Frobenius rigidity in ${\mathbb{A}}^1$-homotopy theory

We study the homotopy fixed points under the Frobenius endomorphism on the stable ${\mathbb{A}}^1$-homotopy category of schemes in characteristic $p>0$ and prove a rigidity result for cellular objects in these categories after inverting $p$. As a consequence we determine the analogous fixed points on the $K$-theory of algebraically closed fields in positive characteristic. We also prove a rigidity result for the homotopy fixed points of the partial Frobenius pullback on motivic cohomology groups in weights at most $1$.