Prime to p fundamental groups and tame Galois actions
Annales de l'Institut Fourier, Volume 50 (2000) no. 4, p. 1099-1126

We show that for a local, discretely valued field F, with residue characteristic p, and a variety 𝒰 over F, the map ρ: Gal (F sep /F) Out (π 1, geom (p ) (𝒰)) to the outer automorphisms of the prime to p geometric étale fundamental group of 𝒰 maps the wild inertia onto a finite image. We show that under favourable conditions ρ depends only on the reduction of 𝒰 modulo a power of the maximal ideal of F. The proofs make use of the theory of logarithmic schemes.

Soit F un corps ayant une valuation complète et discrète, et de caractéristique résiduelle p. Si 𝒰 est une variété sur F, notons π 1, geom (p ) (𝒰) le quotient maximal du groupe étale fondamental de 𝒰 qui est premier à p. Nous considérons l’application ρ: Gal (F sep /F) Out (π 1, geom (p ) (𝒰)) au groupe des automorphismes extérieurs, et nous montrons qu’elle applique le groupe de ramification sauvage sur un groupe fini. Nous montrons que sous certaines conditions ρ dépend seulement de la réduction de 𝒰 modulo une puissance de l’idéal maximal de F. Les preuves utilisent la théorie des schémas logarithmiques.

@article{AIF_2000__50_4_1099_0,
     author = {Kisin, Mark},
     title = {Prime to $p$ fundamental groups and tame Galois actions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     pages = {1099-1126},
     doi = {10.5802/aif.1786},
     zbl = {0961.14014},
     mrnumber = {2001j:14035},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_4_1099_0}
}
Kisin, Mark. Prime to $p$ fundamental groups and tame Galois actions. Annales de l'Institut Fourier, Volume 50 (2000) no. 4, pp. 1099-1126. doi : 10.5802/aif.1786. http://www.numdam.org/item/AIF_2000__50_4_1099_0/

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