On the Galois closure for torsors

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 $\boldsymbol{\pi}(X)$ identifies with the kernel of the map $\boldsymbol{\pi}(S)\to G$.