Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

Details Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber

2008-07-15

arxiv.org/abs/0807.2286v3

Journal de th\'eorie des nombres de Bordeaux, 2010

Mathematics

Algebraic Geometry

19 pages, final version

Scientific paper

We show that the natural morphism $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ between the fundamental group scheme of the generic fiber $X_{\eta}$ of a scheme $X$ over a connected Dedekind scheme and the generic fiber of the fundamental group scheme of $X$ is always faithfully flat. As an application we give a necessary and sufficient condition for a finite, dominated pointed $G$-torsor over $X_{\eta}$ to be extended over $X$. We finally provide examples where $\phi:\pi_1(X_{\eta},x_{\eta})\to \pi_1(X,x)_{\eta}$ is an isomorphism..

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