ON THE COHOMOLOGICAL DIMENSION OF SOME PRO-p-EXTENSIONS ABOVE THE CYCLOTOMIC Z_p-EXTENSION OF A NUMBER FIELD

摘要

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