$F$-divided bundles on normal $F$-finite schemes

Abstract

In this paper we study $F$-divided bundles on irreducible Noetherian normal $F$-finite $\mathbb{F}_p$-schemes and we show that their Tannakian category is governed by the behaviour at the generic point. In particular, if $U\subset X$ is an open subset of a normal variety defined over an algebraically closed field then the corresponding homomorphism of $F$-divided fundamental groups is faithfully flat. This is analogous to a known fact about the topological fundamental group of an open subset of a normal complex analytic variety. We use this result to show that simply connected, proper, normal varieties in positive characteristic admit no nontrivial $F$-divided bundles. This generalizes an earlier result of H. Esnault and V. Mehta concerning smooth projective varieties, and settles Gieseker's conjecture in a more general setting.