Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

Thanks.

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.

share|cite|improve this question
6  
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 $P_j$ .

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.