Galois scaffolds for p-extensions in characteristic p
[Échafaudages galoisiens pour les p-extensions en caractéristique p]
Elder, G. Griffith1 ; Keating, Kevin2
1 Department of Mathematics University of Nebraska Omaha Omaha, NE 68182 (USA)
2 Department of Mathematics University of Florida Gainesville, FL 32611 (USA)
Annales de l'Institut Fourier, Online first, 27 p.

Soit K un corps local de caractéristique p>0 de corps résiduel parfait et soit G un p-groupe fini. Dans cet article nous utilisons la construction de Saltman d’une G-extension générique d’anneaux de caractéristique p pour construire des G-extensions L/K totalement ramifiées qui ont un échafaudage galoisien. Nous spécialisons cette construction pour produire des G-extensions L/K telles que l’anneau d’entiers 𝒪 L soit libre de rang 1 sur son ordre associé 𝒜 0 , et des extensions telles que 𝒜 0 soit un ordre de Hopf dans l’anneau de groupe K[G].

Let K be a local field of characteristic p>0 with perfect residue field and let G be a finite p-group. In this paper we use Saltman’s construction of a generic G-extension of rings of characteristic p to construct totally ramified G-extensions L/K that have Galois scaffolds. We specialize this construction to produce G-extensions L/K such that the ring of integers 𝒪 L is free of rank 1 over its associated order 𝒜 0 , and extensions such that 𝒜 0 is a Hopf order in the group ring K[G].

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DOI : 10.5802/aif.3712
Classification : 11S15, 11R33, 14L15, 16T05, 11S23
Keywords: generic extensions, ramification, Galois module structure, Galois scaffold, Hopf order
Mots-clés : extension générique, ramification, structure du module galoisien, échafaudage galoisien, ordre de Hopf

Elder, G. Griffith 1 ; Keating, Kevin 2

1 Department of Mathematics University of Nebraska Omaha Omaha, NE 68182 (USA)
2 Department of Mathematics University of Florida Gainesville, FL 32611 (USA) 
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Elder, G. Griffith; Keating, Kevin. Galois scaffolds for $p$-extensions in characteristic $p$. Annales de l'Institut Fourier, Online first, 27 p.

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