The \'etale fundamental group and $F$-divided sheaves in characteristic $p>0$

Abstract

We investigate how the \'etale 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 $f \colon Y \to X$ of smooth projective varieties over $k=\bar{k}$ induces a surjection on the \'etale fundamental groups, then the pullback functor ${\rm Fdiv}(X)\to {\rm Fdiv}(Y)$ 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.