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I'm reading the first chapter of Serre's Galois Cohomology.

On p. 58, He gives two examples of projective profinite groups:

  1. the profinite completion of free (discrete) groups;
  2. the cartesian product over all prime numbers of free pro-p groups.

Looking for other examples I understood solvable projective profinite groups.

Mimicking the construction of projective covers for modules, I found the concept of the projective cover of a profinite group.

Now I'm stuck!

Any suggestions of more examples of projective profinite groups?


EDIT: Actually I'd like to see finitely generated profinite groups. So, in other words, I want to know the finitely generated, closed subgroups of free profinite groups.

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Interesting question. I would give another +1, for "He"! – Louis Burkill Apr 12 '12 at 22:41

I take it that by « projective » you mean "cohomological dimension $\le 1$". Here are some examples from number theory and arithmetic geometry :

1). Let $K$ be an extension of degree $n$ of the field $\mathbf Q_p$ of $p$-adic numbers. Let $\mathcal G$ be the Galois group of the maximal pro-$p$-extension of $K$. If $K$ does not contain a primitive $p$-th root of 1, then $\mathcal G$ is pro-$p$-free on $(n + 1)$ generators. This is a theorem of Shafarevitch. Note that if $K$ contains a primitive $p$-th root of 1, then $\mathcal G$ is a so called Demushkin pro-$p$-group (with $(n + 2)$ generators and one explicit relation)

2). Over a number field much less is known, if not conjectures. Let ‘s call a field « projective » if its absolute Galois group is projective. In the literature on the inverse Galois problem, there is a certain amount of conjectures/results on the minimal projective subfields of $\mathbf Q^{ab}$.The motivation comes from a celebrated conjecture of Shafarevitch, which predicts that the absolute Galois group of $\mathbf Q^{ab}$ should be profree on a countable basis. It follows from the Kronecker-Weber theorem that the field $\mathbf Q^{ab}$ is projective, and it is not difficult to construct strict projective subfields of $\mathbf Q^{ab}$. It is then natural to study and classify minimal (for the inclusion) such objects. Let $\mathcal Z$ be the unique $\hat {\mathbf Z} $- extension of $\mathbf Q$, i.e. the compositum of all the (cyclotomic) $\mathbf Z_p$- extensions of $\mathbf Q$. Since $\mathcal Z$ is totally real, its absolute Galois group contains "the" complex conjugation, hence cannot have finite cohomological dimension, and $\mathcal Z$ cannot be projective. Some recent results (see e.g. Bruno Deschamps, J. of Algebra, 441 (2015), 1-20) :

  • . a subfield $\Omega$ of $\mathbf Q^{ab}$ is minimal projective if and only if $\Omega$ is equal to the compositum of 𝒵 and a 2-cyclic extension of $\mathbf Q$, if and only if $\Omega$ is minimal for the two following properties : it is totally imaginary and contains $\mathcal Z$ . More generally, an algebraic extension of $\mathcal Z$ (with no further hypothesis) is projective if and only if it is totally imaginary

    • . the projective subfields of $\mathbf Q^{ab}$ which are not extensions of any minimal projective subextension are called « abyssal ». The best known result is that a projective abelian field which does not contain $\mathcal Z$ is necessarily abyssal. Infinite families of such fields have been constructed

3). Let $X$ be a smooth irreducible projective curve over $\mathbf C$ (or more generally over an algebraically closed field of characteristic 0). Let $F$ be the function field of $X$ and $S$ be a finite set of points of $X$ consisting of $s$ points $P_j$ and the point at infinity. The Galois group over $F$ of the maximal extension of $F$ unramified outside $S$ is isomorphic to the algebraic fundamental group $\Pi_1{(X-S)}$, and Riemann’s existence theorem implies that this algebraic $\Pi_1$ is the profinite completion of the topological $\Pi_1$. In particular, if $X$ has genus 0, it’s the pro-free product of $s$ copies of $\hat {\mathbf Z} $, which are the inertia subgroups at the points $P_j$ .

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