Moscow Mathematical Journal

Volume 13, Issue 4, October–December 2013 pp. 601–619.

On the Cohomological Dimension of Some Pro-p-Extensions above the Cyclotomic ℤ_p-Extension of a Number Field

Authors: Julien Blondeau (1), Philippe Lebacque (1), and Christian Maire (1)
Author institution: (1) Laboratoire de Mathématiques, UFR Sciences et Techniques, 16 route de Gray, 25030 Besançon
Summary:

Let K̃_S^T 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 G̃_S^T=Gal(K̃_S^T/K).

In this work we adapt the methods developed by Schmidt in order to show that the group G̃_S^T=Gal(K̃_S^T/K) is of cohomological dimension 2 provided the finite set S is well chosen. This group G̃_S^T is in fact mild in the sense of Labute. We compute its Euler characteristic, by studying the Galois cohomology groups Hⁱ(G̃_S^T,𝔽_p), i=1,2. Finally, we provide new situations where the group G̃_S^T is a free pro-p-group.

2010 Mathematics Subject Classification. 11R34, 11R37

Keywords: Mild pro-p-groups, Galois cohomology, restricted ramification, cyclotomic ℤ_p-extension

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