[Motifs et modules sur la cohomologie motivique]
Dans cette Note, nous présentons nos résultats principaux et les techniques utilisées dans notre étude des motifs, qui est basée sur la théorie d'homotopie. Une partie importante de ce travail utilise des modèles hautement structurés pour la théorie d'homotopie stable motivique. Pour tout schéma de base noethérien, séparé et de dimension de Krull finie, ces outils permettent l'étude de la théorie d'homotopie des modules sur la cohomologie motivique. Lorsque le schéma de base est , pour k un corps de caractéristique zéro, la catégorie homotopique obtenue est équivalente à la grande catégorie des motifs introduite par Voevodsky.
In this Note we summarize the main results and techniques in our homotopical algebraic approach to motives. A major part of this work relies on highly structured models for motivic stable homotopy theory. For any noetherian and separated base scheme of finite Krull dimension these frameworks give rise to a homotopy theoretic meaningful study of modules over motivic cohomology. When the base scheme is , for k a field of characteristic zero, the corresponding homotopy category is equivalent to Voevodsky's big category of motives.
Accepté le :
Publié le :
@article{CRMATH_2006__342_10_751_0,
author = {Oliver R\"ondigs and Paul Arne {\O}stv{\ae}r},
title = {Motives and modules over motivic cohomology},
journal = {Comptes Rendus. Math\'ematique},
pages = {751--754},
publisher = {Elsevier},
volume = {342},
number = {10},
year = {2006},
doi = {10.1016/j.crma.2006.03.013},
language = {en},
}
Oliver Röndigs; Paul Arne Østvær. Motives and modules over motivic cohomology. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 751-754. doi : 10.1016/j.crma.2006.03.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.013/
[1] Smoothness, semi-stability and alterations, Publ. Math. IHES, Volume 83 (1996), pp. 51-93
[2] Motivic functors, Doc. Math., Volume 8 (2003), pp. 489-525
[3] Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, Volume 165 (2001), pp. 63-127
[4] On the Picard group of the -stable homotopy category, Topology, Volume 44 (2005), pp. 609-640
[5] Simplicial presheaves, J. Pure Appl. Algebra, Volume 47 (1987), pp. 35-87
[6] Motivic symmetric spectra, Doc. Math., Volume 5 (2000), pp. 445-553
[7] -homotopy theory of schemes, Publ. Math. IHES, Volume 90 (1999), pp. 45-143
[8] O. Röndigs, Functoriality in motivic homotopy theory, Preprint
[9] O. Röndigs, P.A. Østvær, Motivic spaces with transfers, Preprint
[10] O. Röndigs, P.A. Østvær, Modules over motivic cohomology, Preprint
[11] Algebras and modules in monoidal model categories, Proc. London Math. Soc., Volume 80 (2000), pp. 491-511
[12] Stable model categories are categories of modules, Topology, Volume 42 (2003), pp. 103-153
[13] Relative cycles and Chow sheaves, Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., vol. 143, 2000, pp. 10-86
[14] Triangulated categories of motives over a field, Cycles, Transfers, and Motivic Homology Theories, Ann. of Math. Stud., vol. 143, 2000, pp. 188-238
[15] Open problems in the motivic stable homotopy theory I, Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, 2002, pp. 3-34
[16] V. Voevodsky, Cancellation theorem, Preprint
Cité par Sources :
Commentaires - Politique
Motives and homotopy theory in logarithmic geometry
Federico Binda; Doosung Park; Paul Arne Østvær
C. R. Math (2022)