We investigate how the étale fundamental group controls local systems in characteristic p, namely F-divided sheaves. In analogy with Grothendieck-Malcev's results for discrete groups, we show that if a morphism f Y X of smooth projective varieties over k=k induces a surjection on the étale fundamental groups, then the pullback functor Fdiv (X) Fdiv (Y) is fully faithful. If f is surjective and the induced map is an isomorphism, then the functor is an equivalence. These results extend the theorem of Esnault-Mehta on the triviality of F-divided sheaves over simply connected varieties.
Sun et al. (Mon,) studied this question.