Unstable étale motives

Abstract

We prove a rigidity result for certain $p$-complete étale $\mathbf{A}^{1}$-invariant sheaves of anima over a qcqs finite-dimensional base scheme $S$ of bounded étale cohomological dimension with $p$ invertible on $S$. This generalizes results of Suslin--Voevodsky, Ayoub, Cisinski--Déglise, and Bachmann to the unstable setting. Over a perfect field we exhibit a large class of sheaves to which our main theorem applies, in particular the $p$-completion of the étale sheafification of any $2$-effective $2$-connective motivic space, as well as the $p$-completion of any $4$-connective $\mathbf{A}^{1}$-invariant étale sheaf. We use this rigidity result to prove (a weaker version of) an étale analog of Morel's theorem stating that for a Nisnevich sheaf of abelian groups, strong $\mathbf{A}^{1}$-invariance implies strict $\mathbf{A}^{1}$-invariance. Moreover, this allows us to construct an unstable étale realization functor on $2$-effective $2$-connective motivic spaces.