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Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber. (English. French summary) Zbl 1267.14029

Summary: We show that the natural morphism \(\phi :\pi _{ 1 }(X_{ \eta },x_{ \eta })\rightarrow \pi _{ 1 }(X,x)_{ \eta }\) 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 \(\phi :\pi _{ 1 }(X_{ \eta },x_{ \eta })\rightarrow \pi _{ 1 }(X,x)_{ \eta }\) is an isomorphism.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14L15 Group schemes
14A20 Generalizations (algebraic spaces, stacks)
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References:

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