The fundamental groupoid scheme and applications

Hélène Esnault[1]; Phùng Hô Hai[2]

  • [1] Universität Duisburg-Essen Mathematik 45117 Essen (Germany)
  • [2] Universität Duisburg-Essen Mathematik, 45117 Essen (Germany)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2381-2412
  • ISSN: 0373-0956

Abstract

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We define a linear structure on Grothendieck’s arithmetic fundamental group π 1 ( X , x ) of a scheme X defined over a field k of characteristic 0. It allows us to link the existence of sections of the Galois group Gal ( k ¯ / k ) to π 1 ( X , x ) with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine curve.

How to cite

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Esnault, Hélène, and Hai, Phùng Hô. "The fundamental groupoid scheme and applications." Annales de l’institut Fourier 58.7 (2008): 2381-2412. <http://eudml.org/doc/10382>.

@article{Esnault2008,
abstract = {We define a linear structure on Grothendieck’s arithmetic fundamental group $\pi _1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group $\operatorname\{Gal\}(\bar\{k\}/k)$ to $\pi _1(X, x)$ with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine curve.},
affiliation = {Universität Duisburg-Essen Mathematik 45117 Essen (Germany); Universität Duisburg-Essen Mathematik, 45117 Essen (Germany)},
author = {Esnault, Hélène, Hai, Phùng Hô},
journal = {Annales de l’institut Fourier},
keywords = {Finite connection; tensor category; tangential fiber functor; finite connection},
language = {eng},
number = {7},
pages = {2381-2412},
publisher = {Association des Annales de l’institut Fourier},
title = {The fundamental groupoid scheme and applications},
url = {http://eudml.org/doc/10382},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Esnault, Hélène
AU - Hai, Phùng Hô
TI - The fundamental groupoid scheme and applications
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2381
EP - 2412
AB - We define a linear structure on Grothendieck’s arithmetic fundamental group $\pi _1(X, x)$ of a scheme $X$ defined over a field $k$ of characteristic 0. It allows us to link the existence of sections of the Galois group $\operatorname{Gal}(\bar{k}/k)$ to $\pi _1(X, x)$ with the existence of a neutral fiber functor on the category which linearizes it. We apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine curve.
LA - eng
KW - Finite connection; tensor category; tangential fiber functor; finite connection
UR - http://eudml.org/doc/10382
ER -

References

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  1. P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over (Berkeley, CA, 1987) 16 (1989), 79-297, Springer, New York Zbl0742.14022MR1012168
  2. P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II 87 (1990), 111-195, Birkhäuser, Boston, MA Zbl0727.14010MR1106898
  3. P. Deligne, J. Milne, Tannakian Categories, Lectures Notes in Mathematics, Springer-Verlag 900 (1982), 101-228 Zbl0477.14004
  4. H. Esnault, P.H. Hai, X.-T. Sun, On Nori group scheme, Progress in Math., Birkhäuser 265 (2007), 375-396 
  5. Hélène Esnault, Phùng Hô Hai, The Gauss-Manin connection and Tannaka duality, Int. Math. Res. Not. (2006) Zbl1105.14012MR2211153
  6. A. Grothendieck, Revêtements étales et groupe fondamental, SGA 1, Lectures Notes in Mathematics, Springer-Verlag 224 (1971) MR354651
  7. Alexander Grothendieck, Brief an G. Faltings, Geometric Galois actions, 1 242 (1997), 49-58, Cambridge Univ. Press, Cambridge Zbl0901.14002MR1483106
  8. P.H. Hai, A construction of a quotient category, (2006) 
  9. Nicholas M. Katz, On the calculation of some differential Galois groups, Invent. Math. 87 (1987), 13-61 Zbl0609.12025MR862711
  10. Madhav V. Nori, On the representations of the fundamental group, Compositio Math. 33 (1976), 29-41 Zbl0337.14016MR417179
  11. Madhav V. Nori, The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), 73-122 Zbl0586.14006MR682517

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