%0 Journal Article
%T Stratified bundles and ¨¦tale fundamental group
%A H¨¦l¨¨ne Esnault
%A Xiaotao Sun
%J Mathematics
%D 2010
%I arXiv
%X 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 $[\pi_1^{{\rm \acute{e}t}}, \pi_1^{{\rm \acute{e}t}}]$ of the \'etale 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 $ \pi_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.
%U http://arxiv.org/abs/1012.5381v3