Norms in motivic homotopy theory

If $f:S^{\mathrm{\prime }}\to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal “norm” functor $f_\otimes: \mathcal H_(S’) \to\mathcal H_(S)$,where$\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite 'etale, we show that it stabilizes to a functor $f_{\otimes }:\mathcal{S}\mathcal{H}(S^{\mathrm{\prime }})\to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the ${\mathbb{P}}^1$-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 $E_{\mathrm{\infty }}$-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 $H\mathbb{Z}$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson’s mutliplicative transfers on Chow groups and of Voevodsky’s power operations in motivic cohomology.