Norms in motivic homotopy theory - Bachmann Tom (Société mathématique de France, 2021)
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.
Keywords: Géométrie algébrique; Cohomologie; Homotopie; K-théorie