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In this paper, we construct a refined, relative version of the etale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $infty$-category of motivic spaces over a base scheme $S$ to the $infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $infty$-category is generated under cofiltered limits by those spaces whose associated local system on $S$ is $A^1$-invariant. We then construct a new, relative version of their etale realization functor which takes into account the geometry and arithmetic of the base scheme $S$. For example, when $S$ is the spectrum of a field $k$, our functor lands in a certain $infty$-category equivariant for the absolute Galois group. Our construction relies on a relative version of etale homotopy types in the sense of Artin-Mazur-Friedlander, which we also develop in some detail, expanding on previous work of Barnea-Harpaz-Schlank. We then stabilize our functor, in the $S^1$-direction, to produce an etale realization functor for motivic $S^1$-spectra (in other words, Nisnevich sheaves of spectra which are $A^1$-invariant). To this end, we also develop an $infty$-categorical version of the theory of profinite spectra, first explored by Quick. As an application, we refine the construction of the etale $K$-theory of Dwyer and Friedlander, and define its non-commutative extension. This latter invariant should be seen as an $ell$-adic analog of Blancs theory of semi-topological $K$-theory of non-commutative schemes. We then formulate and prove an analog of Blancs conjecture on the torsion part of this theory, generalizing the work of Antieau and Heller.
We obtain geometric models for the infinite loop spaces of the motivic spectra $mathrm{MGL}$, $mathrm{MSL}$, and $mathbf{1}$ over a field. They are motivically equivalent to $mathbb{Z}times mathrm{Hilb}_infty^mathrm{lci}(mathbb{A}^infty)^+$, $mathbb{
If $f:S to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal norm functor $f_otimes: mathcal H_*(S) tomathcal H_*(S)$, where $mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finit
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 hom
We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $mathbb{P}^infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspensi
We prove that the $infty$-category of motivic spectra satisfies Milnor excision: if $Ato B$ is a morphism of commutative rings sending an ideal $Isubset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is equivalent to a pair