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

Marco Antei[1]

  • [1] Laboratoire Paul Painlevé, U.F.R. de Mathématiques Université des Sciences et des Techonlogies de Lille 1 59 655 Villeneuve d’Ascq, France

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 537-555
  • ISSN: 1246-7405

Abstract

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We show that the natural morphism ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η between the fundamental group scheme of the generic fiber X η 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 η to be extended over X . We finally provide examples where ϕ : π 1 ( X η , x η ) → π 1 ( X , x ) η is an isomorphism.

How to cite

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Antei, Marco. "Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 537-555. <http://eudml.org/doc/116419>.

@article{Antei2010,
abstract = {We show that the natural morphism $\varphi :\pi _1(X_\{\eta \},x_\{\eta \}) \rightarrow \pi _1(X,x)_\{\eta \}$ 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 $\varphi :\pi _1(X_\{\eta \},x_\{\eta \})\rightarrow \pi _1(X,x)_\{\eta \}$ is an isomorphism.},
affiliation = {Laboratoire Paul Painlevé, U.F.R. de Mathématiques Université des Sciences et des Techonlogies de Lille 1 59 655 Villeneuve d’Ascq, France},
author = {Antei, Marco},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {537-555},
publisher = {Université Bordeaux 1},
title = {Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber},
url = {http://eudml.org/doc/116419},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Antei, Marco
TI - Comparison between the fundamental group scheme of a relative scheme and that of its generic fiber
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 537
EP - 555
AB - We show that the natural morphism $\varphi :\pi _1(X_{\eta },x_{\eta }) \rightarrow \pi _1(X,x)_{\eta }$ 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 $\varphi :\pi _1(X_{\eta },x_{\eta })\rightarrow \pi _1(X,x)_{\eta }$ is an isomorphism.
LA - eng
UR - http://eudml.org/doc/116419
ER -

References

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