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 .