On the “Galois closure” for torsors

Garuti, Marco

doi:10.1090/S0002-9939-09-09997-3

On the “Galois closure” for torsors
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by Marco A. Garuti PDF

Proc. Amer. Math. Soc. 137 (2009), 3575-3583 Request permission

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

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