The fundamental groupoid scheme and applications
Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2381-2412.

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.

Nous définissons une structure linéaire sur le groupe fondamental arithmétique π 1 (X,x) d’un schéma X défini sur un corps k de caractéristique 0. Cela nous permet de lier l’existence de sections du groupe de Galois Gal(k ¯/k) vers π 1 (X,x) à l’existence d’un foncteur neutre sur la catégorie qui linéarise ce dernier. Nous appliquons cette construction à une courbe affine et aux foncteurs neutres qui proviennent d’un vecteur tangent à l’infini. Nous pouvons ainsi suivre ce point rationnel dans le revêtement universel de la courbe affine.

Received:
Revised:
Accepted:
DOI: 10.5802/aif.2418
Classification: 14F05,  14L17,  18D10
Keywords: Finite connection, tensor category, tangential fiber functor
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Esnault, Hélène; Hai, Phùng Hô. The fundamental groupoid scheme and applications. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2381-2412. doi : 10.5802/aif.2418. https://aif.centre-mersenne.org/articles/10.5802/aif.2418/

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