Fields Institute Communications Volume 00, 0000

Classically projective groups and pseudo classically closed fields

Florian Pop

Universität Bonn

                                 pop@math.uni-bonn.de

This is the content of the preprint with the same title published in May 1990 in the Preprint Reihe Mathematik of the Mathematics Institute, University of Heidelberg. The present variant is almost identical with the one from 1990. The main change is a correction in proof of Lemma 2.4, which was pointed out to me by Matthias Flach. I would also like to give my special thanks to P. Roquette and A. Prestel for several discussions we have had on this subject since the publication of the preprint. Finally, I would like to thank various people who showed interest in this preprint, among others B. Green, Z. Chatzidakis, F. Delon, Ido Efrat, D. Harbater, Yu. Ershov, G. Frey, D.-W. Geyer, Dan Haran, M. Jarden, E. Kani, J. Koenigsmann, F.-V. Kuhlmann, A. Macintyre, L. Moret-Bailly, L. Ribes, J. Schmidt, and many others. Concerning the present manuscript, on the one hand I have tried to take into account as many of their correction suggestions as possible; but on the other hand I did not want to change (too much) the concept of the original preprint version. The proofs are quite sketchy, and the motivation for the (new) concepts is mostly not to be found in the text. (Nevertheless, for the expert, who is familiar with previous work on PAC, PRC, and PpC fields and their Galois theory, all these facts might be quite enlightening...) As for the 10^{th} anniversary of the 1990 Preprint, I am glad to say that the preprint was at least in a couple of cases inspiring... Detailed proofs and explanations are underway.

Introduction

Let K be a PAC (pseudo algebraically closed) field. Then the absolute Galois group G_{K} of K is projective by Ax [A], i.e., group extensions of G_{K} by a finite groups H

1 → H → F ⟶^{π} G_{K} → 1

are split: There exists σ : G_{K} → F such that πσ = id_{G_{K}}. On the other hand, Gruenberg [G] showed that for a profinite projective group G, all group extensions of G by profinite groups H are split. Using Gruenberg’s theorem Lubotzky – van den Dries [L–vdD] solved the inverse absolute Galois problem for projective groups as follows:

2000 Mathematics Subject Classification. Primary 11, 12, 14; Secondary 11: G, S20, U09; 12: D, E20, F, L; 14: E, G.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

For projective profinte groups G there exist P AC fields K with G_{K} ≅ G.

Extending the category of PAC fields, one introduced the PRC (pseudo real closed) fields, the PpC (pseudo p-adically closed) fields and further, fields regularly closed with respect to finitely many henselisations and orderings, see Prestel [Pr2], Ershov [E], Jarden [J], Heinemann–Prestel [H–P], and to some extent studied their properties. The corresponding group theoretical side had its turn only after understanding what the (group theoretical) abstract characterization of the absolute Galois groups of such fields might be. In this process one was led to the concept of real, respectively p-adically projective groups, the absolute Galois group of PRC, respectively PpC field being real, respectively p-adically projective, see Basarab [Ba], [E] and Haran–Jarden [H–J1], [H–J2]. To prove the converse of this assertion, i.e., to solve the inverse absolute Galois problem for real, respectively p-adically projective groups, one has had to work much harder. After some partial results in [E] and [J], the full result was obtained by Haran–Jarden loc. cit.. This was done by developing some sophisticated concepts, namely the Artin– Schreier structures in the real case, respectively the Γ-structures in the p-adic case. See also Basarab loc.cit..

Adopting a different point of view, in the present paper we make an attempt to lay the foundations by which the kind of problems discussed above could be uniformly treated. In Ch 1 we begin by studying what we have called classically projective groups, which is a special class of relatively projective groups. They generalize the real, resp. p-adically projective groups in a natural and non-trivial way. A classically projective group is a profinite group G endowed with a family G of classical subgroups which, in some sense, contains all the obstructions for the splitting of all group extensions of G by finite groups. We show that the classically projective groups are actually strongly relatively projective. This is Theorem 1.2. For the proof we use the results by Herfort–Ribes [H–R], and a characterization of ”big” pro-solvable subgroups of usual profinite free products, see Pop [P2]. Further, dealing with strongly relatively projective groups we first prove the Separating Theorem 1.9. Finally, we give a good enough generalization of Gruenberg’s Theorem which is our Theorem 1.11. To obtain a similar result in the case of real, resp. p-adically projective groups, Haran–Jarden developed in loc. cit. their theory of Artin-Schreier, respectively Γ-structures. Here we take a direct approach. One should also remark that in order to make the machinery developed by Haran–Jarden work, one first needs to prove that the family of obstructions G of the real, resp. p-adically projective group in discussion has a compact fundamental domain with respect to conjugation. This is true for real, resp. p-adically projective groups and also for separably generated groups, see loc. cit. and Haran [H1], but seems not to be generally true. In Ch 2 we introduce and study the so called PCC (pseudo classically closed) fields which include the PAC, PRC and PpC fields. We first review some basic facts about the classical spectra of a field, and we prove mainly that they have good functorial and topological behavior, see Proposition 2.1, and Theorems 2.5, Theorem 2.7, as well as their corollaries. See also [Be], [B–S], [C–R], [Ro] and others. After discussing generalities about the pseudo closed fields, like the Density and Uniqueness Theorem 2.9 and the Overfield Theorem 2.12, in the last paragraph we use an idea from van den Dries [vdD] in order to construct pseudo closed fields starting with some “initial conditions”, see Theorem 2.15. Similar constructions have been realized in the real and p-adic case, but using the very specific situation. See Haran–Jarden [H–J1], [H–J2] and others. Finally, in Ch 3 we are concerned with the relationship between the relatively projective groups and the pseudo closed fields. Here we prove the main result of the paper, by showing that the inverse absolute Galois problem for classically projective groups has a solution in the category of pseudo classically closed fields:

A profinite group is classically projective if and only if it is isomorphic to the absolute Galois group of a pseudo classically closed field.

By Theorem 3.3, the assertion in one direction has a very general character. Namely, the absolute Galois group G_{K} of any pseudo closed field K is relatively projective. (But we cannot prove that G_{K} is strongly relatively projective, and it seems it is not always the case.) The opposite direction, namely the realization of a Gprojective group G as an absolute Galois group of a pseudo closed field is a much more difficult question. For such a realization some obvious restrictions on the family of obstructions G are necessary from the start like for instance: all finite subgroups in G must be isomorphic to G_{ℝ}, and the set all such groups must be τ^{st}-closed (why?) !... But near this kind of obstructions, which are more or less obvious, there exist ones of a much deeper and not yet understood nature, namely those making a profinite group into an absolute Galois group. Theorem 3.4 solves the inverse absolute Galois problem for a strongly relatively projective group if some initial Galois approximation does exist. Applying this and using Theorem 1.6, we get immediately the absolute Galois realization of classically projective groups.

For definitions and facts concerning boolean and quasi-boolean spaces, the étale topology τ^{et} and the strict topology τ^{st} on spaces of subgroups or subfields, and for questions concerning the functors “subgroups” sg( ), respectively “subfields” sf( ), the reader is referred to the Appendix.

1 Relatively projective groups

A) Classically projective groups

A classical field is any finite extension of ℝ or of some ℚ_{p}. A classical Galois group 𝔾 is any profinite group which is isomorphic to the absolute Galois group of a classical field. We say that 𝔾 is real if 𝔾 ≅ G_{ℝ}, respectively p-adic if it is isomorphic to an open subgroup of G_{ℚ_{p}} for some prime number p. It is well known that all classical Galois groups 𝔾 have finite corank, i.e., 𝔾 has only finitely many open subgroups of index ≤ n for every positive bound n.

Definition Let G be a profinite group.

• A subset of closed subgroups G = G^{G} ⊆ sg(G) is called simple classical if it is τ^{st}-closed and consists only of subgroup which are isomorphic to a given classical Galois group 𝔾.

• A subset G ⊆ sg(G) is called classical if it is finite union of simple classical subsets of sg(G).

Taking into account that a classical Galois group is not isomorphic to any of its proper subgroups (see for instance the discussion in the proof of Theorem 1.6), it follows by Appendix, 4.4, that any simple classical subset of sg(G) is τ^{et}-compact. Further, any classical family of subgroups of G is τ^{st}-closed. Let G be an arbitrary profinite group endowed with a subset G ⊆ sg(G). Let further

(ϵ) 1 → H ⟶ F ⟶^{π} G → 1

be a group extension of G by a profinite group H. We say that (ϵ) splits locally, if for all Γ ∈ G, the corresponding ”local extensions”

(ϵ)_{Γ} 1 → H ⟶ π^{−1}(Γ) ⟶^{π} Γ → 1

are split, i.e., there exist (group theoretical) sections of π over each Γ ∈ G. We say that (ϵ) splits globally if (ϵ) is split, i.e., there exists a right inverse σ of π.

A local splitting of (ϵ) is a subset F ⊆ sg(G) which is mapped by π onto G and such that π is injective on all Φ ∈ F. Obviously, (ϵ) splits locally if and only if (ϵ) has local splittings. A global splitting of (ϵ) is any right inverse of π.

Definition Let G be an arbitrary profinite group, and G ⊆ sg(G) a subset.

• We say that G is G-projective if any group extension of G by a finite group, which splits locally, splits globally.

• We say that G is classically projective if there exists a classical family of subgroups G of G such that G is G-projective.

Our next task is to prove that classically projective groups is actually strongly relatively projective.

Definition Let G be an arbitrary profinite group endowed with a family of subgroups G ⊆ sg(G). We say that G is strongly G-projective if for any group extension (ϵ) of G by a finite group as above one has: For every local splitting F of (ϵ) and any τ^{et}-open quasi-compact neighborhood U of con(F) in sg(F) (see Appendix, 4.10, 1)–3) for definitions), there exists a global splitting σ of π with σ(G) ⊆ U.

Intuitively, if G is strongly G-projective, then any local splitting F of any (ϵ) can be better and better approximated by global splittings; or equivalently, the family of all global splittings is dense in the set of all local splittings.

Alternatively, one can define the relative projectivity by means of the so called finite embedding problems.

Definition An embedding problem EP = (γ, α, B) for (G, G) consists of:

i) A diagram of homomorphisms of profinite groups γ : G → A, α : B → A, such that α is surjective.

ii) A quasi τ^{st}-compact subset B ⊆ sg(B) such that every γ|_{Γ} with Γ ∈ G factorizes through α|_{Δ} for some Δ ∈ B. Equivalently, for every Γ there exists Δ ∈ B and β_{Γ} : Γ → Δ such that γ|_{Γ} = α β_{Γ}.

An embedding problem EP as above is called finite if B is a finite group. A solution of EP is a group homomorphism β : G → B with αβ = γ. A strong solution of EP is a solution β satisfying β(G) ⊆ con(B); see Appendix, 4.10, 1)–3) for definitions.

Fact A straightforward verification using fiber products shows: G is (strongly) G-projective if and only if every finite embedding problem for (G, G) has (strong) solutions.

For two embeddings problems EP and EP^{0} we say that EP ≺ EP^{0} if there exists a commutative diagram of the form

B^{0} ⟶^{α^{0}} A^{0}
pr^{B} ↓  ↓^{pr}^{B}
B ⟶^{α} A

with pr_{B}(B^{0}) ⊆ con(B). It is obvious that the relation ≺ defined above is a preorder on the class of all embedding problems for (G, G). Furthermore, if β^{0} is a (strong) solution of EP^{0}, then pr_{B} ∘ β^{0} is a (strong) solution for EP. This observation is in particular very useful when G is quasi τ^{et}-compact (or τ^{st}- closed). Let namely EP be a finite embedding problem for (G, G). Then using fiber products, one easily checks that there exists an open normal subgroup D_{EP} of G with the property: For any normal subgroup D^{0} ⊆ D_{EP} of G, there exists a finite embedding problem EP^{0} = (γ^{0}, α^{0}, B^{0}) for (G, G) with EP ≺ EP^{0} and having the properties

1) γ^{0} is the canonical projection G → G/D^{0}.

2) α^{0} is injective on each Δ^{0} ∈ B^{0}.

Thus this allows to restrict the set of embedding problems we have to check in order show/check that (G, G) is (strongly) relatively projective. Summarizing we have:

Fact 1.1 Let (G, G) be given, and suppose that G is quasi τ^{et}-compact. Then G is (strongly) G-projective if and only if every finite EP = (γ, α, B) for (G, G) has (strong) solutions, provided

1) γ is the canonical projection G → G/ker(γ).

2) α is injective on each Δ ∈ B.

An embedding problem as above is called a special embedding problem.

Theorem 1.2 Let G be relatively projective with respect to a classical family G of subgroups. Then G is strongly G-projective.

Proof We begin with the following general fact, compare Haran–Jarden [H–J2], Section 4.4.

Lemma 1.3 Let G be an arbitrary profinite group, and X = (𝔾_{k})_{k} a finite family of profinite groups of finite corank. For each k let G_{k} ⊆ sg(G) consist of subgroups Γ ≅ 𝔾_{k}, and set G = ∪_{k}G_{k}. Suppose that G is G-projective.

Then any embedding problem EP = (γ, α, B) for (G, G) has solutions, provided A is finite and B is separable as a topological space.

Proof The proof is almost identical with that in loc. cit., therefore we only give a sketch here. Let namely for each Γ ∈ G_{k} consider a fixed isomorphism τ_{Γ} : 𝔾_{k} → Γ. Then giving an embedding problem for (G, G) amounts to giving for each k and Γ ∈ G_{k} a homomorphism ρ_{Γ} : 𝔾_{k} → B such that γ τ_{Γ} = α ρ_{Γ}.

Next let {C_{n}}_{n} be a decreasing sequence of open subgroups of B satisfying the conditions ∩_{n}C_{n} = {1} and C_{0} = ker(α). For n > 0 consider the canonical pro-

jections B ⟶^{α_{n}} B/C_{n}, B/C_{n} ⟶^{α^{n}} A, and B/C_{n} ⟶^{δ^{n}} B/C_{n−1}, and set B_{n} = α_{n}(B).

Then EP_{n} = (γ, α_{n}, B_{n}) is a finite embedding problem for (G, G). For every solution β of EP_{n}, we set D_{k,β} = {β ∘ τ_{Γ} | Γ ∈ G_{k}}. Then D_{k,β} is a subset of the finite set Hom(𝔾_{k}, B/C_{n}). Let T_{k,n} denote the set of all D_{k,β} for β solution of EP_{n}. Obviously, δ^{n}(T_{k,n}) ⊆ T_{k,n−1}. Therefore, denoting T_{n} = ∏_{k} T_{k,n} we get a projective

system of finite sets ⋯ ⟶^{δ^{3}} T_{2} ⟶^{δ^{2}} T_{1} ⟶^{δ^{1}} T_{0}. For any (D_{k,β_{n}})_{k,n} ∈ \underleftarrow{lim} T_{n} one has δ^{n+1}(D_{k,β_{n+1}}) = D_{k,β_{n}}. Hence, for every Γ ∈ G_{k}, there exists Γ_{n} ∈ G_{k} with Γ_{1} = Γ and δ^{n+1}β_{n+1}τ_{Γ_{n+1}} = β_{n} τ_{Γ_{n}} for all n. Therefore, (β_{n} ∘ τ_{Γ_{n}} : 𝔾_{k} → B/C_{n})_{n} is a family of group homomorphisms which is compatible with the canonical projections δ^{n}. Thus there exists a homomorphism ρ_{Γ}^{1} : 𝔾_{k} → B such that α_{n} ρ_{Γ}^{1} = β_{n} ∘ τ_{Γ} for all n. In particular, for n = 1 we get α_{1} ρ_{Γ}^{1} = β_{1} ∘ τ_{Γ}. This being true for every Γ ∈ G, it follows that the family (ρ_{Γ}^{1})_{Γ∈G} gives rise to an embedding problem EP^{1} = (β_{1}, α_{1}, B^{1}) for (G, G). One proceeds by induction on n and obtains a sequence of group homomorphisms β_{n} : G → B/C_{n} which satisfy β_{n} = δ^{n+1}β_{n+1}. Hence, β = \underleftarrow{lim} β_{n} is a solution of EP.

The next ingredient in our proof is also of general nature:

Lemma 1.4 Let G be an arbitrary profinite group of finite corank having the ℓ-cohomological dimension cd_{ℓ}G ≥ 2. Then there exists a finite quotient \overline{G} of G with the property: If G̃ is any quotient of G which itself admits \overline{G} as quotient, then cd_{ℓ}G̃ ≥ 2.

A quotient \overline{G} of G with the property above is called an ℓ-big quotient of G.

Proof Since cd_{ℓ}G ≥ 2 there exists an open normal subgroup D of G and a finite ℓ-torsion G/D-module M such that denoting π_{D} : G → G/D the canonical projection, the inflation map Inf(π_{D}) : H^{2}(G/D, M) → H^{2}(G, M) is non-trivial. Let D_{0} be the intersection of all open subgroups of G of index ≤ (G : D). We claim

that \overline{G} = G/D_{0} does the job. Indeed, let ϕ : G \underrightarrow{ϕ̃} G̃ ⟶^{\overline{ϕ}} \overline{G} be surjective group homomorphisms. By the choice of D_{0}, it follows that any surjective homomorphism π_{D} : G → G/D factor through some surjective homomorphism \overline{G} → G/D. Hence, Hom(G, G/D) and Hom(\overline{G}, G/D) have the same cardinality. On the other hand, since ϕ is surjective, the last set has the same cardinality as Hom(G, G/D) ∘ ϕ. Therefore, the canonical projection π_{D} can be written π_{D} = ϕ_{D} ϕ for some surjective ϕ_{D} : \overline{G} → G/D and so, π_{D} = ϕ_{D} \overline{ϕ} ϕ̃. Now, since Inf(π_{D}) is non-trivial, it follows that the inflation map H^{2}(G/D, M) → H^{2}(G̃, M) defined via ϕ_{D} \overline{ϕ} is non-trivial. Thus cd_{ℓ}G̃ ≥ 2.

We now come to the proof of Theorem 1.2. Set G = ∪_{k}G_{k} with G_{k} = G_{𝔾_{k}}, the simple classical subsets of 𝔾. By a theorem of Tate, see e.g., [Ri] or [Se], one has: cd_{ℓ}𝔾 = 2 for all non-real 𝔾 and all prime numbers ℓ. For each k let us set:

i) \overline{𝔾}_{k} ≅ G_{ℝ}, if 𝔾_{k} is real.

ii) \overline{𝔾}_{k} a fixed ℓ-big quotient of 𝔾_{k} for ℓ in a fixed finite set of prime numbers

X_{k}, if 𝔾_{k} is non-real. Let γ : G → G/D be the canonical projection, and γ(G) and Γ/D the images of G, respectively Γ ∈ G by γ. For every Γ ∈ G_{k}, let D_{Γ} be an open normal subgroup of G such that Γ/D_{Γ} has \overline{𝔾}_{k} as quotient, and let V_{D_{Γ}}(Γ) be the τ^{st}-neighborhood of Γ defined by the D_{Γ}. Then, if Δ ∈ V_{D_{Γ}}(Γ), we have Δ/D_{Γ} = Γ/D_{Γ}, thus Δ/D_{Γ} has \overline{𝔾}_{k} as quotient too. Using this and taking into account that G_{k} is τ^{st}-compact, it follows that there exist open normal subgroups D_{k} of G such that Γ/D_{k} has \overline{𝔾}_{k} as quotient for all Γ ∈ G_{k}. Finally, for any open normal subgroup D ⊂ ∩_{k}D_{k} of G we obtain: Γ/D has \overline{𝔾}_{k} as quotient for all k and Γ ∈ G_{k}. Furthermore, since all G_{k} are τ^{st}-closed, after replacing D by a small enough open normal subgroup of G, we can also suppose that for all Γ, Δ ∈ G we have: If Γ/D = Δ/D then Γ, Δ lie in the same simple set G_{k}.

For each Γ/D ∈ γ(G_{k}), let Φ_{ϕ_{k}} be isomorphic copies of 𝔾_{k} indexed by the finite family of all homomorphisms ϕ_{k} : 𝔾_{k} → Γ/D. Let F_{|G/D|} be the profinite free product on the adjacent set |G/D| of the finite group G/D. Further we set B_{D} = F_{|G/D|} ∗ ∗_{k,ϕ_{k}} Φ_{ϕ_{k}} for the corresponding profinite free product of (profinite)

groups. We define

α_{D} : B_{D} → G/D by |g| ↦ g, α|_{Φ_{ϕ_{k}}} = ϕ_{k}

It is clear that if Δ/D = Γ/D, then the canonical projection γ|_{Δ} factories through α|_{Φ_{ϕ_{k}}} for some ϕ_{k}. Hence, setting B_{D} = {Φ_{ϕ_{k}}}_{k,ϕ_{k}}, we get an embedding problem for (G, G) as follows

1.5 EP_{D} = (γ, α_{D}, B_{D})

The embedding problem EP_{D} has the following universal property: If EP is any special embedding problem for (G, G) having ker(γ) = D, then EP ≺ EP_{D}. Therefore, in order to show that all finite embedding problems for (G, G) has (strong) solutions, it is sufficient to show that all embedding problems of the form EP_{D} have (strong) solutions.

On the other hand, B_{D} is separable. Hence by Lemma 1.3 above, EP_{D} has solutions β. We now show that any such solution β is actually a strong solution. For Γ ∈ G_{k}, set Γ̃ = β(Γ). Then Γ̃ is a quotient of 𝔾_{k}, and α(Γ̃) = Γ/D. Hence, by the definition of D we have:

1) Γ̃ is real if 𝔾_{k} is so.

2) cd_{ℓ}Γ̃ ≥ 2 for ℓ ∈ X_{k}, if 𝔾_{k} is non-real. Now suppose that for non-real 𝔾_{k}, the set X_{k} contains at least 2 primes. Then by Herfort–Ribes [H–R] in the first case, and by Pop [P2] in the second case, it follows that Γ̃ is contained in a conjugate of some Φ_{ϕ_{l}}. Therefore, β is a strong solution of EP_{D}.

We finish these considerations by the following theorem, which is used for the realization of classically projective groups as absolute Galois groups of pseudo classically closed fields.

Theorem 1.6 Let G be relatively projective with respect to a classical family G of subgroups of G. Let G_{k} = G_{𝔾_{k}} denote the simple components of G. Then there exists a homomorphism

φ : G → ∗_{k} 𝔾_{k}

which maps every Γ ∈ G injectively into some conjugate of one of the groups 𝔾_{k} for some k (which is uniquely determined).

Proof The proof is actually only an elaboration of what we have done above. First, let 𝔾 ⊆ G_{ℚ_{p}} be any p-adic classical Galois group. Then the Sylow p-group of 𝔾 is not finitely generated, contrary to the q-Sylow groups of 𝔾, (q ≠ p), which are generated by two elements, see [Se], II. Therefore, there exists a finite quotient, \overline{𝔾} of 𝔾 with the property: If a classical Galois group 𝔾^{1} has \overline{𝔾} as quotient, then 𝔾^{1} ⊆ G_{ℚ_{P}} too.

Second, by results of Demuskin and refinements by Poitou (see for instance loc. cit. and [N]), for any classical Galois group 𝔾 ⊆ G_{ℚ_{p}}, its index in G_{ℚ_{p}} is encoded in the structure of 𝔾 and can be described as follows. For an arbitrary subgroup G of G_{ℚ_{p}}, let G(p) denote its maximal pro-p-quotient. Set δ^{1}(G) = dim H^{1}(G(p), 𝔽_{p}) the minimal number of generators of G(p) (which can be infinite). Further we set δ^{2}(G) = dim H^{2}(G(p), 𝔽_{p}) the minimal number of relations defining G(p) as quotient of a profinite free pro-p-group. If cd_{p}G = 2, then δ^{2}(G) is either 1 or 0, corresponding to the fact that the fixed field of G does or does not contain the p-roots of unity. In both cases d(G) = δ^{1}(G)−δ^{2}(G)−1 equals the index (G_{ℚ_{p}} : G) of G in G_{ℚ_{p}}. In particular, if cd_{p}G = 2, then d is finite if and only if G is open.

Combining these two observations, for every k we can choose \overline{𝔾}_{k} such that with the notation from ii) above the following holds: If 𝔾_{k} ⊆ G_{ℚ_{p}}, then X_{k} also contains p and furthermore, if some Γ has \overline{𝔾}_{k} as quotient then Γ is p-adic.

Let β be any solution of the embedding problem EP_{D} from the proof of Theorem 1.2. Then Γ̃ = β(Γ) is necessarily a classical Galois group. Indeed, let Γ ∈ G_{𝔾_{k}} and Γ̃ be contained in a conjugate of some Φ_{ϕ_{l}} ≅ 𝔾_{l}. Then by the choice of \overline{𝔾}_{l}, \overline{𝔾}_{k}, and of D, it follows that both 𝔾_{l} and 𝔾_{k} are contained in the same G_{ℚ_{p}}. Since p lies in X_{k}, we get cd_{p}Γ̃ ≥ 2. On the other hand, since Γ̃ is a homomorphic image of Γ, one has δ^{1}(Γ) ≥ δ^{1}((Γ̃), and so, d(Γ̃) ≤ d(Γ) + 1. Therefore, d(Γ̃) is finite, and so Γ̃ is isomorphic to an open subgroup of G_{ℚ_{p}}. It remains to show that for a proper choice of the big quotients \overline{𝔾}_{k}, the groups Γ̃ and Γ are actually isomorphic. As we have already remarked, one always has d(Γ̃) ≤ d(Γ) + 1.

Let 𝔾 = 𝔾_{k} be fixed. It is clear that there exist only finitely many isomorphy classes of subgroups G of G_{ℚ_{p}} with d(G) ≤ d(𝔾) + 1.

Claim. There exists a quotient \overline{𝔾} with the property: If G is an open subgroup of G_{ℚ_{p}} such that d(G) ≤ d(𝔾) + 1, G is a quotient of 𝔾, and \overline{𝔾} also is a quotient of G, then 𝔾 and G are isomorphic.

Proof (of the Claim) It is sufficient to prove the claim for any particular G. Suppose that all finite quotients \overline{𝔾} are also quotients of G. Since G and 𝔾 have finite corank it follows that 𝔾 itself is a quotient of G. But then we get surjective homomorphisms 𝔾 → G → 𝔾. Since 𝔾 has finite rank the above maps must be isomorphisms.

Coming back to the proof of Theorem 1.6, we remark that for the above choice of X_{k} and \overline{𝔾}_{k} for all k, it is clear that β is injective on each Γ. To conclude the proof of Theorem 1.6, we remark that with B_{D} as above there exist group homomorphisms B_{D} → ∗_{k} 𝔾_{k} which map all Φ_{ϕ_{k}} isomorphically onto 𝔾_{k} and are trivial on F_{|G/D|}.

B) Strongly relatively projective groups

Convention. Unless explicitly stated, when speaking about a strongly G-projective group G we will always assume that G is a standard G-subspace of sg(G), i.e., G is quasi τ^{et}-compact, G-invariant and G = G_{max}.

Fact 1.7 Let G be strongly G-projective. Then every special embedding problem EP = (γ, α, β) with A finite has strong solutions, i.e., there exists β : G → B with β(G) ⊆ con(B).

Proof For each γ(Γ) and a ∈ A, choose pre-images Δ ∈ con(B) and b ∈ B. The closed subgroup B^{0} of B generated by these Δ and b is finitely generated, hence separable. Moreover, setting B^{0} = con(B) ∩ sg(B^{0}) we get a special embedding problem EP^{0} = (γ, α|_{B^{0}}, B^{0}) for (G, G) satisfying EP ≺ EP^{0}. Hence we can suppose that B is separable. Let {C_{n}}_{n} be a decreasing sequence of open normal subgroups of B with ∩_{n}C_{n} = {1} and C_{0} = ker(α). Let B ⟶^{α_{n}} B/C_{n} ⟶^{δ^{n}} B/C_{n−1} denote the canonical projections. Then (γ, δ^{1}, α_{1}(B)) is an embedding problem for (G, G). Let β_{1} be any strong solution of it. Then β_{1}(G) ⊆ α_{1}(con(B)). Hence, for any β_{1}(Γ) there exists Δ ∈ con(B) such that α_{1}(Δ) = β_{1}(Γ). Let B^{1} ⊆ con(B) be the (finite) collection of all such Δ. Then EP^{1} = (α_{1}, β_{1}, B^{1}) is an embedding problem for (G, G) and α_{1} is injective on all Δ ∈ B^{1}. Now, starting with EP^{1} we take any strong solution β^{2} of (β_{1}, δ^{2}, α_{2}(B^{1})) and so on... Inductively we get a sequence of homomorphisms {β_{n}}_{n} with the properties:

β_{n} : G → B/C_{n} β_{n} = δ^{n+1}β_{n+1}, β_{n}(G) ⊆ α_{n}(con(B)).

It is now obvious that β = \underleftarrow{lim} β_{n} : G → B has the properties we want.

Fact 1.8 Let G be strongly G-projective. For an arbitrary open normal subgroup D of G let γ : G → G/D denote the canonical projection, and let γ(G), Γ/D be the images of G, respectively Γ ∈ G by γ. Let ϕ_{Γ/D} : Δ_{Γ/D} :→ Γ/D be an isomorphic copy of Γ/D, and let B_{D} be the set of all Δ_{Γ/D}. Now consider the usual profinite free product B_{D} = F_{|G/D|} ∗ ∗_{Δ_{Γ/D}} Δ_{Γ/D} and the “canonical projection”

α_{D} : B_{D} → G/D by |g_{D}| ↦ g_{D}, α|_{Δ_{Γ/D}} = ϕ_{Γ/D}

Then α_{D} maps B_{D} bijectively onto γ(G), and it is injective on all Δ_{Γ/D} ∈ B_{D}. Therefore, EP_{D} = (γ, α_{D}, B_{D}) is an embedding problem for (G, G) which satisfies the hypothesis of Fact 1.7. Hence, EP_{D} has strong solutions.

Interesting corollaries of Fact 1.8 are the Separating Theorem 1.9 and the Subgroup Theorem 1.10, see also Haran [H1], Theorem 5.1 for the last assertion. For the proofs we use properties of the usual profinite free products, like the results of Herfort–Ribes [H–R], and Binz–Neukirch–Wenzel [B–N–W].

Theorem 1.9 (Separating Theorem). Let G be strongly G-projective. Then the following holds:

(1) Any distinct elements of G have trivial intersection. Hence the residues set of G consists only of {1} and so, G is separating.

(2) Any Γ ∈ G is self-normalizing, i.e., if Γ^{g} = Γ for some g ∈ G, then g ∈ Γ.

Proof . To (1): Let Γ_{1}, Γ_{2} be elements of G and suppose that there exists g ≠ 1 in their intersection. For every open normal subgroup D of G not containing g let β be a strong solution of the embedding problem EP_{D} considered at Fact 1.8. Since αβ(g) = γ(g) ≠ 1, it follows that β(Γ_{1}) ∩ β(γ_{2}) contains β(g) ≠ 1. On the other hand, β(G) ⊆ con(B). Hence by [H–R] we have: β(Γ_{1}), β(Γ_{2}) are contained in the same maximal element Δ of con(B). Hence γ(Γ_{1}) and γ(Γ_{2}) are both contained in some γ(Γ) = α(Δ). Since D was arbitrary, it follows that there exists some Γ ∈ G containing both Γ_{1} and Γ_{2}. Hence Γ_{1} = Γ_{2}.

The proof of (2) follows the same pattern.

Remark 1) Let G be strongly G-projective. From the theorem above it follows that G is a posteriori separating in the sense of Haran [H1], Definition 3.1. Hence, our notion of strong relative projectivity coincides with the relative projectivity from [H1]. Therefore, the supplementary conditions imposed by the definition in [H1] on G are redundant.

2) Let G be strongly G-projective and suppose that {1} does not lie in the τ^{st}- closure of G. Then by Appendix, Fact 4.4 and 4.5, it follows that G is τ^{et}-compact. In particular, this is the case for G_{max}, where G is a classical family of subgroups of G.

Without proof we mention here the Subgroup Theorem, see Haran [H1], Theorem 5.1. It is the group theoretical counterpart of field theoretic assertion Theorem 2.12, and its proof follows easily using Fact 1.1 and 1.8, and the structure theorem of open subgroups of usual profinite free products from [B–N–W]. For a different proof idea see Haran [H1].

Theorem 1.10 (Subgroup Theorem). Let G be strongly G-projective. Then for every closed subgroup H ⊆ G, the family H = {Γ ∩ H | Γ ∈ G, Γ ∩ H ≠ 1} is a standard H-subspace of sg(H), and H is strongly H-projective.

We are now going to introduce the notion of a cover for (G, G). The covers for (G, G) are group extensions (ϵ) of G endowed with local splittings F of a very special nature.

Definition Let G be an arbitrary profinite group, and G ⊆ sg(G) a quasi τ^{et}- compact subset. A cover (F, F) for (G, G) is a group extension F of G of the form 1 → H → F ⟶^{π} G → 1 endowed with a τ^{et}-compact local splitting F satisfying the following:

i) F is mapped by π bijectively onto G, and the orbit space F^{∧} of F is mapped by π bijectively onto the orbit space G^{∧} of G.

ii) For every Φ, Φ_{1} ∈ F and 1 ≠ h ∈ ker(π), one has Φ^{h} ∩ Φ_{1} = {1}.

We remark that with the notations from Appendix, 4.10, C), the “canonical cover” associated to D, say π = π_{D} : (F_{D}, F) → (G, G), is a cover for (G, G).

Behold, we now come to (one of) the promised generalization of Gruenberg’s [G] result, which is the following

Theorem 1.11 Let G be strongly G-projective with G τ^{et}-compact and Ginvariant. Then for every cover

π : (F, F) → (G, G),

there exists a section σ : G → F of π such that σ(G) ⊆ F^{ker(π)}. Moreover, for any Φ ∈ con(F), either Φ ⊆ σ(G) or Φ ∩ σ(G) = {1}.

Proof . The proof is quite technical. For the beginning, we introduce the notion of a sub-cover, as follows: Let G be an arbitrary group, and G ⊆ sg(G) a quasi τ^{et}-compact subspace. A sub-cover (E, E) of a cover π : (F, F) → (G, G), written (E, E) ≺ (F, F), consists of a subgroup E ⊆ F, and a subset E ⊆ F^{ker(π)} of subgroups of E such that ρ = π|_{E} : (E, E) → (G, G) itself is a cover for (G, G). We say that a cover for (G, G) is minimal if it has no proper sub-covers.

Lemma 1.12 Let (G, G) be as above. They every cover π : (F, F) → (G, G) contains minimal sub-covers.

Proof . We will apply Zorn Lemma. It is clear that ≺ is a pre-order on the set of all sub-covers of (F, F). Let (E_{λ}, E_{λ})_{λ} be a totally ordered subfamily of sub-covers of (F, F). For each λ set D_{λ} = E_{λ}^{ker(ρ_{λ})}. Then ρ_{λ} = π|_{E_{λ}} has the properties:

1) ρ_{λ} maps D_{λ} onto G and D_{λ}^{∧} bijectively onto G^{∧}.

2) D_{λ} is τ^{et}-compact and ker(ρ_{λ}) acts regularly on it.

We set E = ∩_{λ}E_{λ}, D = ∩_{λ} D_{λ} and ρ = π|_{E}. It is clear that ρ has the properties 1), 2) above. Therefore, D contains a τ^{et}-compact fundamental domain E with respect to the action of ker(ρ). One verifies without difficulties that (E, E) is a sub-cover of (F, F).

Lemma 1.13 Under the hypothesis from Lemma above, suppose that G has the properties 1), 2) from the Separating Theorem. Let π : (F, F) → (G, G) be a cover for (G, G). Then for every Δ ∈ env(F) the pre-image of π(Δ) in con(F) is exactly Δ^{ker(π)}. In particular, if G is a G-space, then F^{ker(π)} is an F -space.

Proof . Let us denote H = ker(π). For a fixed Φ_{0} ∈ F, consider π(Φ^{f}) = π(Φ_{0}) for some Φ ∈ F and f ∈ F. Setting Γ = π(Φ), Γ_{0} = π(Φ_{0}) one has Γ^{π(f)} = Γ_{0}. Hence, by the definition of cover, Φ and Φ_{0} are conjugated in F, say Φ_{0} = Φ^{f_{0}}. This in turn implies Γ_{0} = Γ^{π(f)} = Γ^{π(f_{0})}. Since Γ is self-normalizing it follows that π(f) = π(f_{0})g^{0} for some g^{0} ∈ Γ. Hence, f = hf_{0}f^{0} for f^{0} the preimage of g^{0} in Φ and some h ∈ H. With these data we now get: Φ^{f} = Φ^{hf_{0}f^{0}} = Φ_{0}^{h} and so, the pre-image of Γ_{0} in F^{F} is Φ_{0}^{H}.

Now take any Δ, Δ_{1} ∈ con(F) such that π(Δ) = π(Δ_{1}). If Φ, Φ_{1} ∈ F^{F} which contain Δ, respectively Δ_{1}, it follows that Γ = π(Φ) and Γ_{1} = π(Φ_{1}) lie in G and have non-trivial intersection. Hence they are equal and so, Φ_{1} = Φ^{h} for some h ∈ H. From this the desired result follows.

Coming back to the proof of Theorem 1.11. Set H = ker(π). We first prove the existence of right inverses σ of π satisfying σ(G) ⊆ con(F). By the Lemma above, we can suppose that (F, F) is a minimal cover for (G, G). Let C be any open normal subgroup of F. Then H ∩ C is a normal subgroup of F open in H. Consider the canonical projections

F \underrightarrow{π̂} F̃ = F/(H ∩ C) \underrightarrow{π̃} G

and denote H̃ = π̂(H), F̃ = π̂(F) and Φ̃ = π̂(Φ) for Φ ∈ F. By Lemma 1.13, for every Δ ∈ con(F) the pre-image of π(Δ) in con(F) is exactly Δ^{H}. Using this fact, a simple verification shows for any Δ̃ ∈ (F̃) the pre-image of π̃(Δ̃) in con(F̃) is exactly Δ̃H̃. Moreover, since H̃ = ker π̃ is finite and H̃ ∩ Δ̃ = {1} for all Δ̃ ∈ con(F̃), it follows from the τ^{st}-compactness of con(F̃) that there exist open normal subgroups C̃ of F̃ such that C̃Δ̃ ∩ H̃ = {1} for all Δ̃ ∈ F̃.

Let U ⊆ sg(F̃) consist of all subgroups Δ̃_{1} of F̃ which are contained in C̃Δ̃ for some Δ̃. Then con(U) = U and U is a quasi τ^{et}-compact open neighborhood of con(F̃). A simple verification shows that if C̃ is small enough, then the pre-image of every Γ ∈ G in U is actually exactly Φ̃H̃, for any pre-image Φ̃ of Γ. Therefore, if σ̃ is any right inverse of π̃ with σ̃(G) ⊆ U, then σ̃(G) ⊆ F̃ H̃.

Let E be the pre-image of σ̃(G) by π̂, and E_{1} the pre-image of σ̃(G) in F^{H}. We show that E_{1} contains a subfamily E such that (E, E) is a cover for (G, G). For this we first observe that E is an open subgroup of F (indeed, it is the pre-image by π̂ of the open subgroup σ̃(G) of F̃). Secondly, we remark that every Φ ∈ F^{H} = F^{F} is conjugated to some Δ ∈ E_{1}. Applying Appendix, 4.9, B), it follows that there exists an E-invariant τ^{et} compact subset D of E_{1} with the property: If D^{h} ∩ D ≠ ∅ for some h ∈ F then h lies in E. From this it follows that ρ = π|_{E} has the properties 1), 2) from the proof of the first Lemma above with respect to D. One proceeds as in the proof of Lemma, and obtains a τ^{et}-compact subset E ⊆ D such that (E, E) is a sub-cover of (F, F). ¿From the minimality of (F, F) it follows that E = F and so, ker(π) = {1}.

It remains to prove that for every Φ_{1} ∈ con(F), either Φ_{1} ∩ σ(G) = {1}, or Φ_{1} ⊆ σ(G). It is sufficient to consider the case Φ_{1} = Φ^{h} for some Φ ∈ F and h ∈ H. Set Γ = π(Φ) and suppose Δ = Φ^{h} ∩ σ(G) ≠ {1}. Then Φ_{0} = σ(Γ) contains Δ, hence Φ_{0} ∩ Φ^{h} ≠ {1}. On the other hand, Φ^{h} and Φ_{0} are both pre-images of Γ in con(F). By the definition of cover it follows that Φ_{0} = Φ^{h}.

The proof of Theorem 1.11 is finished.

2 Pseudo closed fields

A) Localities of a field

Definition Let K be an arbitrary field. A locality of K any separable (but non-separably closed) overfield L ⊆ K^{alg} of K which is real closed or henselian with respect to some non-trivial valuation.

We denote the family of all localities of K by loc(K).

We say that a locality of K is classical, if it is a real or p-adically closed field, p being an arbitrary rational prime number.

Each locality L of K defines a V -topology τ on K, and we denote by K^{τ} the separable closure of K in the τ-completion of K. One has: L is a minimal element of loc(K) if and only if L = K^{τ}.

A valuation v of a field K is called minimal (maximal) if it has no proper coarsennings (if it is not a proper coarsening of another valuation of K; or equivalently, if the residue field Kv of v is algebraic over a finite field). If v is maximal and K henselian with respect to it, then by general valuation theory, G_{K} is pro-solvable.

We now give some examples of prominent families of localities of a field. The henselian Spectrum of a field Let lochens(K) denote the space of all proper henselisations of K in its algebraic closure. Then G_{K} acts continuously on lochens(K) and the henselian spectrum of K is defined to be the orbit space of lochens(K) with respect to the conjugation. General valuation theory now gives: if L|K is an algebraic extension of K, then lochens(L) consists of all non-separably closed compositae LK^{h} in a fixed algebraic closure of K.

The henselian spectrum of a field has a very complicated topology, and it is in general very difficult to describe it and its closure in sf(K). The classical spectra of a field For a family X of classical fields we denote by loc^{X} (K) the set of all localities of K which are elementarily equivalent to elements in X. If X contains just one element 𝕂, then we write loc^{𝕂}(K) for loc^{X} (K). It is clear that for every X, the space loc^{X} (K) is a G_{K}-invariant subspace of loc(K).

Proposition 2.1 For every finite set X as above, the space loc^{X} (K) is τ^{st}- closed in the space sf(K).

Proof . It is sufficient to prove the assertion for any particular loc^{𝕂}(K). Let an overfield L ⊂ K^{alg} of K lie in the τ^{st}-closure of loc^{𝕂}(K). For every finite Galois extension K_{1}|K, by our assumption on L it follows that

X_{K_{1}} = {K ∈ loc^{𝕂}(K) | K ∩ K_{1} = L ∩ K_{1}} ≠ ∅.

Therefore, U′ = (X_{K_{1}})_{K_{1}} is a filter base on loc^{𝕂}(K). Let U be a ultrafilter containing U′. Then (K^{∗}, v^{∗}) = ∏_{K}(K, v)/U is elementary equivalent to 𝕂. Moreover, the mapping L → K^{∗}, x ↦ (x_{K})/U with x_{K} = x if x ∈ K ∩ L and x_{K} = 0 otherwise, defines an embedding of L into K^{∗}. Let Λ denote the relative algebraic closure of L in K^{∗}. By the elementary theory of the classical fields, see for instance [Pr1] and [P–R], it follows that Λ is elementary equivalent to 𝕂, i.e., it lies in loc^{𝕂}(K).

For every field R, let R^{abs} denote the absolute subfield of R. By loc. cit. it follows that K^{abs} ∈ loc^{𝕂}(K^{abs}), and K^{alg} = K(K^{abs})^{alg} for every K in loc^{𝕂}(K). Further, from the continuity of the canonical restriction map ϕ_{K} : sf(K) → sf(K^{abs}), it follows that L^{abs} lies in the τ^{st}-closure of loc^{𝕂}(K^{abs}) in sf(ℚ). On the other hand, loc^{𝕂}(K^{abs}) is τ^{st}-closed. Thus L^{abs} ∈ loc^{𝕂}(K^{abs}). Since Λ^{abs} also lies in loc^{𝕂}(K^{abs}), and further L^{abs} ⊆ Λ^{abs}, we obtain L^{abs} = Λ^{abs}.

On the other hand, G_{L} lies in the τ^{st}-closure of {G_{K} | K ∈ loc^{𝕂}(K)} in sg(G_{K}). Hence every finite quotient of G_{L} is a quotient of some G_{K} ≃ G_{𝕂}. Since the latter group has finite corank, it follows that G_{L} itself is a quotient of G_{𝕂}, i.e., there exists a surjective homomorphism φ : G_{𝕂} → G_{L}. Thus we get a surjective homomorphism

φ̃: G_{𝕂} ⟶^{φ} G_{L} ⟶^{π_{L}} G_{Λ^{abs}} ≅ G_{𝕂}.

Since G_{𝕂} has finite corank it follows that φ̃ is an isomorphism. In particular, the canonical projection π_{L} is an isomorphism. Since G_{Λ} ⊆ G_{L} is also mapped by π_{L} isomorphically onto G_{L^{abs}} = G_{Λ^{abs}}, it follows that G_{L} = G_{Λ}, hence L = Λ.

Corollary 2.2 The topologies τ^{et} and τ^{st} coincide on loc^{𝕂}(K) for every 𝕂.

Proof By Appendix, 4.4, and Proposition 2.1, as loc^{𝕂}(K) = loc^{𝕂}(K)_{min}.

Remark For a given finite family X of classical fields, the space loc^{X} (K) does not have nice behavior as G_{K}-space. The reason for this is that the elements of loc^{X} (K) are in some sense “too big”. We are going to remedy this by working with classical closures instead of classical localities.

Definition A classical locality of K is called classical closure of K if it is minimal with respect to the inclusion in the set of all classical localities of K.

Theorem 2.3 For any classical locality L of K there exists a unique classical closure L_{0} of K which is contained in L.

Proof If L ≡ ℝ then everything is clear. Suppose L ≡ 𝕂, with 𝕂, a classical p-adic field. Let Λ ⊆ L be a classical locality of K. We show that Λ is p-adically closed and L|Λ is a finite extension. It is clear that Λ is non-real, hence it is p′- adically closed with respect to a finite rational prime number p′. By [P–R], §3, one has: L^{abs} is p-adically closed, and Λ^{abs} is p′-adically closed. Since Λ ⊆ L, it follows that Λ^{abs} ⊆ L^{abs}. Hence p = p′, and [L^{abs} : Λ^{abs}] is finite. But then L_{1} = ΛL^{abs} is p-adically closed and L_{1}^{abs} = L^{abs}. Therefore, L = L_{1}, and [L : Λ] = [L^{abs} : Λ^{abs}].

¿From this we deduce that any strictly descending sequence (L_{λ})_{λ} of classical localities of K which are contained in L is finite and has the same length as (L_{λ}^{abs})_{λ}. Hence L contains classical closures Λ_{0} of K. For any such classical closure (Λ, v_{Λ}), by [P1], (1.10), we get: The restriction of v_{L} to Λ is exactly v_{Λ}. Denote by v their common restriction to K. Let (K^{h}, v^{h}) be the unique henselisation of (K, v) inside L. Then (K^{h}, v^{h}) is also contained in every classical closure of K which lies in L. Hence the problem is reduced to the case where K is henselian with respect to a p-adic valuation v. We now claim:

Lemma 2.4 Let K be a henselian field with respect to some p-adic valuation, and Λ_{1}, Λ_{2} ⊆ L be p-adically closed over-fields of K. Then Λ = Λ_{1} ∩ Λ_{2} is a p-adically closed field too.

Proof We use an elaborated version of the characterization of p-adically closed fields from [P–R], §3, which reads as follows:

Theorem. Let K be a formally p-adic henselian field. Then K is p-adically closed if and only if for every t ∈ K and any natural number n there exist a_{n} ∈ K^{abs} and x_{n} ∈ K such that t = a_{n}x_{n}^{n}. Moreover, if K is p-adically closed, then the mapping

K^{×} → K̂ = lim_{⟵} K^{×}/n ≅ lim_{⟵} K^{abs}^{×}/n

is a group homomorphism, and K^{×}/K^{abs}^{×} is a ℚ-vector space.

Since Λ contains K, by the assumptions on K it follows that Λ is henselian, and Λ^{abs} = Λ_{1}^{abs} ∩Λ_{2}^{abs}. By the above Theorem, to show that Λ is p-adically closed, it is sufficient to prove that Λ^{×}/Λ^{abs}^{×} is divisible. For t ∈ Λ^{×} and n ≥ 1, set t = b_{n}x_{n}^{n} = c_{n}y_{n}^{n} = d_{n}z_{n}^{n} in L, Λ_{1}, and Λ_{2} respectively. Then (b_{n})_{n}, (c_{n})_{n}, and (d_{n})_{n} define the same element t̂ of L̂; further, if Λ̂′_{i} is the image of Λ̂_{i} in L̂, then t̂ lies in Λ̂′_{1} ∩ Λ̂′_{2}. Finally, let Λ̂′ be the image of Λ̂^{abs} in L̂. Then by local class field theory, in L̂ it holds (Λ̂′_{1} ∩ Λ̂′_{2} : Λ̂′) < ∞.

Therefore, there exists d ≥ 1 such that t̂^{d} lies in Λ̂′. Equivalently, there exists (a_{n})_{n} in Λ̂ defining t̂^{d}. Equivalently, t^{d}/a_{n} is an n^{th} power in Λ_{i} and L. Thus we can write

t^{d}/a_{n} = x_{n}^{n} = y_{n}^{n} = z_{n}^{n}

for some x_{n} ∈ Λ_{1}, y_{n} ∈ Λ_{2}, and z_{n} ∈ L. In particular, there exist n^{th} roots of unity ζ_{n} and ζ′_{n} in L, such that x_{n}ζ_{n} = z_{n} = yζ′_{n}. Taking into account that the group of roots of unity in L is finite, say of order m ≥ 1, it follows that

t^{dm}/a_{n}^{m} = (x^{m})_{n}^{n} = (y^{m})_{n}^{n} = (z^{m})_{n}^{n}

where x_{n}^{m} = y_{n}^{m} = z_{n}^{m} are equal, thus elements of Λ. ¿From this it follows that a_{n}^{m} are elements of Λ too. Thus finally, t^{md} (mod Λ^{abs}^{×}) is divisible in Λ^{×}/Λ^{abs}^{×}.

Now the proof of the Theorem follows immediately from the Lemma above and the comments preceding it.

For a fixed finite family X of classical fields we denote by clos^{X} (K) the set of all classical closures of K which are contained in elements of loc^{X} (K). It is clear that clos^{X} (K) is closed by conjugation, and we have:

Theorem 2.5 The space clos^{X} (K) is τ^{et}-compact and the canonical mapping

cl : (loc^{X} (K), τ^{st}) → (clos^{X} (K), τ^{et}), L ↦ L_{0}.

is continuous.

Proof It is sufficient to consider the case when X is the family of all classical subfields of some given classical field K. But in this case loc^{𝕂}(K) is τ^{st}-closed by Proposition 2.1, and it does not meet its residue set. To proceed apply Appendix, 4.4.

The orbit space of (clos^{X} (K), τ^{et}) with respect to the conjugation is denoted by Spec^{X} (K) and is called the (classical) X -spectrum of K. The correspondence

K → clos^{X} (K) → Spec^{X} (K)

has good functorial properties. Special cases of this definition but in a more general context can be found in [Be], [B–S], [C–R], [Ro], and others.

We now want to analyze more deeply the action of G_{K} on the space of all classical localities of K.

Lemma 2.6 For every classical locality L of K, its normalizer in G_{K} is exactly the normalizer of L in the absolute Galois group of the unique closure L_{0} contained in K. The canonical projection

π_{K} : G_{K} → G_{K^{abs}}

maps the normalizer of any L bijectively onto the normalizer of L^{abs} in the absolute Galois group of L_{0}^{abs}.

Proof From the elementary theory of the classical fields, see [Pr1], [P–R], it follows that there exists a valuation (possibly trivial) w of K with the properties:

i) If w is non-trivial, then it is henselian and wK is divisible.

ii) Kw is dense in K and elementary equivalent to K.

If w is trivial then the assertion of the Lemma is well known. Now suppose that w is non-trivial. Let g ∈ G_{K} lie in the normalizer of L, i.e., g(L) = L. Denote by M the fixed field of g in K. Then L|M is a pro-cycle Galois extension. Hence by general valuation theory, it follows that M is henselian with respect to the restriction v of w to M. We show that the value group vM of v is divisible. Suppose the contrary. Let t ∈ L and take any rational prime number q such that v(t) is not divisible by q. Then for every natural number m there exist a_{m} ∈ L^{abs} and x_{m} ∈ L with t = a_{m}x^{q^{m}}. Thus u_{m} = t/a_{m} has the property: v(t) = v(u_{n}) = q^{m} ·v(x_{m}) for all m. Therefore, for every m, the polynomial X^{q^{m}} − u_{m} is irreducible over M and has a root x_{m} in L. Since L|M is Galois, it follows that all the roots of these polynomials are in L. Hence L contains all the q^{m} roots of unity. Contradiction! Since the residue field Mv ⊆ Lw has characteristic 0, and vM is divisible, all the algebraic extensions of M are inert. Therefore, the normalizer of L in G_{K} is isomorphic to the normalizer of Lw in G_{Mv}. To proceed, apply the case when w is trivial.

Summarizing we have:

Theorem 2.7 For any finite family X of classical fields, the space clos^{X} (K) of all classical closures of K which are contained in elements of loc^{X} (K) is τ^{et}- compact and consists only of self-normalizing elements.

Proof Apply Lemma above and Theorem 2.5.

B) Pseudo classically closed fields

Definition Let K be an arbitrary field. A simple classical family of localities of K is a τ^{st}-closed subset K = K_{𝕂} of loc^{𝕂}(K) for some classical field 𝕂. A classical family of localities of K is a finite union of simple classical families of localities of K, i.e., a τ^{st}-closed subset of some loc^{X} (K).

Definition Let K be an arbitrary field and K ⊆ loc(K) a family of localities of K. We say that K is pseudo K-closed if every absolutely irreducible affine variety V |K has K-rational points, provided it has regular L-rational points for any L ∈ K.

We say that a field K is pseudo classically closed, if K is pseudo closed with respect to a classical family K of localities of K.

Let K be an arbitrary field. For an affine variety V |K, let V (K) denote the set of K-rational points of V, and V_{r}(K) the set of regular K-rational points of V. The next lemma shows that an absolutely irreducible affine variety over a pseudo closed field K which locally has regular rational points, has also regular K-rational points.

Lemma 2.8 Let K ⊆ sf(K) be a family of separable algebraic extensions of K. Then the following assertions are equivalent:

(1) For every absolutely irreducible affine variety V |K, the set V (K) is nonempty, provided V_{r}(L) is non-empty for all L ∈ K.

(2) For every absolutely irreducible affine variety V |K, the set V_{r}(K) is nonempty, provided V_{r}(L) is non-empty for all L ∈ K.

Proof This is a straightforward assertion.

Before going further, we recall that a subspace K ⊆ sf(K) is called a standard G_{K}-subspace, if it is quasi τ^{et}-compact, closed under conjugation by G_{K}, and K = K_{min}.

Theorem 2.9 (Density and Uniqueness Theorem). Let K be a pseudo Kclosed, with K some standard G_{K}-subspace of loc(K). Then K is dense in every L ∈ K, and moreover, K = lochens(K). Therefore, K depends only on K. It holds:

1) Every L ∈ K is self-normalizing.

2) The compositum of any tow distinct elements of K equals the separable closure of K.

Proof The main technical step is the following Lemma, comp. with [H–P]:

Lemma 2.10 Let K be an arbitrary field and K ⊆ sf(K) satisfying the equivalent conditions from above lemma. Let τ be any V -topology on K, and let K^{τ} denote the relative separable closure of K in the completion of K with respect to τ. Then there exists an element L_{0} in the τ^{st}-closure \overline{K} of K in sf(K) which is K-embeddable in K^{τ}.

Proof Comp. with [H–P]: Claim: Let f(X) ∈ K[X] be a separable polynomial. If f(X) has zeros in every L ∈ K, then f(X) has zeros in K^{τ}. Indeed, suppose that f(X) has no zeros in K^{τ}. Let c = f(b) be a fixed element in f(K). Since 0 ∉ f(K), it follows that 1 − c/f(K) is τ-bounded, hence there exist d ∈ K and a neighborhood U of 0 with the properties:

i) f(K^{τ}) ∩ U = ∅.

ii) d · (1 − c/f(K^{τ})) ⊆ U.

Set h(X, Y) = f(Y)(1 − d^{−1}f(X)) − c. The affine variety V |K defined by K[V] = K[X, Y]/(h) is absolutely irreducible, and V_{r}(L) is non-empty for every L ∈ K. Indeed, if a ∈ L is any zero of f(X), a direct verification shows that (a, b) is regular point of V. Hence, by our assumption on K, it follows that V (K) is non-empty. Considering any point (x, y) in V (K), we obtain h(x, y) = 0, hence f(y) = d · (1 − c/f(x)). This contradicts i), ii).

We now finish the proof of the Lemma. If no L ∈ \overline{K} can be K-embedded in K^{τ}, then for every L, there exists some a_{L} ∈ L such that the irreducible polynomial of a_{L} over K has no zeros in K^{τ}. For each L, denote by U_{L} the set of all L′ which contain a_{K}. Since L lies in U_{L}, it follows that (U_{L})_{L} is a τ^{et}-open covering of \overline{K}. Let (U_{L_{k}})_{k} be a finite sub-covering of it, and let f_{k}(X) denote the irreducible polynomial of a_{L_{k}} over K. Then the lowest common multiple f(X) = lcm(f_{k}(X))_{k} is a separable polynomial over K and has zeros in all L ∈ K, but not in K^{τ}. This is a contradiction.

We now come to the proof of Theorem 2.9. Let v be a non-trivial valuation of K, and τ the V -topology on K defined by v. Suppose that the separable closure K^{τ} of K in the v-completion of K is not separably closed. By the above Lemma, there exists an element L′ of K which is K-embeddable in K^{τ}. If τ′ denotes the V -topology defined by the valuation v′ of L′, and one defines correspondingly L′^{τ}, then L′^{τ} ⊆ L′ ⊆ K^{τ}. By [P–Z], it follows that τ = τ′ and K^{τ} = L′^{τ}. Therefore, L′^{τ} = L′ = K^{τ}, and so K^{τ} lies in K.

The remaining assertions of the theorem now follow easily from the one proved and general valuation theory.

Corollary 2.11 Let K be a pseudo K-closed field, with K a G_{K}-invariant classical family of localities of K. Then K_{min} is the space of all classical closures of K, and so depends only on K.

The next theorem shows how pseudo closed fields behave under algebraic extensions.

Theorem 2.12 (Overfield Theorem). Let K be a pseudo K-closed field with K a standard G_{K}-subspace of loc(K). Let M|K be an algebraic extension, and let M denote the family of all non-separably closed localities Λ of M of the form Λ = ML (L ∈ K). Then M is a standard G_{M}-subspace of loc(M), and moreover, M is pseudo M-closed.

Proof It is obvious that M is G_{M}-invariant; and using Theorem 2.9 combined with Appendix, 4.7, (3), it follows that M is standard.

We show that M is pseudo M-closed. Let V |M be an absolutely irreducible affine variety. Then there exists a finite sub-extension K_{1}|K of M|K, and an absolutely irreducible affine variety W |K_{1} such that V = W ×_{K} M. If K_{1} denotes the family of all non-separably closed localities L_{1} = LK_{1} of K_{1}, then M is exactly the family of all non-separably closed localities of M of the form Λ = L_{1}M. Obviously, K_{1} is quasi τ^{et}-compact and G_{K_{1}}-invariant, and by Heinemann–Prestel [H–P], we know that K_{1} is pseudo K_{1}-closed. Hence, we can assume that K_{1} = K. Now, let V_{r}(Λ) ≠ ∅ for all Λ ∈ M. Since for every Λ there exists L ∈ K such that Λ = LM, we have: For every L, there exists a finite extension K_{L}|K contained in L, and a finite extension M_{L}|K contained in M, such that V has regular M_{K}K_{L}-rational points. By construction, (U_{K_{L}})_{L} is a τ^{et}-open covering of K. Therefore, there exist finitely many L_{m} (m = 1, …, n) such that: For every L, there exists m with K_{L_{m}} ⊆ L. Let M_{0} be the compositum of the fields M_{L_{m}}. Then M_{0}|K is finite, and M_{0} is pseudo M_{0}-closed, where L_{0} is the set of all non-separably closed localities of M_{0} of the form Λ_{0} = LM_{0}.

We now check that V has Λ_{0}-rational regular points for every Λ_{0} ∈ M_{0}. Namely, for every Λ_{0} = LM_{0}, there exists m such that K_{L_{m}} ⊆ L. Further, since M_{L_{m}} are contained in M_{0}, it follows that K_{L_{m}}M_{L_{m}} ⊆ Λ_{0}. Thus the K_{L_{m}}M_{L_{m}}- rational points of V are also Λ_{0}-rational points. Hence this last set is non-empty.

Therefore, V has M_{0}-rational points, and so, L-rational points. The proof is finished.

Corollary 2.13 (1) An algebraic extension of a PRC field is also PRC

(comp. [Pr2]).

(2) An algebraic extension M of a P pC field K is P pC if and only if for any p-adic closure L of K, either M ⊂ L or LM is separably closed.

(3) An algebraic extension M of a pseudo classically closed field K is pseudo classically closed if and only if there exists a positive bound c > 0 such that for every classical closure L of K, either LM is separably closed or [LM : K] < c. In particular, a finite extension of a pseudo classically closed field is itself pseudo classically closed.

Corollary 2.14 Let K be pseudo K-closed. Let M|K be an algebraic extension of K with pro-solvable absolute Galois group G_{M}. Suppose that cd_{q}G_{M} ≥ 2 for at least two rational prime numbers q. Then M is a locality of K, hence there is some L ∈ K which is K-embeddable in M and moreover, this L is unique.

In particular, if all L ∈ K are endowed with maximal valuations v, and G_{L} is isomorphic to a classical Galois group 𝔾 ⊆ G_{ℚ_{p}}, then L is p-adically closed.

Proof Let M be the prolongation of K to M. Then M is pseudo M-closed. In particular, its Brauer group Br L satisfies a local-global principle with respect to M, i.e., the canonical mapping

Br M → _{∏Λ}Br Λ

is injective. In particular, cd_{q}G_{Λ} ≥ 2 for at least two rational prime numbers q, and corresponding Λ ’s. But then M is itself henselian, see [P1], (1.13). Since by Theorem 2.9, all Λ are henselisations of M, it follows that {M} = M. In other words: M = LM for some L ∈ K. The uniqueness follows by Theorem 2.9.

For the second assertion we remark that the valuation of L is maximal (it prolonging the valuation of the unique K which is contained in L). Further, apply [P1], (E.9), and §2.

C) Construction of pseudo closed fields

We will now solve the problem of constructing pseudo closed fields starting from some “initial conditions” as follows. Let K be an arbitrary field, and K a family of localities of K. Under supplementary conditions on K we will construct a field extension L|K endowed with a family of localities L such that L is pseudo closed with respect to L, and L is closely related to the family of localities K. In the above context, in the case of a finite family K, such extensions L|K were constructed by Heinemann–Prestel [H–P], and other special cases have been dealt with by Ershov [E], Haran–Jarden [H–J1], [H–J2], and others. The idea we use can be found in van den Dries [vdD]. Therefore we will develop here the things only as far as to be in the position to make the machinery from loc.cit. work. We further will consider from the start that the fields we deal with are perfect fields.

For an arbitrary field extension L|K let π_{LK} : G_{L} → G_{K} be the canonical projection, and ρ_{LK} : sf(L) → sf(K), L′ ↦ K′ := L′ ∩ K^{alg} the canonical restriction map. In general, ρ_{LK} does not map localities of L to localities of K, as L could have valuations which are trivial on K. Nevertheless, if Λ is a maximal locality of L, then Λ ∩ K^{alg} is a locality of K (unless K is algebraic over a finite field). This is the case for instance if Λ is a classical locality of Λ.

Definition Let Λ|L be a field extension. We will say that Λ dominates L if Λ is henselian with respect to a non-trivial valuation v having the properties:

(i) v is trivial on L, and the residue field of v is Λv = L.

(ii) Λ has only inert extension.

In particular the canonical projection π_{ΛL} : G_{Λ} → G_{L} is an isomorphism, and sg(G)_{Λ} → sg(G)_{L} and sf(Λ) → sf(L) are homeomorphisms.

We remark that if in the above context L is itself is henselian to some non-trivial valuation v_{L}, then setting v_{Λ} = v ∘ v_{L}, it follows that (L, v_{L}) is existentially closed in (Λ, v_{Λ}) as valued fields. In particular, if L is elementarily equivalent to some classical field 𝕂, then every field Λ dominating L is also elementarily equivalent to 𝕂.

Definition Let K be a perfect field, and K a quasi τ^{et}-compact set of localities of K. A cover of (K, K) is a field extension L|K with L a perfect field together with a τ^{et}-compact family L of localities of L such that:

(i) ρ_{LK} maps L and its orbit space L^{∧} with respect to the conjugation bijectively onto K, respectively K^{∧}.

(ii) Every Λ dominates the corresponding L = ρ_{LK}(Λ) = Λ ∩ K^{alg}.

(iii) For any Λ, Λ_{1} ∈ L and g ∈ ker(π_{LK}) one has: Λ^{g}Λ_{1} = L^{alg}.

We recall that a subspace K ⊆ sf(K) is called special, if K^{G_{K}} is τ^{et}-compact and contains only self-normalizing over-fields of K.

Theorem 2.15 Let K be a perfect field endowed with a special G_{K}-subspace of sf(K). Then there exist covers (^{∗}K,^{∗}K) for (K, K) such that ^{∗}K is pseudo ^{∗}K-closed, and moreover, ^{∗}K is special.

In particular, ^{∗}K consists of all minimal non-separably closed henselisations of ∗K, and its orbit space ^{∗}K^{∧} is mapped by ρ_{^{∗}KK} homeomorphically onto K^{∧}.

Proof The key technical step in the proof is the following

Lemma 2.16 In the context of the above Theorem, let V → K be an absolutely irreducible variety, and K(V) its field of rational functions. Suppose that V has regular L-rational points for every L ∈ K. Then there exits a regular field extension R|K containing K(V), and a space R of localities of R such that (R, R) is a cover for (K, K). Moreover, R is special.

Proof For every x_{L} ∈ V_{r}(L), let U_{L} be the τ^{et}-open subset of K consisting of all L′ such that x_{L} ∈ V (L′). It is obvious that {U_{L}}_{L} is a τ^{et}-open covering of K. Since K is τ^{et}-compact, there exists a finite Galois extension N|K and finitely many L_{l} such that the covering defined by the basic τ^{et}-open subsets {U_{N}(L_{l})}_{L_{l}} is a refinement of the given τ^{et}-open covering. In other words, for every L, there exists l such that N_{l} := N ∩ L_{l} ⊆ L, and V_{r}(N_{l}) ≠ ∅. By standard set topology arguments, we can suppose that the finite open covering {U_{N}(L_{l})}_{l} of K has a finite refinement (K_{l})_{l} consisting of disjoint τ^{et}-open subsets of K.

For every l, let x ∈ V_{r}(N_{l}) be a regular N_{l}-rational point of V. Then O_{V,x} is a regular ring with residue field κ(x) = N_{l}. Therefore there exits a valuation w of K(V) with value group ≅ ℤ^{d}, where d = dim(V), and residue field κ(x) = N_{l}. Let K(V)^{h} be a fixed henselisation of K(V) with respect to w in a fixed algebraic closure of K(V), and finally Ω a maximal purely ramified extension of K(V)^{h}. For the sake of simplicity, we denote the canonical prolongation of w to Ω again by w.

Lemma 2.17 Let R^{0} be an arbitrary field, and R^{1} = R^{0}((X_{i})_{i}) the rational function field in I variables over R^{0}, with I having cardinality |I| ≥ |R^{0}|. Then for every algebraic extension R′|R^{0}, there exits a valuation v^{1} of R^{1} which is trivial on R^{0} and has the properties: The residue field of v^{1} is R^{1}v^{1} = R′, and the values v^{1}(X_{i}) are rationally independent in the value group of v^{1}.

In particular, if R^{0} is perfect, one inductively constructs a purely transcendental extension R^{∗} = ∪_{n}R^{n} of R^{0} of transcendence degree depending only on |R^{0}| such that: If R′|R^{0} is any given algebraic extension, then there exits a valuation v^{∗} on R^{∗} which is trivial on R^{0} and satisfies:

(1) R^{∗}v^{∗} = R′, and the henselisation R^{∗}^{h} of (R^{∗}, v^{∗}) has only inert extensions.

(2) R^{∗}|R^{0} has a transcendence basis T such that v^{∗}(t), (t ∈ T), are rationally independent in the value group of v^{∗}.

Proof The proof is quite straightforward, and therefore left to the reader.

Coming back to the proof of Lemma 2.16: We apply the Lemma above with the hypothesis: R^{0} = K(V), and R′ = Ω. Let (R^{∗}^{h}, v^{∗}^{h}) a fixed henselisation of R with respect to v^{∗}. By construction, R^{∗}^{h} has only inert extension, and therefore ρ_{R^{∗}^{h}Ω} : sf(R^{∗}^{h}) → sf(Ω) is a homeomorphism in both τ^{et} and τ^{st}. We further let denote v = v^{∗}^{h} ∘ w the compositum of v^{∗}^{h} with the valuation w of Ω. Then by general valuation theory, R^{∗}^{h} is also a henselisation of R^{∗} with respect to (the restriction of) v (to R^{∗}), and further R^{∗}^{h}v = (R^{∗}^{h}v^{∗}^{h})w = Ωw = N_{l}. In particular, (R^{∗}^{h}, v^{h}) has inert extension only, and ρ_{R^{∗}^{h}N_{l}} : sf(R^{∗}^{h}) → sf(N_{l}) is a homeomorphism in both τ^{et} and τ^{st}.

For every L ∈ K_{l}, let Π be its pre-image by ρ_{R^{∗}^{h}N_{l}} in sf(R^{∗}^{h}). If we endow Π with the unique prolongation v_{Π}^{∗} of v^{∗}^{h} to Π, then Π dominates L. Further, we will endow every Π with the the unique prolongation v_{Π} of v^{h} to Π, and view it as a locality (Π, v_{Π}) of R and of R^{∗}^{h}.

Let R_{l} be the set of all the localities (Π, v_{Π}) of the above form. Then ρ_{R^{∗}^{h}N_{l}} maps R_{l} homeomorphically onto K_{l} both in τ^{et} and τ^{st}.

Claim. Every Π ∈ R_{l} is self-normalizing in G_{R^{∗}}.

Indeed, let H_{Π} be the normalizer of Π in G_{R^{∗}}. Then, since the residue field Πv_{Π} = L is not separably closed, by the generalization of a Theorem of F. K. Schmidt [P1], (1.9), it follows that H_{Π} is contained in the decomposition group of v_{Π}. On the other hand, v_{Π} is simply the prolongation of v^{h} to Π, thus H_{Π} is mapped via π_{R^{∗}^{h}N_{l}} isomorphically onto the normalizer of G_{L} = π_{R^{∗}^{h}N_{l}}(G_{Π}) in G_{K}. Since by hypothesis, L is self-normalizing in G_{K}, thus in G_{N_{l}} too, it follows that H_{Π} is mapped onto G_{L}. Thus H_{Π} = G_{Π}, i.e., Π is self-normalizing.

To finish the proof of Lemma 2.16, we first suppose that K is G_{K} invariant. Setting G = G_{K} and G = {G_{L} | L ∈ K}, we apply Appendix, 4.10, B) with notations as there: For each k, let Z_{k} ⊂ G_{R} be a pre-image of X_{k} by π_{RK}, and K_{k} = {L | G_{L} ∈ G_{k}}. For every l = k, one defines the valuation v_{l} = v and ρ_{l} = ρ_{R^{∗}^{h}N_{l}} as above; and for the conjugates gK_{l}g^{−1}, (g ∈ X_{k}), of K_{l} one works with the conjugated valuations v ∘ h^{−1}, where h ∈ Z_{k} is the pre-image of g ∈ X_{k}. Then (hR_{k}h^{−1})_{k,h_{∈}Y_{k}} are τ^{et}-compact disjoint subsets of sf(R^{∗}); and obviously, their union R gives rise to a covering (R^{∗}, R) → (K, K).

For the general case, when K is not necessarily G_{K} invariant, we first make the construction for K′ = K^{G_{K}} instead of K. If now R denotes the pre-image of K in R′, then R is special.

The next step in proving Theorem 2.15 is the following

Lemma 2.18 In the context of Theorem 2.15 there exists a cover (K^{1}, K^{1}) of (K, K) with the following property: Every absolutely irreducible variety V → K has a K^{1}-rational point, provided V has a regular L-point for every L ∈ K. Moreover, K^{1} is a special subset of sf(K^{1}).

Proof We use the idea from van den Dries, compare [vdD], p. 28. Let (V_{λ})_{λ} be the family of all absolutely irreducible varieties over K, the index set being well ordered. Setting K_{0} = K, K_{0} = K, using Lemma 2.16, one constructs inductively a chain of pairs (K_{λ}, K_{λ}) such that for μ ≤ ν, the resulting (K_{ν}, K_{ν}) is a cover for (K_{μ}, K_{μ}), and moreover, the following holds:

i) If λ is not a limit ordinal, then V_{λ} has a regular K_{λ}-rational point, provided V_{λ} has regular L_{λ−1}-rational points for all L_{λ−1} ∈ K_{λ−1}

ii) If λ is a limit ordinal, then setting L = ∪_{μ<λ}K_{μ}, and L = \underleftarrow{lim} K_{μ}, the following holds: If V_{λ} has a regular Λ-point for all Λ ∈ L, then V_{λ} has a K_{λ}-rational point.

Indeed, if λ is not a limit ordinal, then the existence/construction of (K_{λ}, K_{λ}) follows immediately from Lemma 2.16. If now λ is a limit ordinal, then it follows that the above (L, L) is a cover for all (K_{μ}, K_{μ}). Then one uses again Lemma 2.16, with hypothesis (L, L) in order to get (K_{λ}, K_{λ}).

Finally set K^{1} = ∪_{λ}K_{λ} and K^{1} = \underleftarrow{lim} K_{λ}. It is clear that (K^{1}, K^{1}) has the desired properties.

We finally come to the proof of Theorem 2.15. We set K^{0} = K and K^{0} = K. Using last Lemma above, we get a chain of covers (K^{n}, K^{n}) with the following properties:

j) K^{n+1}|K^{n} is a regular field extension for every n ≥ 0.

jj) If V → K^{n} is an absolutely irreducible variety which has regular L^{n}-rational

points for all L^{n} ∈ K^{n}, then V has K^{n+1}-rational points. It now follows without difficulty that K^{∗} endowed with K^{∗} = \underleftarrow{lim} K^{n} has the

desired properties.

As a corollary we have:

Theorem 2.19 Let K be an arbitrary field, and K a τ^{et}-compact family of classical closures of K. Then there exists a regular field extension K^{∗}|K endowed with a τ^{et}-compact set K^{∗} of classical localities of K, such that (K^{∗}, K^{∗}) is a cover for (K, K), and K^{∗} is pseudo K^{∗}-closed.

In particular, this applies in the case K is the set of all classical closures of K which are contained in the elements of some classical family clos^{X} (K) of localities of K.

Proof Clear from Theorem 1.6 and the Theorem 2.15 above.

3 Classically projective groups as absolute Galois groups

A) The absolute Galois group of a pseudo closed field

Lemma 3.1 Let N|M be an arbitrary Galois extension, and X a finite set on which G := Gal(N|M) acts. We let N[X] be the polynomial ring in X variables on which G acts by g(ax) = g(a)gx for a ∈ N and x ∈ X. Then the fixed ring of G in N[X] is a polynomial ring M[T] over M, and moreover, N[T] = N[X].

Proof Let {X_{k}}_{k} be the orbits of X with respect to the action of G. If the assertion of Lemma is true for every N[X_{k}], then it is true for N[X]. Hence it is sufficient to consider the case r = 1, i.e., G acts transitively on X. Fixing x ∈ X, we let G_{x} be its stabilizer, and N_{x} the fixed field of G_{x} in N. Let a be a primitive element of N_{x}|M. Remark that for 0 ≤ k ≤ n − 1, the elements, a^{k}x have the same stabilizers, namely G_{a^{k}x} = G_{x} = G_{a}. Let σ_{m} be a system of representatives for G/G_{x}. For each k as above, let t_{k} = ∑ σ_{m}(a^{k}x) be the relative trace of a^{k}x. Denote by T the set of all t_{k}. We claim that this T does the job. Indeed, by their definition, all t_{k} are G invariant. Next, the determinant of the system of equations defining σ_{m}(x) in relation to t_{k} is the Vandermonde one defined by (σ_{m}(a))_{m}, hence it does not vanish. Therefore, σ_{m}(x) are linear forms in t_{k} with coefficients in N, hence N[T] = N[X].

The above construction will be applied in the following way:

3.2 Let K be an arbitrary field endowed with a family K of separable overfields and set G = {G_{L} | L ∈ K}. Let (γ_{K} : G_{K} → A, α : B → A, B) be any finite embedding problem for (G_{K}, G). Thus for every Γ ∈ G, there exits a local splitting β_{Γ} : Γ → Δ ⊂ B such that (γ_{K})|_{Γ} = α β_{Γ}. Moreover, let us suppose that the conditions 1), 2) from Fact 1.1 are satisfied. We denote by N the fixed field of D_{K} = ker(γ_{K}) in K^{sep}, and identify γ_{K} : G_{K} → G_{K}/D_{K} with the canonical restriction map pr_{N} : G_{K} → G = Gal (N|M). Further, let X = |B| be the underlying set of B viewed as B-space by the multiplication from the left. Set R = N(X) the rational function field on X variables over N. We identify B with a subgroup of Aut(R) by setting h(ax) = a(h)(a)hx for all h ∈ B, a ∈ N and x ∈ X. Let S = R^{B} denote the fixed field of B in R. Then R|S is a Galois extension with Galois group Gal (R|S) = B; S|K is a regular extension; and the canonical projection

pr_{RN} : B = Gal (R|S) → Gal (N|K) = G_{K}/D_{K}

is exactly α. Further, for a fixed Γ ∈ G, let N_{Γ} be the fixed field of Γ in N, and B_{Γ} = β_{Γ}(Γ) a pre-image in B of γ_{K}(Γ) = Γ/D_{K}. Then Γ acts on X = |B| by g(x) := β_{Γ}(g) x for g ∈ Γ, x ∈ |B|. Moreover, since (γ_{K})|_{Γ} = αβ_{Γ} we get g(a) = β_{G}(g)(a) for all a ∈ N and g ∈ Γ. By Lemma above we get: The fixed ring of Γ in N[X] with respect to the action of B is a polynomial ring N_{Γ}[T] and moreover, N[X] = N[T]. Therefore, the fixed field S_{Γ} = N_{Γ}(T) of Γ in R is a rational function field over N_{Γ} and furthermore, R = S_{Γ}N.

Theorem 3.3 Let K be a pseudo K-closed with K ⊆ loc(K) quasi τ^{et}-compact, and set G = {G_{L} | L ∈ K}. Then G_{K} is G-projective.

Proof We show that any special finite embedding problem as mentioned above EP = (γ_{K}, α, B) for (G_{K}, G) has solutions. Namely, in the above construction, for any Γ ∈ G, we have S ⊆ S_{Γ} ⊆ R. We let V be any absolutely irreducible affine variety over K with S = K[V]. Correspondingly, let V_{Γ} be an absolutely irreducible affine variety over N_{Γ} with N_{Γ}[T] = N_{Γ}[V_{Γ}]. By the lemma above, V_{Γ} is isomorphic to the (dim V)-dimensional affine space over N_{Γ}. Let L be the locality of K with Γ = G_{L}. Then the inclusion L[V] ⊆ L[V_{Γ}], gives on the geometry side a projection pr_{Γ} : V_{Γ}(L) → V (L). Taking into account that V_{Γ}(L) is Zariski dense, it follows that its image in V (L) is Zariski dense too. In particular, V_{r}(L) is non-empty. Since K is pseudo K-closed, it follows that V also has regular K-rational points. Using any regular K-rational points x of V, one gets a valuation v on S = K(V) such that its valuation ring O_{v} dominates the local ring O_{V,x} of x, and its residue field Sv = κ(x) = K. Let S^{h} be any henselisation of (S, v) and Ω a maximal, totally ramified extention of S^{h}. Then the canonical projection π_{S} : G_{S} → G_{K} induces by restriction an isomorphism π : G_{Ω} → G_{K}. Therefore, denoting by pr : G_{S} → Gal (R|S) = B, the canonical projection we get a group homomorphism

β : G_{K} ⟶^{π^{−1}} G_{Ω} ↪ G_{S} ⟶^{pr} Gal (R|S) = B.

We claim that β is a solution of the embedding problem EP. Namely, we have α β = α ∘ pr ∘ π^{−1}. On the other hand, α is by construction exactly the canonical projection B = Gal (R|S) → Gal (N|K) by construction. Thus α ∘ pr is the canonical projection G_{S} → Gal (R|S) → Gal (N|K). Hence, we get: αβπ^{−1} = γ_{K}π^{−1}.

B) Strongly relatively projective groups as absolute Galois groups

We start with the following:

Definition Let G be an arbitrary profinite group and G a set of closed subgroups of G. A Galois approximation for (G, G) is a pair (K, K) consisting of a field K endowed with a subspace K ⊆ sf(K) together with a surjective group homomorphism

κ : G ⟶ G_{K}

of G onto the absolute Galois group G_{K} of K which has the properties: i) κ is injective on each Γ ∈ G. ii) The fixed field K_{Γ} of κ(Γ) in K^{sep} lies in K. We say that a Galois approximation κ is classical if K_{Γ} is a classical locality of

K for every classical subgroup Γ ∈ G.^{1}

If μ : G → G_{M} is another Galois approximation of (G, G), we say that μ dominates κ, written κ ≺ μ, if the following holds:

j) K ⊆ M and κ = π μ, where π is the canonical projection G_{M} → G_{K}.

jj) K_{Γ} is dominated by Λ_{Γ} for all Γ ∈ G.

It is obvious that ≺ is an ordering on any family of Galois approximations for (G, G). Furthermore, if κ ≺ μ, then ker(μ) ⊆ ker(κ).

Theorem 3.4 Let G be strongly G-projective with G τ^{et}-compact. Suppose that there exists a Galois approximation

κ : G → G_{K}

for (G, G) with K a perfect field. Then there exists λ : G → G_{L} a Galois approximation for (G, G) such that κ ≺ λ, and λ is an isomorphism, and L is pseudo L = {Λ_{Γ} | Γ ∈ G}-closed. In particular, L|K is regular.

Proof We can suppose that G is G-invariant. An obvious argument using Zorn Lemma shows that there exist Galois approximations

μ : G ⟶ G_{M}

for (G, G) such that κ ≺ μ and any other Galois approximation μ_{1} for (G, G) satisfies: If μ ≺ μ_{1}, then ker(μ) = ker(μ_{1}). We now show that any such μ is actually an isomorphism. Equivalently, it is sufficient to show that H = ker(μ) is the trivial group. Mutatis mutandis, we can suppose that κ is maximal in the above sense.

Let D be an arbitrary open subgroup of G. Using the construction from 3.2, we shall construct a Galois approximation μ for (G, G) with ker(μ) ⊆ D. Let namely consider the commutative diagram

G ⟶^{κ} G_{K}
↓^{γ}  _{γ_{K}} ↓
G/D ⟶^{α} G_{K}/D_{K}

where D_{K} = κ(D) and the mappings are the canonical projections. Let N be the fixed field of D_{K} and for Γ ∈ G denote by N_{Γ} the fixed field of γ_{K}κ(Γ) = αγ(Γ). Since κ is injective on all Γ ∈ G, and G is τ^{et}-compact, after replacing D by a sufficiently small open normal subgroup of G, we can suppose that α is injective on all B in γ(G). Then (γ_{K}, α, γ(G)) is a special embedding problem for (G_{K}, κ(G)). Hence, we can apply the construction 3.2 and obtain:

1It was conjectured that the condition ii) is always satisfied, i.e. if G_{K} is a classical Galois group, then K is a classical field. In the case G_{K} ≅ G_{ℝ} it is the famous Artin-Schreier Theorem.

This conjecture is now completely proved by Neukirch, Pop, Koenigsmann, Efrat.

For every Γ ∈ G, the fixed field S_{Γ} of κ(Γ) in R = N(X), is a rational function field S_{Γ} = N_{Γ}(T) over N_{Γ}. Let v be the valuation

v : S_{Γ} → ℤ^{T}

with ℤ^{T} ordered lexicographically, and v(t) = (the t^{th} element of the canonical basis of ℤ^{T}) for any t ∈ T. Then the residue field of v is S_{Γ}v = M_{Γ}. Denote by S^{h} some henselisation of (S_{Γ}, v) (together with a prolongation of v, which we again denote by v). Finally let Ω_{Γ} be a fixed maximally purely ramified extension of S^{h} (together with a prolongation of v, which we again denote by v). Thus, the canonical projection

π_{Γ} : G_{Ω_{Γ}} ⟶ G_{M_{Γ}}

is an isomorphism. Now consider any Γ_{1} ∈ G with Γ_{1}/D ⊆ Γ/D and so, αγ(Γ_{1}) ⊆ αγ(Γ). Then M_{Γ} ⊆ M_{Γ_{1}}, hence we can consider the pre-image Δ_{1} of κ(Γ_{1}) in G_{Ω_{Γ}}. Let Σ_{1} be the fixed field of Δ_{1} in the separable closure of Ω_{Γ} endowed with the unique prolongation v_{Σ_{1}} of v to Σ_{1}. By construction, it follows that φ_{SK}(Σ_{1}) = K_{1} and moreover, Σ_{1} dominates K_{1}. If now S_{1} is the family of all Σ_{1} as above, when Γ varies in G we show that there exist Galois approximations μ : (G, G) → (S, S) with S ⊆ con(S_{1}), κ ≺ μ and ker(μ) ⊆ D.

For the open normal subgroup D of G let namely

π = π_{D} : (F_{D}, F) ⟶ (G, G)

be a “a canonical cover” for (G, G) associated with D as in the Appendix, 4.10, C). Further, we let g_{S} be a pre-image of κ(g) in G_{S} (such pre-images do exist, because S|K is regular, thus G_{S} → G_{K} is surjective). From the universal property of the usual profinite product, it follows that there exists a unique homomorphism φ : F_{D} → G_{S} such that φ |_{D_{k}}= π_{Γ_{k}}^{−1} ∘ κ |_{D_{k}}, and φ(|x|) = g_{S}. Obviously, ϕ maps F into con(S_{1}) and makes the following diagram commutative:

F_{D} ⟶^{ϕ} G_{S} ⟶^{pr_{R}} Gal (R|S) = B
π↓  ^{π_{S}} ↓  ^{α}↓
G ⟶^{κ} G_{S} ⟶^{pr_{N}} Gal (N|K) = A

On the other hand, since π is a cover for (G, G), there exists a right inverse σ of π with σ(G) ⊆ F^{F}. Therefore, μ = ϕ ∘ σ is a Galois approximation for (G, G) with κ ≺ μ and ker(μ) ⊆ D. The maximality of κ implies: ker(κ) = ker(μ). Hence, ker(μ) ⊆ D. Since D was arbitrary, it follows that ker(κ) = {1}.

Now let us consider an arbitrary Galois approximation μ : G → G_{M} satisfying κ ≺ μ, and ker(μ) = {1}. Let us denote G_{M} = κ(G), and M = {Λ_{Γ} | Γ ∈ G}. Then (M, M) satisfies the hypothesis of Theorem 2.12, and moreover, M is a special G_{M}-subspace of sf(M). Let (M^{∗}, M^{∗}) be as in Theorem 3.2, and set G_{M^{∗}} = {G_{Λ^{∗}} | Λ^{∗} ∈ M^{∗}}. Then the canonical projection G_{M^{∗}} → G_{M} defines in fact a cover in the sense of Section 1, B)

π_{M^{∗} M} : (G_{M^{∗}}, G_{M^{∗}}) → (G_{M}, G_{M})

for (G_{M}, G_{M}) = μ(G, G). Since G_{M} is strongly G_{M}-projective, we can apply Theorem 1.11, and get a right inverse σ^{∗} : G_{M} → G_{M^{∗}} of π^{∗} such that σ^{∗}(G_{M}) ⊆ G_{M^{∗}}, and for all Δ^{∗} ∈ con(G_{M^{∗}}) either Δ^{∗} ⊆ σ^{∗}(G_{M}), or Δ^{∗} ∩ σ^{∗}(G_{M}) = {1}. In field theoretical terms this means: Let L, Λ^{∗} be the fixed fields of σ^{∗}(G_{M}), respectively Δ^{∗} in the algebraic closure of M^{∗}. Then either Λ^{∗} contains L or their compositum Λ^{∗}L is algebraically closed. Applying Theorem 2.12, it follows that L is pseudo closed with respect to L = σ^{∗}(M). Finally, λ = σ^{∗}μ is a Galois approximation for (G, G) with κ ≺ λ.

Without proof we give the following corollary:

Theorem 3.5 Let G be strongly G-projective with G = G_{max}, and τ^{s}-compact, and suppose that all Γ ∈ G are isomorphic to the decomposition group Z_{v} of a valuation v of some field. If Z_{v} has finite corank, then (G, G) has Galois approximations. Hence, it can be realized as absolute Galois group of a pseudo closed field.

We conclude this section by the main result of the paper:

Theorem 3.6 A profinite group G is classically projective if and only if it is isomorphic to the absolute Galois group G_{K} of a pseudo classically closed field K.

Proof In one direction, one applies Theorem 3.3 together with Theorem 1.2. In the other direction, apply Theorem 1.2 combined with Theorem 3.4, and Theorem 1.6 (which gives one Galois approximations for classically relatively projective groups).

4 Appendix: Boolean and quasi-boolean G-spaces

Let X be an arbitrary topological space. For any x ∈ X, and any subspace Y ⊆ X we define:

1) env(x) the set of all x′ ∈ X which have x as specialization, i.e., x ∈ \overline{\{x′\}}.

2) env(Y) = ∪_{y∈Y} env(y), the envelope of Y.

3) gen(Y), the residue set of Y, i.e., the set of all y ∈ Y having more than a closed point as specialization in Y.

It is clear that any neighborhood U of x contains env(x), and any neighborhood U of Y contains env(Y). Further, it is obvious that env(gen(Y)) = gen(Y).

Let X be a finite set. Giving a T_{1}-topology on X is the same as giving an ordering ≤ on X. Indeed, if the topology is given, then for x, x′ we define: x′ ≤ x if and only if x′ ∈ env(x). In particular, the sets env(x) (x ∈ X) constitute a basis of the given topology. Conversely, given an ordering ≤ on X, the family of all sets env(x) defined as above is a basis of a T_{1}-topology on X.

A mapping f : X → Y between two finite T_{1}-topological spaces is continuous if and only if it is order preserving.

Definition We say that a topological space is quasi-boolean if it is a projective limit of finite T_{1} topological spaces. For technical reasons we usually denote a quasiboolean topology by τ^{et}. We say that a topological space is boolean if it is projective limit of discrete finite spaces.

Equivalently, a quasi-boolean topological space is a T_{1} quasi-compact topological space which has a topology basis consisting of quasi-compact sets. A boolean space is a compact topological space which has a topology basis consisting of compact sets.

The category of all boolean spaces can be viewed as a completion of the category of all quasi-boolean spaces in the following sense: For every quasi-boolean topological space (X, τ^{et}) there exists on X a unique smallest Hausdorff topology τ^{st} containing τ^{et} such that τ^{et} ↦ τ^{st} has functorial behavior, i.e.:

Any τ^{et}-continuous map f : (X, τ^{et}) → (Y, τ^{et}) is also τ^{st}-continuous.

In particular, if (X, τ^{et}) = \underleftarrow{lim}(X_{i}, τ^{et}) with X_{i} finite and T_{1}, then τ^{st} is the projective limit of the discrete topology on X_{i}.

Directly from the definition it follows that any τ^{st}-closed subset Y ⊆ X of a quasi-boolean topological space X endowed with the τ^{et}-subspace topology, is also a quasi-boolean topological space. Moreover, if (X, τ^{et}) = \underleftarrow{lim}(X_{i}, τ^{et}) with X_{i} finite and T_{1}, then (Y, τ^{et}) = \underleftarrow{lim}(Y_{i}, τ^{et}) with Y_{i} ⊆ X_{i} the image of Y in X_{i} by the canonical projection φ_{i} : X → X_{i}.

Therefore, the next assertions on a quasi-boolean topological space (X, τ^{et}) also hold for any τ^{st}-closed subspace of it, but endowed with the τ^{et}-induced topology.

Lemma 4.1 (1) For every x ∈ X there exist minimal elements x_{0} and maximal elements x^{0} of X with the property x_{0} ≤ x ≤ x^{0}. We denote by X_{max} and X_{min} the set of all maximal, respectively minimal elements of X.

(2) Let x_{1}, x_{2} be arbitrary elements of X. Then there exists x ∈ X_{min} with x ≤ x_{1}, x_{2} if and only if any τ^{et}-neighborhoods U_{1}, U_{2} of x_{1}, x_{2} have nonempty intersection.

Lemma 4.2 Let Y ⊆ X be an arbitrary subset. Then the following assertions are equivalent:

(1) Y is quasi τ^{et}-compact.

(2) env(Y) is quasi τ^{et}-compact.

(3) env(Y) is intersection of τ^{et}-open quasi-compact sets.

(4) Y has a basis of τ^{et}-open quasi-compact neighborhoods.

As an immediate corollary we have

Lemma 4.3 For an arbitrary subspace Y ⊆ X the following assertions are equivalent:

(1) Y is quasi τ^{et}-compact.

(2) env(Y) is τ^{st}-compact. In particular, if Y is quasi τ^{et}-compact, then Y_{max} is defined, and env(Y) = env(Y_{max}).

Definition We say that a subspace Y ⊆ X is standard if it is quasi τ^{et}-compact and Y = Y_{max}.

Lemma 4.4 Let Y ⊆ X be an arbitrary subspace and denote by \overline{Y} its τ^{st}- closure. Then the following assertions are equivalent:

(1) Y is τ^{et}-compact.

(2) Y = (\overline{Y})_{max} and \overline{Y} ∩ gen(Y) = ∅

Thus, if Y is τ^{et}-compact, then for any y ∈ \overline{Y} there exists a unique y_{0} ∈ Y with y ≤ y_{0}.

Proof 1) ⇒ 2). By (4.3) it follows that \overline{Y} ⊆ env(Y). Take y′, y′′ ∈ Y and y ∈ X with y ≤ y′, y′′. Then for any τ^{et}-open neighborhoods U′ and U′′ of y′ respectively y′′ in X it follows that U′ ∩ U′′ is also a τ^{et}-neighborhood of y. In particular, if y lies in \overline{Y} then U′ ∩ U′′ ∩ Y ≠ ⊘. Therefore, U′ ∩ Y and U′′ ∩ Y have non-empty intersection, hence y′ = y′′.

2) ⇒ 1) By (4.3) it follows that Y = (\overline{Y})_{max} is quasiτ^{et}-compact. We show that it is also Hausdorff. Let y′, y′′ be points of Y such that any τ^{et}-open quasicompact neighborhoods U′ and U′′ of them in Y have non-empty intersection. Now by (4.3) it follows that \overline{Y} ∩ env(U′) ∩ env(U′′) is a τ^{st}-closed non-empty set for every U′ and U′′. Therefore, the intersection of all the sets of this form is also non-empty. On the other hand, this intersection equals Z = \overline{Y} ∩ env(y′) ∩ env(y′′). Now for every y ∈ Z one has y ≤ y′, y′′. Therefore, y′ = y′′.

Taking into account that any τ^{et}-closed subset of X is also τ^{st}-closed, by (4.4) one has the following

Corollary 4.5 Let Y ⊆ X be a standard subspace of X. Then for every τ^{st}-closed subset D of X which does not meet gen(\overline{Y}) the set D ∩Y is τ^{et}-compact.

Definition Let (X, τ^{et}) be a quasi-boolean topological space. We say that X is separating if gen(X) is τ^{st}-closed.

Lemma 4.6 Let X be a quasi-boolean topological space. The following assertions are equivalent:

(1) X is separating.

(2) Any y ∈ X_{max} has a τ^{st}-neighborhood U with U ∩ gen(X) = ∅.

(3) Any y ∈ X_{max} has a basis of τ^{st}-clopen neighborhoods U which do not meet gen(X).

Examples of quasi-boolean topological spaces: sg(G) and sf(K).

Let X be an arbitrary boolean space and let expX denote the family of all τ^{st}-closed subsets of X. Let X = \underleftarrow{lim}(X_{i}, φ_{ji}) be a realization of X as projective limit of finite T_{1} spaces and φ_{i} : X → X_{i} the canonical projections. Then expX = \underleftarrow{lim} expX_{i}.

Let us endow all expX_{i} with the ordering defined by the inclusion. The corresponding T_{1}-topology on X_{i} is called the étale topology, denoted τ^{et}. It is obvious that all φ_{ji} respect these orderings, or equivalently, they are τ^{et}-continuous. We define τ^{et} on expX as being the projective limit of the étale topology on X_{i}. A basis of quasi-compact τ^{et}-open neighborhoods of any Y ∈ expX is

U_{i}(Y) = {Y_{i} ∈ expX | φ_{i}(Y_{1}) ⊆ φ_{i}(Y)}, ∀i

One shows without difficulty that τ^{et} does not depend on the concrete realizations X = \underleftarrow{lim}(X_{i}, φ_{ji}) of X as projective limit of finite T_{1} spaces.

The strict topology constructed from τ^{et} is denoted as usually by τ^{st}. We remark that a basis of compact τ^{st}-open neighborhoods of any Y ∈ expX is

V_{i}(Y) = {Y_{i} ∈ expX | φ_{i}(Y_{i}) = φ_{i}(Y)}, ∀i

Further, τ^{st} does not depend on the concrete realizations X = \underleftarrow{lim}(X_{i}, φ_{ji}) of X as projective limit of finite T_{1} spaces.

Finally, we remark that

X → (expX, τ^{et}), X → (expX, τ^{st})

are covariant functors from the category of all boolean spaces into the category of all quasi-boolean spaces, respectively boolean spaces. Namely, for every given continuous mapping φ : X → Y of boolean spaces one defines

expφ : expX → expY by (expφ)(X′) = φ(X′)

and these mappings are τ^{et} and τ^{st}-continuous.

An interesting situation arises when X is a profinite group G viewed as boolean space. To emphasize that we forget about the group structure of G we shall write X = |G|. |G| has a canonical realization as a projective limit of finite discrete spaces, namely that defined by

(∗) G = \underleftarrow{lim} (G/D′, γ_{D′′D′})

where D′ runs over all open normal subgroups of G and γ_{D′′D′} is the canonical projection G/D′′ → G/D′ for D′′ ⊆ D′.

A prominent subspace of exp|G| is the space of all closed subgroups of G, which we denote by sg(G) (Remark: we now write G to emphasize that we consider the projective group and not only its topological structure). sg(G) is τ^{st}-closed in exp|G|, hence (∗) also defines the topologies τ^{et} and τ^{st} on sg(G).

We remark that for every Γ ∈ sg(G), the family

U_{D}(Γ) = {Γ_{1} ∈ sg(G) | DΓ_{1} ⊆ DΓ}, (D ⊂ G open)

is a basis of quasi-compact τ^{et}-open neighborhoods of Γ. Further

V_{D}(Γ) = {Γ_{1} ∈ sg(G) | DΓ_{1} = DΓ} (D ⊂ G open)

is a basis of compact τ^{st}-open neighborhoods of Γ ∈ sg(G).

It is clear that sg( ) defines a co-variant functor from the category of all profinite groups to the category of quasi-boolean spaces, i.e.,

For any morphism of profinite groups φ : F → G the mapping

sg(φ) : sg(F) → sg(G), sg(φ)(Γ) = φ(Γ)

has functorial properties. We shall usually denote sg(φ) simply by φ.

Now suppose that G acts continuously and faithfully on a discrete field L. Then by definition, the orbit Gx of any element x ∈ L is finite (or equivalently, the stabilizer G_{x} ⊆ G of any element x ∈ L is an open subgroup of G). Therefore, L is a Galois extension of the fixed field L_{0} = L^{G}, and G is canonically isomorphic to the Galois group Gal (L|L_{0})) of L|L_{0}.

Definition 1) In the above context, we denote the space of all subextensions of L|L_{0} by sf(L|L_{0}). Via the canonical isomorphism of Gal (L|L_{0}) with G, we endow sf(L|L_{0}) with the topologies τ^{et} and τ^{st}.

2) For an arbitrary field K, we denote by sf(K) the set of all algebraic extensions of K in some separable closure K^{sep} of K, endowed with the topologies τ^{et} and τ^{st} as above.

We remark that for any Λ ∈ sf(L|L_{0}), the family

U_{M}(Λ) = {Λ_{1} ∈ sf(L|L_{0}) | M ∩ Λ ⊆ M ∩ Λ_{1}} M|L_{0} ⊆ L|L_{0} finite

is a basis of quasi-compact τ^{et}-open neighborhoods of Λ. Further,

V_{M}(Λ) = {Λ_{1} ∈ sf(L|L_{0}) | M ∩ Λ = M ∩ Λ} M|L_{0} ⊆ L|L_{0} finite

is a basis of compact τ^{st}-open neighborhoods of Λ ∈ sf(L|L_{0}).

Proposition 4.7 The following holds:

(1) For any injective morphism of profinite groups ι : H → G, the induced mapping

ι : sg(H) → sg(G) defined by Δ ↦ ι(Δ)

is a τ^{et} and τ^{st}-immersion.

(2) For every surjective morphism of profinite groups φ : F → G the induced mapping

φ : sg(F) → sg(G) defined by Φ ↦ φ(Φ)

is a τ^{et}- and τ^{st}-projection.

(3) If G_{1} ⊆ G is a subgroup of G then the mapping

π : sg(G) → sg(G_{1}) defined by Γ ↦ Γ ∩ G_{1}

is τ^{et}-continuous. Corresponding assertions holds for sf(L|L_{0}).

Proposition 4.8 Let G be an arbitrary profinite group acting continuously on a boolean space T.

A) For x ∈ T, let G_{x′} be an open subgroup of G containing G_{x} for some x′ ∈ Gx and further, U_{x′} an open neighborhood of G_{x′}x′. Then there exists a finite family {x_{k}}_{k} of elements x′, and clopen disjoint G_{x_{k}}-invariant neighborhoods U_{k} ⊆ U_{x_{k}} of x_{k} with the properties:

(1) {GU_{k}}_{k} is a disjoint clopen covering of T.

(2) For all m, x ∈ U_{m} and g ∈ G, if gx ∈ ∪_{k}U_{k}, then g ∈ G_{m} and hence

gx ∈ U_{m}.

B) Let H be an open normal subgroup of G such that all G_{x} are conjugate to subgroups of H. Then there exists a boolean H-subspace U ⊆ T with the properties:

(1) GU = T

(2) If gx = y for some g ∈ G and x, y ∈ U, then g ∈ H.

In particular, the canonical projection T/H → T/G has continuous sections.

Proof The proof of A) follows applying the following Lemma.

Lemma 4.9 Let T be a boolean topological G-space. For some x ∈ T let H be an arbitrary open subgroup of G containing G_{x}. Then any neighborhood of Hx contains an H-invariant clopen neighborhood U of Hx which has the property:

g_{1}U ∩ g_{2}U ≠ ∅ ⇔ g_{1}U = g_{2}U ⇔ g_{1}H = g_{2}H, ∀g ∈ G

Proof Fix an open neighborhood U′′ of x containing Gx. For each clopen neighborhood U′ of x set B′_{U′} = {(g, x′) ∈ G × U′ | gx′ ∉ U′′}. By the continuity of the action of G on T, it follows that B′_{U′} is closed. On the other hand, since Gx ⊆ U′′ it follows that ∩_{U′}B′_{U′} is empty. Thus, there exists U′ with GU′ ⊆ U′′.

Working with H instead of G it follows that any neighborhood U′′ of Hx contains HU′ for some clopen neighborhood U′ of x. We now claim: There exist such U′ that U = HU has the properties we want. Indeed, let g_{1}H ≠ g_{2}H. Since G_{x} ⊆ H, we deduce g_{1}Hx ∩ g_{2}Hx = ∅. Therefore, there exists a clopen neighborhood U′ of x such that g_{1}HU ∩ g_{2}HU′ = ∅. Taking into account that there exist only finitely many classes gH, we deduce the existence of U′ with the desired properties.

To prove B) we use once again Lemma 4.9. Namely, for any x take x′ ∈ Gx with G_{x′} ⊆ H. Further, set G_{x′} = H and U_{x′} an arbitrary neighborhood of Hx′.

With the notations from A) we then have: U = ∪_{k}U_{k} is a boolean H-space and if gx = y for some g ∈ G and x, y ∈ U, then g ∈ H.

Let G be an arbitrary profinite group acting continuously on a boolean space X. Then G acts in a canonical way on exp(X), and this action is τ^{et} and τ^{st} continuous.

In particular, we can consider X = |G| endowed with the inner conjugation

G × |G| → |G|, (g, |x|) ↦ |gxg^{−1}|

which is a τ^{et} and τ^{st} continuous left action of G on |G|. In this way we make sg(G) into a τ^{et} and τ^{st} G-space. We shall always consider sg(G) endowed with this action of G.

4.10 Conventions and Notations

1) For x ∈ |G|, G ⊆ sg(G) and g ∈ G we sometimes write x^{g} instead of gxg^{−1}, and also G^{g} instead of gGg^{−1}.

2) For G′ subgroup of G and G ⊆ sg(G), we denote by G^{G′} the least G′-invariant subset of sg(G) containing G.

3) For G ⊆ sg(G), we denote con(G) = env(G^{G}) = (env(G))^{G}. Next, we fix some notations and facts which will be through out the paper.

Let G be an arbitrary profinite group. A) A subset G ⊆ sg(G) is called special if it is τ^{et}-compact and consists only

of self-normalizing elements, i.e., it Γ^{g} = Γ for some Γ ∈ G and g ∈ G, then g ∈ Γ.

B) Let G ⊆ sg(G) be a special G-subspace and D a fixed open normal subgroup of G. By (4.9) and the definition of G, there exist finitely many Γ_{k} ∈ G, τ^{et}-compact disjoint subsets D_{k} ⊆ V_{D}(Γ_{k}) of G such that setting D_{k} = DΓ_{k} the following holds:

(1) D_{k} ⊆ sg(D_{k}) is D_{k}-invariant.

(2) If D_{k} ∩ D_{ι}^{g} ≠ ∅ for some k, l and g ∈ G, then k = 1 and g ∈ D_{k}.

(3) For each k let X_{k} be a set of representatives for G/D_{k}. Then the

finite family (G_{k}^{g})_{k,g∈X_{k}} represents a disjoint covering of G.

C) With the notations from B) we construct a “canonical cover” associated to D in the following way: For every x ∈ G/D, let g ∈ G be a pre-image of it. Further, for each k, let C_{k} = ι_{k}(D_{k}) be an isomorphic copy of D_{k}, and set C_{k} = ι_{k}(D_{k}) the family D_{k} viewed in sg(C_{k}). Now consider the following usual profinite free product together with the canonical projection

π = π_{D} : F_{D} = F_{|G/D|} ∗ ∗_{k}C_{k} → G

where F_{|G/D|} is the profinite free group on the set |G/D|, and π is defined by π(|g|) = g and π|_{C_{k}} = ι_{k}^{−1}. Further, for every k let Y_{k} be a pre-image of X_{k} in F_{D}. Then hC_{k}h^{−1} (k; h ∈ Y_{k}) are τ^{et}-compact disjoint subsets of sg(F_{D}), and we let F be their union. It is clear that π maps F τ^{et}-homeomorphically onto G. Moreover, using the result of Herfort–Ribes [H-R], and taking into account that all Γ ∈ G are self-normalizing it follows that all Φ ∈ F are self-normalizing. Hence, F is a special subspace of sg(F_{D}).

• We say that π : (F_{D}, F) → (G, G) is a “canonical cover” associated to D. The same holds for sf( ).

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