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.
On the "Galois closure" for torsors
GARUTI, MARCO-ANDREA
2009
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.File in questo prodotto:
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