Documents Hoyois, Marc 1 results

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. (4e de couv.)

Sommaire :
1. Introduction
2. Preliminaries
3. Norms of pointed motivic spaces
4. Norms of motivic spectra
5. Properties of norms
6. Coherence of norms
7. Normed motivic spectra
8. The norm-pullback-pushforward adjunctions
9. Spectra over profinite groupoids
10. Norms and Grothendieck's Galois theory
11. Norms and Betti realization
12. Norms and localization
13. Norms and the slice filtration
14. Norms of cycles
15. Norms of linear ∞-categories
16. Motivic Thom spectra
A. The Nisnevich topology
B. Detecting effectivity
C. Categories of spans
D. Relative adjunctions
Table of notation
Bibliography[-]