On the \lq \lq Galois closure\rq \rq for torsors

Abstract

We show that a tower of torsors under affine group schemes can be dominated by a torsor. Moreover, if the base is the spectrum of a field and the structure group schemes are finite, the tower can be dominated by a finite torsor. As an application, we show that if X is a torsor under a finite group scheme G over a scheme S which has a fundamental group scheme, then X has a fundamental group scheme too and that this group π(X) identifies with the kernel of the map π(S) → G.