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 [π1, π1] of the étale fundamental group π1 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 π1 is an Abelian without p-power quotient. This answers positively a conjecture by Gieseker [4, page 8]