We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel-Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoub's and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology.

Abstract. Let S be an essentially smooth scheme over a field of characteristic exponent c. We prove that there is a canonical equivalence of motivic spectra over Swhere HZ is the motivic cohomology spectrum, MGL is the algebraic cobordism spectrum, and the elements an are generators of the Lazard ring. We discuss several applications including the computation of the slices of Z[1/c]-local Landweber exact motivic spectra and the convergence of the associated slice spectral sequences.

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, 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 $E_\infty$-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.

We establish a relative version of the abstract "affine representability" theorem in A 1 -homotopy theory from Part I of this paper. We then prove some A 1 -invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in A 1 -homotopy theory. 14F42; 14L10, 55R15, 20G15Published: XX Xxxember 20XX All rings considered in this paper will be assumed unital. We use the symbol S for a quasi-compact, quasi-separated base scheme, Sm S for the category of finitely presented smooth S-schemes, and Sm aff S ⊂ Sm S for the full subcategory of affine schemes (in the absolute sense). We also reuse some terminology and notation introduced in [9],

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 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∞-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.