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Norms in motivic homotopy theory
If f:S ->S is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor f :H (S )->H (S), where H (S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f :SH(S )->SH(S), where S...
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| Auteurs principaux : | , |
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| Format : | Livre |
| Langue : | anglais |
| Titre complet : | Norms in motivic homotopy theory / Tom Bachmann & Marc Hoyois |
| Publié : |
Paris :
Société mathématique de France
, C 2021 |
| Description matérielle : | 1 vol. (ix-207 p.) |
| Collection : | Astérisque ; 425 |
| Sujets : | |
| Documents associés : | Fait partie de l'ensemble:
Astérisque |
| Résumé : | If f:S ->S is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor f :H (S )->H (S), where H (S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f :SH(S )->SH(S), where SH(S) is the P1-stable motivic homotopy category over S. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic E-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum HZ, the homotopy K-theory spectrum KGL, and the algebraic cobordism spectrum MGL. The normed spectrum structure on HZ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology. |
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| Notes : | Résumés en anglais et en français |
| Historique des publications : | N° de : "Astérisque", ISSN 0303-1179, (2021) n°425 |
| Bibliographie : | Bibliographie p. [199]-207 |
| ISBN : | 978-2-85629-939-5 |