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

Adrian Langer; Lei Zhang

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arXiv · 2510.10582

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

Adrian Langer·Lei Zhang

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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.

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BibTeX RIS

Adrian Langer, Lei Zhang. 2025-10-12. $F$-divided bundles on normal $F$-finite schemes. https://arxiv.org/abs/2510.10582

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