Stratified bundles and étale fundamental group

v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ of the étale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

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