivic functor formalism Chapter 10. Further applications 10.1. The Hopkins-Morel isomorphism 10.2. The dual motivic Steenrod algebra Appendix A. (Semi) model structures Appendix B. Pullback of cycles Appendix C. An explicit periodization of MZ Bibliography