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On the Galois closure for torsors

Author: Marco A. Garuti
Journal: Proc. Amer. Math. Soc. 137 (2009), 3575-3583
MSC (2000): Primary 14L15, 14F20
Published electronically: June 25, 2009
MathSciNet review: 2529863
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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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Additional Information

Marco A. Garuti
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

Received by editor(s): February 14, 2008
Received by editor(s) in revised form: October 29, 2008
Published electronically: June 25, 2009
Communicated by: Ted Chinburg
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.