Relative Étale Realizations of Motivic Spaces and Dwyer-Friedlander $K$-Theory of Noncommutative Schemes

Abstract: 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 Blanc's theory of semi-topological $K$-theory of non-commutative schemes. We then formulate and prove an analog of Blanc's conjecture on the torsion part of this theory, generalizing the work of Antieau and Heller.