# Relative \'{E}tale Realizations of Motivic Spaces and Dwyer-Friedlander
$K$-Theory of Noncommutative Schemes

Research paper by **David Carchedi, Elden Elmanto**

Indexed on: **12 Oct '18**Published on: **12 Oct '18**Published in: **arXiv - Mathematics - Algebraic Geometry**

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#### 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.