Extensions of pro-pgroups of cohomological dimension two

Abstract

The purpose of this note is to extend Brumer's characterization of pro-pgroups of cohomological dimension two ([1], corollary 5·3) to presentations more general than free ones. The result is then used to rid a proof in [7] of certain field theoretic ingredients. As a by-product we complete a result of Tsvetkov [6] about group extensions obtained by omitting relations.

Free-by-Demushkin pro-p groups

“…1.2] for such Galois groups with 3 (topological) generators the second cohomology with coeficients in F p has dimension 3 over F p and therefore cannot be 1 relator. In fact, later [14,Remark,p. 210] Würfel observed that the answer to his question is affirmative if the natural epimorphism G → G/N splits.…”

Dimension-Preserving Extensions of Pro-p-Groups

“…Therefore, TT is surjective and i 2 is injective. • We are now concerned with the question whether an extension satisfying the conditions of Proposition 1 and where T is finite, splits; if it does, then G is a free pro-/?-product of the form G ~ FII V with F pro-/?-free ( [6], Remark 1). By a theorem of Serre [4] one knows that G contains torsion (if T ^ 1), so the answer is "yes" for T ~ Z / (/?).…”

“…For x G £* n) let M\ denote the Z p -submodule generated by all left-normed commutators [jc^i),... ,*7r(n)] (TT a permutation). By (6), the basic ones among these commutators form a Z p -basis of A/f. If 7 G T and Ix^x, then basic commutators coming from lx or x, respectively, are different.…”

Oriented pro-$\ell$ groups with the Bogomolov-Positselski property

“…which satisfies the following three properties: N is a free pro-ℓ group; Ḡ is a Demushkin group; and for every subgroup S of G containing N , the inflation map (5.1) inf 2 S,N : H 2 (S/N, F ℓ ) −→ H 2 (S, F ℓ ) is an isomorphism (this last property is shown to hold in the proof of [35,Cor. 2]).…”