An \'etale realization which does not exist

TL;DR

The paper proves that for certain fields, a functor from the -homotopy category to a Galois-equivariant category satisfying natural hypotheses cannot exist, highlighting limitations of -realization functors.

Contribution

It establishes a non-existence result for -realization functors under natural hypotheses over infinite Galois groups, clarifying the boundaries of such realizations.

Findings

No -realization functor exists under specified conditions for infinite Galois groups.

The result does not contradict existing realizations into pro-spaces or spectra without certain enrichments.

Restrictions are identified on the enrichment of endomorphisms of the unit in these categories.

Abstract

For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the Morel--Voevodsky $\mathbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an \'etale realization functor. For example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$-spaces and the $\mathbb{R}$-realization functor constructed by Morel--Voevodsky. This result does not contradict the existence of \'etale realization functors to (pro-)spaces, (pro-)spectra or complexes of modules with actions of the absolute Galois group when the endomorphisms of the unit is not enriched in a certain sense. It does restrict enrichments to representation rings of Galois groups.