On the cohomological dimension of some pro-p-extensions above the cyclotomic Z_p-extension of a number field - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Moscow Mathematical Journal Année : 2013

On the cohomological dimension of some pro-p-extensions above the cyclotomic Z_p-extension of a number field

Résumé

Let $\KST$ be the maximal pro-$p$-extension of the cyclotomic $\Z_p$-extension $\K^{cyc}$ of a number field $\K$, unramified outside the places above $S$ and totally split at the places above $T.$ Let $\GST=\Gal(\KST/\K).$ In this work we adapt the methods developed by Schmidt in \cite{schmidt2} in order to show that the group $\GST=\Gal(\KST/\K)$ is of cohomological dimension $2$ provided the finite set $S$ is well chosen. This group $\GST$ is in fact \textit{mild} in the sense of Labute \cite{labute}. We compute its Euler characteristic, by studying the Galois cohomology groups $H^i(\GST,\fq_p)$, $i=1,2.$ Finally, we provide new situations where the group $\GST$ is a free pro-$p$-group.
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Dates et versions

hal-00854964 , version 1 (28-08-2013)

Identifiants

  • HAL Id : hal-00854964 , version 1

Citer

Julien Blondeau, Philippe Lebacque, Christian Maire. On the cohomological dimension of some pro-p-extensions above the cyclotomic Z_p-extension of a number field. Moscow Mathematical Journal, 2013, 13, pp.601-619. ⟨hal-00854964⟩
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