Stratified bundles and étale fundamental group

Abstract

For a smooth projective variety X over an algebraically closed field of characteristic p > 0, we show that all irreducible stratified bundles on X have rank 1 if and only if the commutator [⇡1, ⇡1] of the ´etale fundamental group ⇡1 of X is a pro-p-group, and we prove that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if ⇡1 is Abelian without p-power quotient. This answers positively a conjecture by Gieseker [4, page 8].