P-ADICALLY PROJECTIVE GROUPS AS ABSOLUTE GALOIS GROUPS^{∗}

by Dan Haran

School of Mathematics, Tel Aviv University

Ramat Aviv, Tel Aviv 69978, Israel e-mail: haran@post.tau.ac.il

and Moshe Jarden

School of Mathematics, Tel Aviv University

Ramat Aviv, Tel Aviv 69978, Israel e-mail: jarden@post.tau.ac.il

and Florian Pop

Department of Mathematics, University of Pennsylvania

Philadelphia, PA 19104-6395, USA e-mail: pop@math.upenn.edu

MR Classification: 12E30 Directory: \Jarden\Diary\HJPb

15 November, 2003

* Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation and by the European Community’s Human Potential Programme under Contract HPRN-CT-2000-00114, GTEM

Introduction We address one of the major problems of Galois theory: the classification of absolute Galois groups among all profinite groups. Specifically, we consider a profinite group G equipped with a subset G of subgroups each of which is isomorphic to an absolute Galois group. We assume that the pair (G, G) satisfies a local global principle with respect to finite embedding problems and a topological condition. The problem is now to classify those pairs for which G is isomorphic to an absolute Galois group of a field K and K satisfies a local-global principle for points on smooth varieties.

The case we consider here has a classical nature. Let F be a finite set of classical local fields of characteristic 0. Each 𝔽 ∈ F is either the field ℝ of real numbers or a finite extension of the field ℚ_{p} of p-adic numbers for some prime number p. We assume F is closed under Galois isomorphism: That is, if 𝔽 and 𝔽′ are finite extensions of ℚ_{p}, 𝔽 ∈ F, and Gal(𝔽′) ≅ Gal(𝔽), then 𝔽′ ∈ F. Here Gal(𝔽) denotes the absolute Galois group of 𝔽.

Main Theorem: Let F be a finite set of classical local fields of characteristic 0 which is closed under Galois isomorphism and let G be a profinite group. Then G is isomorphic to the absolute Galois group of a PFC field K if and only if G is F-projective and Subgr(G, Gal(𝔽)) is strictly closed in Subgr(G) for each 𝔽 ∈ F.

The notions appearing in the Main Theorem: For each 𝔽 ∈ F let AlgExt(K, 𝔽) be the set of all algebraic extensions F of K which are elementarily equivalent to 𝔽. We say that K is PFC (pseudo-F-closed) if it satisfies the following local-global principle: Let V be a smooth absolutely irreducible variety over K. Suppose V (F) ≠ ∅ for each F ∈ ⋃_{F ∈F} AlgExt(K, 𝔽). Then V (K) ≠ ∅.

The notation Subgr(G) stands for the space of all closed subgroups of G. Subgr(G) is the inverse limit of the discrete finite spaces Subgr(G/N) where N ranges of all open normal subgroups of G. Thus, Subgr(G) is a profinite space. We refer to its topology as strict and write Subgr(G, Gal(𝔽)) = {Γ ∈ Subgr(G) | Γ ≅ Gal(𝔽)}.

Finally, G is F-projective if it satisfies the following local-global principle: Let α: B → A be an epimorphism of finite groups and φ: G → A a homomorphism. Suppose for each 𝔽 ∈ F and each Γ ∈ Subgr(G, Gal(𝔽)) there is a homomorphism γ_{Γ}: Γ → B satisfying α ∘ γ_{Γ} = φ|_{Γ}. Then there is a homomorphism γ: G → B with α ∘ γ = φ.

The Main Theorem generalizes several known special cases:

1. First suppose F is the empty set. Then a PFC field is a PAC field [FrJ, Chap. 10] and an F-projective group is a projective group. The Main Theorem is due in this case to Lubotzky-v.d.Dries [FrJ, Cor. 20.16] and Ax [FrJ, Thm. 10.17].

2. If F = {ℝ}, then a PFC field is a PRC field and an F-projective group is a real projective group is a projective group G such that the set of all involutions of G is closed. See [HaJ1, p. 450, Thm.] for the Main Theorem in this case.

3. If F = {ℚ_{p}}, then a PFC field is a PpC field and an F-projective group is p-adically projective G such that the set of all subgroups of G which are isomorphic to Gal(ℚ_{p}) is strictly closed in Subgr(G). See [HaJ2, p. 148, Thm.] for the Main Theorem in this case.

4. The general case is proved in the unpublished and not fully worked out manuscript [Pop2]. The current work is based on ideas of [Pop2] and applies results of [HJPa].

None of those papers goes as far as we do in this work and equips K in the Main Theorem with a set of valuations satisfying the “block approximation theorem”. We refer the reader to Section 10, in particular to Theorem 10.3 for the exact result. Here is suffices to say that each F ∈ AlgExt(K, F) is the Henselian closure of K at a discrete valuation v_{F} and the family {v_{F} | F ∈ AlgExt(K, F)} satisfies a very strong independence-density property.

In the rest of the introduction we describe the structure of the proof of the Main Theorem and the stronger version of the Main Theorem which we actually prove.

Denote the set of all separable algebraic extensions of a field by AlgExt(K). The map F ↦ Gal(F) is a bijection of AlgExt(K) onto Subgr(Gal(K)). It transfers the strict topology of Subgr(Gal(K)) to the strict topology of AlgExt(K). We prove that AlgExt(K, 𝔽) is strictly closed in AlgExt(K) for each 𝔽 ∈ F. In particular, this holds if K is PFC. In this case the existence of points on smooth varieties over K translates into F-projectivity.

Conversely, let G be a profinite group. Suppose G is F-projective and Subgr(G, Gal(𝔽)) is strictly closed in Subgr(G) for each 𝔽 ∈ F. Put C = {Gal(𝔽) | 𝔽 ∈ F}. Let Subgr(G, C) be the set of all H ∈ Subgr(G) which are isomorphic to some Γ in C. Denote the set of all maximal elements in Subgr(G, C) by Subgr(G, C)_{max}. For each Γ ∈ C we construct a finite quotient Γ̄ such that the set {Γ̄ | Γ ∈ C} is a “system of big quotients” of C in a sense made precise in Section 6. This allows us to prove that G is “strongly G-projective” (Proposition 6.5). Thus, every G-embedding problem for G which is locally solvable is globally solvable (Section 6). In particular, there is a homomorphism κ of G into the free product B^{∗} = ∏ ∗_{Γ∈C} Γ which maps each H ∈ Subgr(G, C) isomorphically into a conjugate of some Γ ∈ C. By a theorem of Geyer we may identify B^{∗} with the absolute Galois group of an algebraic extension K_{0} of ℚ. Denote the fixed field of κ(G) in ℚ̃ by K_{1}. By Proposition 6.5, G = (G, Subgr(G, C)_{max}) is a “proper projective group structure” (Section 5). Let Gal(K_{1}) = (Gal(K_{1}), Subgr(Gal(K_{1}, C)). Then κ extends to a Galois cover κ: G → Gal(K_{1}) of group structures (Proof of Proposition 10.3). By the main result of HJPa, there is a field K and an isomorphism φ: G → Gal(K) such that res ∘ φ = κ. Moreover, every F ∈ AlgExt(K, F) is either real closed or elementarily equivalent to some 𝔽 ∈ F. In particular, F has a “P -adic valuation” v_{F}. The system of fields F ∈ AlgExt(K, F) and valuations v_{F} satisfies a strong version of the weak approximation theorem which we call the “block approximation theorem”. In particular, K is PFC (Theorem 10.3).

1. The Étale and the Strict Topologies of Subgr(G)

Let G be a profinite group. Denote the collection of all closed (resp. open, open normal) subgroups of G by Subgr(G) (resp. Open(G), OpenNormal(G)). We introduce two topologies on Subgr(G) and relate them to each other.

For each H, N ∈ Open(G) with N ◁ G let

V(H, N) = {A ∈ Subgr(G) | AN = HN}.

The collection of all V(H, N) is a basis for a topology on Subgr(G) which we call the strict topology. When G is finite, the strict topology is the discrete topology. In general, Subgr(G) ≅ \underleftarrow{lim} Subgr(G/N) with N ranging over all open normal subgroups

of G. Thus, Subgr(G) is a profinite space under the strict topology. We use the adverb “strictly” as a replacement for “in the strict topology”. For example, for a subset G of Subgr(G) we say G is strictly open (resp. closed, compact, Hausdorff) if it is open (resp. closed, compact, Hausdorff) in the strict topology. Likewise, for a function f from a topological space X into Subgr(G), we say f is strictly continuous if f is continuous when Subgr(G) is equipped with the strict topology. We denote the strict closure of a subset G of Subgr(G) by StrictClosure(G).

If U_{1}, …, U_{m} ∈ Open(G), then U = ⋂_{i=1}^{m} U_{i} is open and Subgr(U) is ⋂_{i=1}^{m} Subgr(U_{i}). Therefore {Subgr(U) | U ∈ Open(G)} is a basis for a topology on Subgr(G) which we call the étale topology. As above, for a subset G of Subgr(G) we say G is étale open (closed, compact, Hausdorff, etc) if G is open (closed, compact, Hausdorff, etc) in the étale topology. Likewise, for a function f from a topological space X into Subgr(G) we say f is étale continuous if f is continuous when Subgr(G) is equipped with the étale topology.

The envelope of a subset G of Subgr(G) is the set of all H_{0} ∈ Subgr(G) which are contained in some H ∈ G. We denote it by Env(G) and use it to relate the strict topology and the étale topology of Subgr(G) to each other:

Lemma 1.1: A subset G of Subgr(G) is étale compact if and only if Env(G) is strictly closed.

Proof: Suppose first Env(G) is strictly closed, hence strictly compact. Let U_{i}, i ∈ I, be open subgroups of G with G ⊆ ⋃_{i∈I} Subgr(U_{i}). Then Env(G) ⊆ ⋃_{i∈I} Subgr(U_{i}). Each of the sets Subgr(U_{i}) is strictly open [HJPa, Remark 1.2]. Hence, I has a finite subset I_{0} with Env(G) ⊆ ⋃_{i∈I_{0}} Subgr(U_{i}). Thus, G ⊆ ⋃_{i∈I_{0}} Subgr(U_{i}). Therefore, G is étale compact.

Conversely, suppose G is étale compact. Consider A ∈ Subgr(G) ∖ Env(G) and H ∈ G. Then A ≰ H. Hence, there is N_{H} ∈ OpenNormal(G) with A ≰ HN_{H}. Thus, A is not in the étale open neighborhood Subgr(HN_{H}) of H.

The collection of all Subgr(HN_{H}) covers G. Since G is étale compact, there are H_{1}, …, H_{n} ∈ G and N_{1}, …, N_{n} ∈ OpenNormal(G) with G ⊆ ⋃_{i=1}^{n} Subgr(H_{i}N_{i}) and A ∉ ⋃_{i=1}^{n} Subgr(H_{i}N_{i}). In addition, ⋃_{i=1}^{n} Subgr(H_{i}N_{i}) is strictly closed. Hence, StrictClosure(G) ⊆ ⋃_{i=1}^{n} Subgr(H_{i}N_{i}). Thus, A belongs to the strictly open set Subgr(G) ∖ ⋃_{i=1}^{n} Subgr(H_{i}N_{i}) which is disjoint from Env(G). Therefore, A is not in StrictClosure(Env(G)). It follows that Env(G) is strictly closed.

Corollary 1.2: Let G be an étale compact subset of Subgr(G). Then StrictClosure(G) is contained in Env(G).

For a profinite group G, a closed subgroup H, and a subset G of Subgr(G) let G^{H} = {Γ^{h} | Γ ∈ G, h ∈ H}. Put Con(G) = Env(G^{G}) = Env(G)^{G}.

Lemma 1.3: Let G be an étale compact subset of Subgr(G). Then each of the sets G^{G}, Env(G), and Con(G) is étale compact.

Proof: The set G^{G} is the image of the compact space G × G under the étale continuous map (Γ, g) ↦ Γ^{g}. Hence, G^{G} is étale compact.

By Lemma 1.1, Env(G) is strictly closed, hence étale compact [HJPa, Remark 1.2]. Therefore, by the first paragraph, Con(G) = Env(G)^{G} is étale compact.

Lemma 1.4: Let H be a closed subgroup of G. Then ÉtaleClosure({H}) = {B ∈ Subgr(G) | H ≤ B}.

Proof: First suppose B ∈ ÉtaleClosure({H}). Then H belongs to each étale open neighborhood of B. In other words, if U ∈ Open(G) and B ≤ U, then H ≤ U. Hence, H ≤ B.

Conversely, suppose H ≤ B. Then, H belongs to each basic étale open neighborhood Subgr(U) of B. Therefore, B ∈ ÉtaleClosure({H}).

Lemma 1.5: Let φ: G → H be an epimorphism of profinite groups and G_{0} a closed subgroup of G. The set {B ∈ Subgr(G) | φ(G_{0}) ≤ φ(B)} is étale closed.

Proof: By [HJPa, Remark 1.1(b)], the map φ: Subgr(G) → Subgr(H) induced by φ is étale continuous. By Lemma 1.4, the set {C ∈ Subgr(H) | φ(G_{0}) ≤ C} is étale closed. Its inverse image in Subgr(G) is {B ∈ Subgr(G) | φ(G_{0}) ≤ φ(B)}, so it is étale closed.

Lemma 1.6: Let G be an étale compact subset of Subgr(G). Then each A ∈ G is contained in a maximal element of G.

Proof: By Zorn’s lemma, it suffices to prove that each ascending chain G_{0} in G is bounded by an element of G. Consider B_{1}, ⋯, B_{n} ∈ G_{0}. Then the B_{i} are comparable. Assume B_{1} ≤ B_{2} ≤ … ≤ B_{n}. By Lemma 1.4, B_{n} ∈ ⋂_{i=1}^{m} ÉtaleClosure({B_{i}}). Thus, G∩ ⋂_{i=1}^{n} ÉtaleClosure({B_{i}}) ≠ ∅. Since G is étale compact, G ∩ ⋂_{B∈G_{0}} ÉtaleClosure({B}) is nonempty. Each element of the latter set is a bound of G_{0}.

If a subset G of Subgr(G) contains groups A and B with A < B, then G is not étale Hausdorff. Thus, removing all nonmaximal elements from G is the only way to make G étale Hausdorff while preserving the essential information stored in G. We denote the set of all maximal elements of G by G_{max}.

Lemma 1.7: Let G be an étale compact subset of Subgr(G). Then G_{max} is étale compact.

Proof: By Lemma 1.6, Env(G) = Env(G_{max}). Hence, by Lemma 1.1, G_{max} is étale compact.

2. Relatively Projective Groups

Pairs (G, G) consisting of a profinite group and a subset G of Subgr(G) which satisfy a local global principle for finite embedding problems naturally arise from pairs (K, X) consisting of a field K and a set X of separable algebraic extensions of K which satisfy a local global principle for points on absolutely irreducible varieties (Section 3). We prove in Proposition 3.1 that G is “G-projective” in a sense we now explain:

Let G be a profinite group and G a subset of Subgr(G). An embedding problem for G is a pair

(1)  (φ: G → A, α: B → A),

where φ is a homomorphism and α is an epimorphism of profinite groups. The embedding problem is finite if A and B are finite. We call (1) a G-embedding problem if it is locally solvable; that is

(2) for each Γ ∈ G there exists a homomorphism γ_{Γ}: Γ → B with α ∘ γ_{Γ} = φ|_{Γ}. We say (1) is a rigid G-embedding problem if

(3) for each Γ ∈ G there is B_{0} ∈ Subgr(B) such that α: B_{0} → φ(Γ) is an isomorphism. A solution of (1) is a homomorphism γ: G → B with γ ∘ α = φ. We say G

is G-projective if every finite G-embedding problem for G is solvable. Our definition generalizes the definition of a “projective group”. Indeed, “G is ∅-projective” means “G is projective”. We refer to G as relatively projective if G is G-projective for a subset G of Subgr(G).

If G ⊆ G′ ⊆ Subgr(G) and G is G-projective, then G is G′-projective. Moreover, if G is étale compact, then by Lemma 1.6, each group in G is contained in a group of G_{max}. Hence, under the assumption that G is étale compact, G is G-projective if and only if G is G_{max}-projective. Hence, G is G-projective if and only if G is G^{G}-projective.

Suppose (1) is a rigid G-embedding problem and Γ and B_{0} are as in (3). Put A_{0} = φ(Γ) and γ_{Γ} = (α|_{A_{0}})^{−1} ∘ φ|_{Γ}. Then γ_{Γ}: Γ → B is a homomorphism satisfying α ∘ γ_{Γ} = φ|_{Γ}. Thus, every rigid G-embedding problem is a G-embedding problem. The next lemma establishes a sort of converse to this statement:

Lemma 2.1: Let G be a profinite group and G an étale compact subset of Subgr(G). Let (1) be a finite G-embedding problem for G. Then:

(a) There exists a commutative diagram

\xymatrix{ & G \ar[d]_{φ̂} \ar[dd]^{φ} \\ B̂ \ar[r]^{α̂} \ar[d]_{β} & Â \ar[d]_{φ̄} \\ B \ar[r]^{α} & A}

in which φ̂ is an epimorphism and (φ̂: G → Â, α̂: B̂ → Â) is a finite rigid Gembedding problem.

(b) If every finite rigid G-embedding problem (1) for G in which φ is an epimorphism is solvable, then G is G-projective.

Proof of (a): Consider Γ ∈ G. Choose a homomorphism γ_{Γ} with α ∘ γ_{Γ} = φ|_{Γ}. Then Ker(γ_{Γ}) is an open subgroup of Γ. Choose N_{Γ} ∈ OpenNormal(G) with N_{Γ} ≤ Ker(φ) and Γ ∩ N_{Γ} ≤ Ker(γ_{Γ}). Then Subgr(ΓN_{Γ}) is an étale open neighborhood of Γ in Subgr(G) and γ_{Γ} extends to a homomorphism γ′_{Γ}: ΓN_{Γ} → B with kernel Ker(γ_{Γ})N_{Γ}.

Since G is étale compact, there are Γ_{1}, …, Γ_{m} ∈ G with G ⊆ ⋃_{i=1}^{m} Subgr(Γ_{i}N_{Γ_{i}}). Then N = ⋂_{i=1}^{m} N_{Γ_{i}} ∈ OpenNormal(G) and N ≤ Ker(φ). Let Â = G/N, φ̂: G → Â the quotient map, and φ̄: Â → A the map induced by φ. Then φ = φ̄ ∘ φ̂. Now consider the fiber product B̂ = B ×_{A} Â with the projection maps α̂: B̂ → Â and β: B̂ → B on the coordinates. Since α is surjective, so is α̂.

For each i put N_{i} = N_{Γ_{i}} and γ_{i} = γ′_{Γ_{i}}: Γ_{i}N_{i} → B. Then there is a homomorphism γ̂_{i}: Γ_{i}N_{i} → B̂ satisfying α̂ ∘ γ̂_{i} = φ̂|_{Γ_{i}N_{i}} and β ∘ γ̂_{i} = γ_{i} [FrJ, Prop. 20.6]. Put B̂_{i} = γ̂_{i}(Γ_{i}N_{i}).

Claim: α̂: B̂_{i} → φ̂(Γ_{i}N_{i}) is an isomorphism. It suffices to prove that α̂ is injective on B̂_{i}. Indeed, consider b ∈ B̂_{i} with α̂(b) = 1. Choose g ∈ Γ_{i}N_{i} with γ̂_{i}(g) = b. Then φ̂(g) = α̂(γ̂_{i}(g))) = 1, so g ∈ N ≤ N_{i}. Thus, β(b) = γ_{i}(g) = 1. Therefore, b = 1, as desired.

Now consider Γ ∈ G. Choose i with Γ ≤ Γ_{i}N_{i}. Then φ̂(Γ) ≤ φ̂(Γ_{i}N_{i}). By the Claim, α̂ maps α̂^{−1}(φ̂(Γ))∩B̂_{i} isomorphically onto φ̂(Γ). Hence, (φ̂: G → Â, α̂: B̂ → Â) is a finite G-embedding problem for G satisfying the rigidity condition.

Proof of (b): Consider a finite G-embedding problem (1) for G. Let (φ̂: G → Â, α̂: B̂ → Â) be the embedding problem given by (a). By assumption, it has a solution γ̂. Then γ = β ∘ γ̂ solves (1).

3. Pseudo Closed Fields Let K be a field and X a subset of the set SepAlgExt(K) of all separable algebraic extensions of K. By an absolutely irreducible variety over K we mean a geometrical integral scheme of finite type over K. We say K is pseudo-X -closed (abbreviated PX C) if it satisfies the following condition:

(1) Every smooth absolutely irreducible variety over K, with an F -rational point for each F ∈ X, has a K-rational point. If V is an arbitrary absolutely irreducible variety over K, then the Zariski open

subset V_{simp} of all simple points of V is also an absolutely irreducible variety over K. Hence, (1) is equivalent to the following condition:

(2) Every absolutely irreducible variety over K, with a simple F -rational point for each F ∈ X, has a K-rational point.

Note that K is P∅C if and only if K is PAC [FrJ, Chap. 10].

Under a mild topological assumption on X, the PX C property of K results in a relative projectivity of Gal(K). The topology in question is the étale topology of the space SepAlgExt(K). This space stands in a bijective correspondence with Subgr(Gal(K)). Thus, SepAlgExt(K) inherits the étale topology from that of Subgr(Gal(K)). Basic étale open subsets of SepAlgExt(K) are SepAlgExt(L) with L/K finite and separable.

If X ⊆ X′ ⊆ SepAlgExt(K) and K is PX C, then K is PX′C. Denote the set of all minimal fields in X by X_{min}. If X is étale compact, then by Lemma 1.6, each field in X contains a minimal field in X. Hence, K is PX C if and only if K is PX_{min}C.

Proposition 3.1: Let K be a field and X a subset of SepAlgExt(K). Put G = {Gal(K′) | K′ ∈ X }. Suppose X is étale compact and K is PX C. Then Gal(K) is G-projective.

Proof: By Lemma 1.3, G^{G} is étale compact. If we prove that Gal(K) is G^{G}-projective, it will follow that Gal(K) is G-projective. We may therefore assume, G is Gal(K)- invariant.

By Lemma 2.1, it suffices to solve every finite rigid G-embedding problem

(3)  (φ: Gal(K) → A, α: B → A)

where φ is an epimorphism.

Let L be the fixed field of Ker(φ) in K_{s}. Then identify A with Gal(L/K) and φ with res_{K_{s}/L}. Next use [HJPa, Lemma 6.2] to construct a finitely generated regular extension E of K and a finite Galois extension F of E containing L with these properties: (4a) B = Gal(F/E) and α is the restriction map res_{F/L}: Gal(F/E) → Gal(L/K). (4b) Let L_{0} be a field between K and L and F_{0} a field between E and F which contains

L_{0}. Suppose res_{F/L}: Gal(F/F_{0}) → Gal(L/L_{0}) is an isomorphism. Then F_{0} is a purely transcendental extension of L_{0}.

Since E/K is finitely generated and regular, one may view E as the function field of an absolutely irreducible smooth affine variety V over K [FrJ, Cor. 9.23].

Now let {L_{i} | i ∈ I} = {K′ ∩ L | K′ ∈ X }. By rigidity, choose for each i ∈ I a field F_{i} between E and F containing L_{i} such that res_{F/L}: Gal(F/F_{i}) → Gal(L/L_{i}) is an isomorphism. By (4b), F_{i} is a purely transcendental extension of L_{i}. Hence, V (L_{i}) ≠ ∅. Therefore, V (K′) ≠ ∅ for each K′ ∈ X. Since K is PX C-closed, V has a K-rational point, which by assumption is simple. By [JaR, Cor. A2], E has a valuation which is trivial on K and with K as its residue field. [HJPa, Lemma 7.4] gives an algebraic extension E′ of E such that res_{E′_{s}/K_{s}}: Gal(E′) → Gal(K) is an isomorphism. Denote its inverse by γ′. Then γ = res_{E′_{s}/F} ∘ γ′ solves (3).

Again, let K be a field and X a subset of SepAlgExt(K). For each algebraic extension L of K let X_{L} = {K′L | K′ ∈ X }. If [L : K] < ∞, then L is PX_{L}C [Jar1, Lemma 7.2]. The same result holds for arbitrary L if X is strictly closed [Jar1, Lemma 7.4]. Here we prove that L is PX_{L}C under the weaker condition that X is étale compact.

Proposition 3.2 (Extension theorem): Let K be a field, X an étale compact subset of SepAlgExt(K), and L a separable algebraic extension of K. Suppose K is PX C. Then X_{L} is étale compact and L is PX_{L}C.

Proof: The map K′ ↦ K′L from X to X_{L} is étale continuous. Since X is étale compact, so is X_{L}.

Next let V be a smooth absolutely irreducible variety over L with V (K′L) ≠ ∅ for each K′ ∈ X. Choose a finite subextension K_{1} of L/K over which V is already defined. Denote the set of all finite subextensions of L/K_{1} by E. For each E ∈ E let T_{E} = {K′ ∈ X | V (K′E) ≠ ∅}.

Claim: T_{E} is étale open in X. Indeed, let K′ ∈ T_{E}. Then V (K′E) ≠ ∅. Hence, K′/K has a finite subextension K′_{0}/K with V (K′_{0}E) ≠ ∅. The open neighborhood X ∩ SepAlgExt(K′_{0}) of K′ in X is contained in T_{E}. Therefore, T_{E} is open in X.

Since L = ⋃_{E∈E} E, we have ⋃_{E∈E} T_{E} = X. Since X is étale compact, E has a finite subset E_{0} with ⋃_{E∈E_{0}} T_{E} = X. Let F be the union of all E ∈ E_{0}. Then T_{E} ⊆ T_{F} for each E ∈ E_{0}, so X = T_{F}. Thus, V (K′F) ≠ ∅ for each K′ ∈ X. By [Jar1, Lemma 7.4] F is PX_{F} C. Hence, V (F) ≠ ∅, so V (L) ≠ ∅. Consequently, L is PX_{L}C.

4. Strongly Projective Groups

Consider a profinite group G and a subset G of Subgr(G). Suppose G is G-projective. If G is empty, then G is projective, so every embedding problem for G is solvable [FrJ, Lemma 20.4]. Unfortunately, we are able to solve an arbitrary G-embedding problem in the general case only if we impose a strong condition on the global solution of each finite embedding problem: The solution has to map every local group Γ ∈ G into a subset of Subgr(B), given in advance, which is closed under conjugations and taking subgroups. In addition, we have to assume that every Γ ∈ G is maximal in G and 1 ∉ ÉtaleClosure(G). Section 5 shows that when these conditions are fulfilled, G, G naturally give rise to a proper projective group structure G = (G, X, G_{x})_{x∈X} in the sense of [HJPa, Section 4]. By [HJPa, Prop. 4.2], every embedding problem for G is solvable.

Consider again a profinite group G and a subset G of Subgr(G). A G-embedding problem for G with local data is a triple

(1)  (φ: G → A, α: B → A, B)

where A and B are profinite groups, B is a strictly closed subset of Subgr(B) which is closed under conjugations and taking closed subgroups, φ is a homomorphism, and α is an epimorphism. In addition, we assume α is B-rigid. That is:

(2) φ(G) ⊆ α(B) and α is injective on each B_{0} ∈ B. Call (1) finite if B is finite.

Occasionally we achieve B as above in the following way. Let B_{0} be a subset of Subgr(B). Then B = Con(B_{0}) is the set of all subgroup of B which are contained in B_{0}^{b} for some B_{0} ∈ B_{0} and b ∈ B. By Lemma 1.1, B is strictly compact (hence closed) if B_{0} is étale compact. In particular, this is the case if B_{0} is finite.

A solution of (1) is a homomorphism γ: G → B with γ(G) ⊆ B. Call G strongly G-projective if every finite G-embedding problem (1) for G with local data has a solution. If in addition G is étale compact, then G is G-projective.

Indeed, let (φ: G → A, α: B → A) be a rigid G-embedding problem in the sense of Section 2, (3). For each Γ ∈ G choose B_{Γ} ∈ Subgr(B) such that α: B_{Γ} → φ(Γ) is an isomorphism. Let B = Con(B_{Γ} | Γ ∈ G). Then (1) is a G-embedding problem for G with local data and α is B-rigid. By assumption, there exists a homomorphism γ: G → B with α ∘ γ = φ. By Lemma 2.1(b), G is G-projective. Moreover, by Lemmas 1.6 and 1.7, G is étale compact and G is strongly G_{max}-projective Example 4.1: Free product of finitely many profinite groups. Consider a free product G = ∏ ∗_{i=1}^{n} G_{i} of finitely many profinite groups. Put G = {G_{1}, …, G_{n}}. Then G is strongly G-projective.

Indeed, let (1) be a finite embedding problem for G with local data. Then φ maps each G_{i} onto a subgroup A_{i} of A and there is B_{i} ∈ B which α maps isomorphically onto A_{i}. Then γ_{i} = (α|_{B_{i}})^{−1} ∘ (φ|_{G_{i}}) is an epimorphism of G_{i} onto B_{i}. Extend γ_{1}, …, γ_{n} to a homomorphism γ: G → B. Then γ solves embedding problem (1).

Remark 4.2: Suppose G is étale compact and G is a strongly G-projective group. An obvious modification of Lemma 2.1 proves (1) is solvable even if α is not necessarily rigid but satisfies the weaker condition instead:

(3) For each Γ ∈ G there is B_{0} ∈ B and a homomorphism γ_{0}: Γ → B_{0} with α∘γ_{0} = φ|_{Γ}. However, we do not use (3) in the definition of strong projectivity because all embedding problems which we use in this work satisfy the condition “α is B-rigid”.

Lemma 4.3: Let G be a strongly G-projective group with G ⊆ Subgr(G). Then every embedding problem with local data (1) such that A is finite and rank(B) ≤ ℵ_{0} is solvable.

Proof: Put N_{0} = Ker(α) and identify A with B/N_{0} and α with the quotient map B → B/N_{0}. Choose a descending sequence N_{i} ∈ OpenNormal(B) with N_{i} ≤ Ker(α), i = 1, 2, 3, …, and ⋂_{i=1}^{n} N_{i} = 1. For j ≥ i let α_{ji}: B/N_{j} → B/N_{i} and β_{i}: B → B/N_{i} be the quotient maps. For each i, B/N_{i} = β_{i}(B) is closed under conjugation and taking subgroups. The map α is injective on each B_{0} ∈ B, so α_{i+1,i} is injective on B_{0}/N_{i+1}. Therefore, we may inductively construct a sequence of homomorphisms γ_{i}: G → B/N_{i} satisfying: γ_{0} = φ, γ_{i}(G) ⊆ B/N_{i}, and α_{i+1,i} ∘ γ_{i+1} = γ_{i}, i = 1, 2, 3, ….

The γ_{i}’s define a homomorphism γ: G → B with β_{i} ∘ γ = γ_{i}, i = 0, 1, 2, …. Since B is strictly closed, B = \underleftarrow{lim} B/N_{i}. Hence, γ(G) ⊆ B. Therefore, γ is a solution of (1).

Free products of finitely many profinite groups have some nice properties:

Lemma 4.4 ([HeR, Prop. 2 and Thm. B’]): Let G = ∏ ∗_{i∈I} G_{i} be the free profinite product of finitely many profinite groups G_{i}. Then G_{i}^{g} ∩ G_{j} ≠ 1 implies i = j and g ∈ G_{i}.

Lemma 4.4 carries over to strongly G-projective groups:

Proposition 4.5: Let G be a profinite group and G an étale compact subset of Subgr(G) which is closed under conjugation. Suppose G is strongly G-projective. Then:

(a) Γ_{1} ∩ Γ_{2} = 1 for all distinct Γ_{1}, Γ_{2} ∈ G_{max}.

(b) N_{G}(Γ) = Γ for each nontrivial Γ ∈ G_{max}.

Proof: Consider an epimorphism φ: G → A with A finite. Write φ(G) = {A_{i} | i ∈ I} with I finite. For each i ∈ I choose an isomorphic copy B_{i} of A_{i}. Choose a large positive integer e and put B = F̂_{e} ∗ ∏ ∗_{i∈I} B_{i}. Then there is an epimorphism α: B → A which maps B_{i} isomorphically onto A_{i}. Let B = Con(B_{1}, …, B_{n}). Then, (1) is a G-embedding problem for G with local data. By Lemma 4.3, there is a homomorphism γ: G → B with α ∘ γ = φ and γ(G) ⊆ B.

Proof of (a): Assume Γ_{1} ∩ Γ_{2} ≠ 1. Choose N_{0} ∈ OpenNormal(G) with Γ_{1}N_{0} ≠ Γ_{2}N_{0}. Consider N ∈ OpenNormal(G) with N ≤ N_{0}. Put A = G/N and let φ: G → G/N be the quotient map. Then let B, B_{i}, B, α, and γ be as above. In particular, γ(Γ_{i}) ∈ B, i = 1, 2. Hence, there are j, k ∈ I and b_{j}, b_{k} ∈ B with γ(Γ_{1}) ≤ B_{j}^{b_{j}} and γ(Γ_{2}) ≤ B_{k}^{b_{k}}. Also, α(γ(Γ_{1}) ∩ γ(Γ_{2})) ⊆ φ(Γ_{1}) ∩ φ(Γ_{2}) ≠ 1, hence γ(Γ_{1}) ∩ γ(Γ_{2}) ≠ 1, so

B_{j}^{b_{j}} ∩ B_{k}^{b_{k}} ≠ 1. By Lemma 4.4, B_{j}^{b_{j}} = B_{k}^{b_{k}}. Consider Γ_{N} ∈ G with φ(Γ_{N}) = α(B_{j}^{b_{j}}). Then, φ(Γ_{i}) ≤ φ(Γ_{N}), i = 1, 2. It follows that the set G_{N} = {Γ ∈ G | φ(Γ_{1}), φ(Γ_{2}) ≤ φ(Γ)} is nonempty. By Lemma 1.5, G_{N} is étale closed. If N_{1}, …, N_{m} are open normal subgroups of G and N = ⋂_{j=1}^{m} N_{j}, then G_{N} ⊆ ⋂_{j=1}^{m} G_{N_{j}}. Hence, since G is weakly compact, ⋂_{N∈OpenNormal(G)} G_{N} ≠ ∅. Each Γ in this intersection belongs to G and satisfies Γ_{1} ∩ Γ_{2} ≤ Γ. Since Γ_{1} and Γ_{2} are maximal in G, Γ_{1} = Γ = Γ_{2}, in contradiction to assumption.

Proof of (b): Let g ∈ G with Γ^{g} = Γ. Choose N_{0} ∈ OpenNormal(G) with Γ ≰ N_{0}. Consider N ∈ OpenNormal(G) with N ≤ N_{0}. Put A = G/N and let φ: G → G/N be the quotient epimorphism. Then let B, B_{i}, B, and α be as in the first paragraph of the proof. In particular, there is i ∈ I and b ∈ B with γ(Γ) ≤ B_{i}^{b}. Also, γ(Γ)^{γ(g)} = γ(Γ) and γ(Γ) ≠ 1 so B_{i}^{b} ∩ B_{i}^{bγ(g)} ≠ 1. By Lemma 4.4, γ(g) ∈ B_{i}^{b}. Choose Γ_{N} ∈ G with φ(Γ_{N}) = α(B_{i}^{b}). Then φ(Γ) ≤ φ(Γ_{N}) and φ(g) ∈ φ(Γ_{N}).

Again, by Lemma 1.5, the nonempty set

G′_{N} = {Γ′ ∈ G | φ(Γ) ≤ φ(Γ′), φ(g) ∈ φ(Γ′)}

is étale closed. Since G is étale compact, there is Γ′ which belongs to all G′_{N}. It satisfies Γ ≤ Γ′ and g ∈ Γ′. Since Γ is maximal in G, we have Γ = Γ′. Therefore, g ∈ Γ.

Lemma 4.6: Let G be a profinite group and G an étale compact subset of Subgr(G) which is closed under conjugation. Suppose 1 ∉ StrictClosure(G) and G is strongly G-projective. Then:

(a) G is strongly G_{max}-projective.

(b) G_{max} is étale compact Hausdorff.

(c) N_{G}(Γ) = Γ for each Γ ∈ G_{max}.

Proof of (a): By Lemma 1.7, G_{max} is étale compact. Suppose (1) is a finite G_{max}- embedding problem with local data for G. In particular, φ(G_{max}) ⊆ α(B) and α is injective on each B_{0} ∈ B. We prove (1) is a G-embedding problem with local data for G. To this end let Γ_{0} ∈ G. By Lemma 1.6, Γ_{0} is contained in some Γ ∈ G_{max}. Choose B_{1} ∈ B with α(B_{1}) = φ(Γ). Let B_{0} = B_{1} ∩ α^{−1}(φ(Γ_{0})). Then B_{0} ∈ B, and α maps B_{0} isomorphically onto φ(Γ_{0}), as needed.

Since G is strongly G-projective, there is a homomorphism γ: G → B with α∘γ = φ and γ(G) ⊆ B, so γ(G_{max}) ⊆ B. Thus, G is strongly G_{max}-projective.

Proof of (b): By (a) and Proposition 4.5(a), Γ_{1} ∩ Γ_{2} = 1 for all distinct Γ_{1}, Γ_{2} ∈ G_{max}. Hence, by [HJPa, Cor. 1.4], G_{max} is étale Hausdorff.

Proof of (c): By assumption, each Γ ∈ G_{max} is nontrivial. Hence, by Proposition 4.5(b), N_{G}(Γ) = Γ.

Our next goal is to prove under the assumptions of Lemma 4.6 that G_{max} is a profinite space in the étale topology. By definition, a profinite space X is an inverse limit of discrete finite spaces. In particular, X has a basis consisting of open-closed sets. Conversely, every compact Hausdorff space which has a basis consisting of open-closed sets is profinite (See also [RiZ, Thm. 1.1.12] for the connection with condition “X is totally disconnected”.)

Lemma 4.7: Let X be a compact Hausdorff space and G a profinite group which acts continuously on X. Suppose X/G has a basis consisting of open-closed sets. Then X is profinite.

Proof: Let x ∈ X and W an open neighborhood of x. We have to find an open-closed neighborhood of x in W.

Part A: G is finite. Let S = {σ ∈ G | x^{σ} = x}. Write G = ⋃ · _{i=1}^{m} Sσ_{i} with σ_{1} = 1. Then x^{σ_{1}}, …, x^{σ_{m}} are the distinct conjugates of x. Since X is Hausdorff, there are open neighborhoods V_{1}, …, V_{m} of x in W such that V_{1}^{σ_{1}}, …, V_{m}^{σ_{m}} are disjoint. Put V = ⋂_{i=1}^{m} ⋂_{σ∈S} V_{i}^{σ}. This is an S-invariant open neighborhood of x in W and V^{σ_{1}}, …, V^{σ_{n}} are disjoint.

The quotient map π: X → X/G is continuous and open. In particular, π(V) is an open neighborhood of π(x) in X/G. By assumption, there is an open-closed neighborhood Ū of π(x) in X/G with Ū ⊆ π(V). Then, U = π^{−1}(Ū) is an openclosed G-invariant neighborhood of x in X and U ⊆ π^{−1}(π(V)) = ⋃ · _{i=1}^{m} V^{σ_{i}}. Therefore U = ⋃ · _{i=1}^{m} U ∩ V^{σ_{i}}. Since the sets U ∩ V^{σ_{i}} are open, they are also closed in U, and hence in X. Thus U ∩ V = U ∩ V^{σ_{1}} is an open-closed neighborhood of x contained in W.

Part B: G is arbitrary. The action X × G → X is continuous and x^{1} ∈ W. Hence, x has an open neighborhood V and G has an open normal subgroup N with V^{N} ⊆ W. Let ν: X → X/N be the quotient map. Then ν(V) is an open neighborhood of ν(x) in X/N. Since X is compact Hausdorff, so is X/N [Bre, Thm. 3.1(1)]. The finite group G/N acts on X/N continuously and X/G = (X/N)/(G/N). Thus, by Part A, X/N is profinite. Therefore, ν(x) has an open-closed neighborhood Ū in X/N with Ū ⊆ ν(V). Therefore U = ν^{−1}(Ū) is an open-closed neighborhood of x in X and U ⊆ V^{N} ⊆ W, as desired.

Proposition 4.8: Let G be a profinite group and G an étale compact G-invariant subset of Subgr(G). Suppose 1 ∉ StrictClosure(G) and G is strongly G-projective. Then G_{max} is étale profinite.

Proof: By Lemma 4.6, G is strongly G_{max}-projective and G_{max} is étale Hausdorff compact. We may therefore replace G by G_{max}, if necessary, to assume G = G_{max} and prove that the étale topology of G has a basis consisting of étale open-closed sets.

Let π: Subgr(G) → Subgr(G)/G be the quotient map modulo conjugation. Put a bar over each group in Subgr(G) and each subset of Subgr(G) to denote their images under π. By Lemma 4.7, it suffices to prove that the étale topology of Ḡ has a basis consisting of étale open-closed sets. Thus, given Γ_{0} ∈ G and H ∈ Open(G) with Γ_{0} ≤ H, it suffices to find a G-invariant étale open-closed subset U_{0} of Subgr(G) with Γ̄_{0} ∈ Ū_{0} ⊆ \overline{Subgr(H)}. The construction of U_{0} breaks up into three parts.

Part A: An open normal subgroup of G. Since 1 ∉ ÉtaleClosure(G), there is N_{0} ∈ OpenNormal(G) which contains no Γ ∈ G. Consider the étale open subset H = ⋃_{g∈G} Subgr(H^{g}) and the étale closed subset H′ = G ∖ H of G. Both H and H′ are Ginvariant. Since G is étale compact and Hausdorff, so is H′. Each Γ ∈ H′ is not contained in H, so Γ ≠ Γ_{0}. Since G = G_{max}, Γ_{0} ≰ Γ. Therefore, there is N_{Γ} ∈ OpenNormal(G) with N_{Γ}Γ_{0} ≰ N_{Γ}Γ and N_{Γ}Γ ≰ N_{Γ}H.

The set Subgr(N_{Γ}Γ) is an étale open neighborhood of Γ in Subgr(G). Since H′ is étale compact, there are Δ_{1}, …, Δ_{m} ∈ H′ with H′ ⊆ ⋃_{i=1}^{m} Subgr(N_{Δ_{i}}Δ_{i}). Then N = N_{0} ∩ ⋂_{i=1}^{m} N_{Δ_{i}} is an open normal subgroup of G, NΓ ≠ N for each Γ ∈ G, and NΓ_{0} ≰ NΓ, NΓ ≰ NH for each Γ ∈ H′.

Part B: G-embedding problem for G with local data. Put A = G/N and let φ: G → A be the quotient map. By Part A, (4a) φ(Γ) ≠ 1 for each Γ ∈ G and (4b) φ(Γ_{0}) ≰ φ(Γ) and φ(Γ) ≰ φ(H) for each Γ ∈ H′.

Now choose Γ_{1}, …, Γ_{n} ∈ H′ such that φ(Γ_{1}), …, φ(Γ_{n}) represent the conjugacy classes in A of the maximal elements of φ(H′). Let B_{0} be an isomorphic copy of φ(H) and B_{i} an isomorphic copy of φ(Γ_{i}), i = 1, …, n. Choose a positive integer e ≥ rank(A). Put B = F̂_{e} ∗ ∏ ∗_{i=0}^{n} B_{i}. Then B is finitely generated and there is an epimorphism α: B → A which maps B_{0} isomorphically onto φ(H) and B_{i} isomorphically onto φ(Γ_{i}), i = 1, …, n. Let B = Con(B_{0}, …, B_{n}). Then

(5)  (φ: G → A, α: B → A, B)

is a G-embedding problem for G with local data. By Lemma 4.3, there is a homomorphism γ: G → B with α ∘ γ = φ and γ(G) ⊆ B.

Part C: Partition of G. For each i let B′_{i} be an identical copy of B_{i}. Let B′ = ∏_{i=0}^{n} B′_{i} be the direct product of B_{0}, …, B_{n}. Let β: B → B′ be the epimorphism which maps F̂_{e} to 1 and each B_{i} identically onto B′_{i}. Put γ′ = β ∘ γ. For each i put U_{i} = {Γ ∈ G | γ′(Γ) ≤ B′_{i}}. Then U_{i} is a G-invariant étale open subset of G. Therefore, Ū_{i} is an étale open subset of Ḡ.

Claim C1: Ḡ = ⋃ · _{i=0}^{m} Ū_{i}. Indeed, since γ(G) ⊆ B, there are for each Γ ∈ G an i between 0 and m and a b ∈ B with γ(Γ) ≤ B_{i}^{b}. Hence, γ′(Γ) ≤ B′_{i} and Γ̄ ∈ Ū_{i}.

Moreover, by (4a), α(γ(Γ)) = φ(Γ) ≠ 1. Hence, γ(Γ) ≠ 1. Since β is injective on B_{i}^{b}, we have γ′(Γ) ≠ 1. Hence, γ′(Γ) ≰ B′_{j}, so Γ̄ ∉ Ū_{j} for all j ≠ i.

It follows that each Ū_{i} is a étale open-closed subset of Ḡ. In particular, so is Ū_{0}.

Claim C2: Γ̄_{0} ∈ Ū_{0}. Indeed, γ(Γ_{0}) ≤ B_{i}^{b} with 0 ≤ i ≤ n and b ∈ B (Claim C1). Assume i ≥ 1. Then φ(Γ_{0}) = α(γ(Γ_{0})) ≤ α(B_{i})^{α(b)} = φ(Γ_{i}^{g}) for some g ∈ G and Γ_{i}^{g} ∈ H′ (by the choice of Γ_{i}). Hence, by (4b), φ(Γ_{0}) ≰ φ(Γ_{i}^{g}). This contradiction proves that i = 0, γ′(Γ) ≤ B′_{0}, and Γ̄_{0} ∈ Ū_{0}.

Claim C3: Ū_{0} ⊆ H̄. Indeed, consider Γ ∈ U_{0}. By Part B, γ(Γ) ≤ B_{i}^{b} with 0 ≤ i ≤ n and b ∈ B. If i ≥ 1, then γ′(Γ) ≤ B′_{i} and Γ̄ ∈ Ū_{i}, in contradiction to Claim C1. Hence, γ(Γ) ≤ B_{0}^{b}. Therefore, φ(Γ^{g}) ≤ φ(H) for some g ∈ G with φ(g) = α(b)^{−1}. Since H′ is G-invariant, (4b) implies Γ ∉ H′. Consequently, Γ ∈ H, as desired.

Finally, observe that H̄ = \overline{Subgr(H)} to conclude the proof of the proposition.

5. Projective Group Structures

The crucial step of going from solvability of finite G-embedding problems for a profinite group G to solvability of arbitrary G-embedding problems occurs in the category of “profinite groups structures”. We recall the definition of this concept from [HJPa, Section 2].

A profinite group structure is a data G = (G, X, G_{x})_{x∈X} where G is a profinite group, X is a profinite space on which G acts continuously from the right, and G_{x} is a closed subgroup of G, x ∈ X. These objects must satisfy the following conditions: (1a) The map x ↦ G_{x} from X into Subgr(G) is étale continuous. (1b) G_{x^{g}} = G_{x}^{g} for all x ∈ X and g ∈ G. (1c) {g ∈ G | x^{g} = x} ≤ G_{x} for each x ∈ X. The structure G is finite if both G and X are finite.

A morphism of group structures φ: (G, X, G_{x}) → (H, Y, H_{y})_{y∈Y} is a couple consisting of a homomorphism φ: G → H and a continuous map φ: X → Y such that φ(G_{x}) ≤ H_{φ(x)} and φ(x^{g}) = φ(x)^{φ(g)} for all x ∈ X and g ∈ G. The morphism φ is an epimorphism if φ(G) = H, φ(X) = Y, and for each y ∈ Y, there is x ∈ X with φ(G_{x}) = H_{y}. We call φ a cover if φ(G) = H, φ(X) = Y, φ: G_{x} → H_{φ(x)} is an isomorphism for each x ∈ X, and φ(x) = φ(x′) implies x^{k} = x′ for some k ∈ Ker(φ).

An embedding problem for G is a pair (φ: G → A, α: B → A) where A and B are profinite group structures, φ is a morphism, and α is a cover. A solution of the problem is a morphism γ: G → B satisfying α ∘ γ = φ. The problem is finite if both B and A are finite. We say G is projective if every finite embedding problem for G is solvable. Then every embedding problem for G is solvable [HJPa, Prop. 4.2].

Lemma 5.1: Let G = (G, X, G_{x})_{x∈X} be a projective group structure. Put G = {G_{x} | x ∈ X}. Then G is strongly G-projective.

Proof: By definition, G is the image of the compact space X under the étale continuous map x ↦ G_{x}. Hence, G is étale compact.

Now consider a finite G-embedding problem

(2)  (φ: G → A, α: B → A, B)

for G with local data. Replace A by A_{0} = φ(G), B by B_{0} = α^{−1}(φ(G)), and B by B_{0} = B ∩ Subgr(B_{0}), if necessary, to assume φ is surjective. By [HJPa, Lemma 3.8], φ: G → A extends to an epimorphism φ of G onto a finite group structure A = (A, I, A_{i})_{i∈I}. Choose a set of representatives I_{0} for the A-orbits of I. For each i ∈ I_{0} there exists x ∈ X with φ(x) = i and φ(G_{x}) = A_{i}. The rigidity condition (2) of Section 4 gives B′ ∈ B which α maps isomorphically onto A_{i}. Hence, by [HJPa, Lemma 4.5], there is a group structure B = (B, J, B_{j})_{j∈J} and α: B → A extends to a cover α: B → A with B_{j} ∈ B for all j ∈ J. In particular, φ: J → I is an epimorphism with finite fibers, so J is finite, hence B is finite. Since G is projective, there is a morphism γ: G → B with α ∘ γ = φ. Its group component γ: G → B solves embedding problem (2). Consequently, G is strongly G-projective.

Lemma 5.2: Let G be a profinite group and G an étale compact subset of Subgr(G). Suppose G = ⋃ · _{i∈I} G_{i} is a partition of G into finitely many disjoint open-closed subsets. Then there exists an open normal subgroup N of G such that if φ: G → A is an epimorphism with Ker(φ) ≤ N and Γ, Γ′ ∈ G satisfy φ(Γ) ≤ φ(Γ′), then there is i ∈ I with Γ, Γ′ ∈ G_{i}.

Proof: Let Γ ∈ G. There Γ belongs to a unique G_{i}. Since G_{i} is open in G, there is an open normal subgroup N_{Γ} of G with G ∩ Subgr(ΓN_{Γ}) ⊆ G_{i}. Thus, G ⊆ ⋃_{Γ∈G} Subgr(ΓN_{Γ}). Since G is étale compact, there are Γ_{1}, …, Γ_{m} ∈ G with G ⊆ ⋃_{j=1}^{m} Subgr(Γ_{j}N_{Γ_{j}}).

For each 1 ≤ j ≤ m there is a unique i(j) ∈ I with G ∩ Subgr(Γ_{j}N_{Γ_{j}}) ⊆ G_{i(j)}. Put N = ⋂_{j=1}^{m} N_{Γ_{j}}. Let φ: G → A be an epimorphism with Ker(φ) ≤ N and let Γ, Γ′ ∈ G with φ(Γ) ≤ φ(Γ′). Choose j between 1 and m with Γ′ ∈ Subgr(Γ_{j}N_{Γ_{j}}). Then Γ′ ∈ G_{i(j)}. Hence, Γ ≤ Γ′Ker(φ) ≤ Γ′N ≤ Γ′N_{Γ_{j}} ≤ Γ_{j}N_{Γ_{j}}. Therefore, Γ ∈ G_{i(j)}, as desired.

Let G = (G, X, G_{x})_{x∈X} be a group structure. Put G = {G_{x} | x ∈ X}. We say G proper if the map x ↦ G_{x} of X onto G is an étale homeomorphism.

Proposition 5.3: Let G = (G, X, G_{x})_{x∈X} be a proper group structure. Let G = {G_{x} | x ∈ X}. Suppose G is strongly G-projective. Then G is projective.

Proof: Consider a finite embedding problem

(3)  (φ: G → A, α: B → A)

for G with A = (A, I, A_{i})_{i∈I}. The solution of this problem breaks up into three parts.

Part A: A partition of G. Consider the partition X = ⋃ · _{i∈I} φ^{−1}(i) into open-closed sets. For each i ∈ I let G_{i} = {G_{x} | φ(x) = i}. Since the map x ↦ G_{x} is an étale homeomorphism, G = ⋃ · _{i∈I} G_{i} is a partition of G into étale open-closed sets. Lemma 5.2 gives an open normal subgroup N of G such that if φ̂: G → Â is an epimorphism with ker(φ̂) ≤ N, then

(4) x, y ∈ X and φ̂(G_{x}) ≤ φ̂(G_{y}) imply φ(x) = φ(y).

By [HJPa, Lemma 3.8] there are a morphism φ̄: Â → A of finite group structures and an epimorphism φ̂: G → Â such that φ = φ̄ ∘ φ̂ and Ker(φ̂) ≤ N. In particular,

(4) holds.

The fiber product B̂ = (B̂, Ĵ, B̂_{j})_{j∈Ĵ} = B ×_{A} Â fits into a commutative diagram

\xymatrix{ & G \ar[d]^{φ̂} \\ B̂ \ar[r]^{α̂} \ar[d]_{β} & Â \ar[d]^{φ̄} \\ B \ar[r]^{α} & A}

in which α̂ is a cover [HJPa, Lemma 2.12(c)]. Let B̂ = Con({B̂_{j} | j ∈ Ĵ}). Then (φ̂: G → Â, α̂: B̂ → Â, B̂) is an embedding problem for G with a local data. Since G is strongly G-projective, there exists a homomorphism γ̂: G → B̂ such that α̂ ∘ γ̂ = φ̂ and (5) for each x ∈ X there is j ∈ Ĵ with γ̂(G_{x}) ≤ B̂_{j}.

Part B: The map γ̂: X → Ĵ. Consider the open normal subgroup K = Ker(γ̂) of G. For each y ∈ X, the open subgroup G_{y}K of G contains S_{y} = {σ ∈ G | y^{σ} = y}. Also, V_{y} = {x ∈ X | G_{x} ≤ G_{y}K} of y is an open neighborhood of y in X which is G_{y}K-invariant. Indeed, if σ ∈ G_{y}, κ ∈ K, and x ∈ V_{y}, then

G_{x^{σκ}} = G_{x}^{σκ} ≤ (G_{y}K)^{σκ} = (G_{y}^{σ}K^{σ})^{κ} = (G_{y}K)^{κ} = G_{y}K,

whence x^{σκ} ∈ V_{y}.

By [HJPa, Lemma 3.6], there are y_{1}, …, y_{m} ∈ X and open-closed subsets X_{1}, …, X_{m} of X such that the following holds for each k between 1 and m: (6a) X_{k} is G_{y_{k}}K-invariant and y_{k} ∈ X_{k} ⊆ V_{y_{k}}. (6b) X = ⋃ · _{k=1}^{m} ⋃ · _{τ∈T_{k}} X_{k}^{τ}, where G = ⋃ · _{τ∈T_{k}} G_{y_{k}}Kτ and 1 ∈ T_{k} ⊆ G.

Define γ̂: X → Ĵ as follows. For each k between 1 and m use (5) to choose γ̂(y_{k}) ∈ Ĵ with γ̂(G_{y_{k}}) ≤ B̂_{γ̂(y_{k})}. Then let

(7)  γ̂(y^{τ}) = γ̂(y_{k})^{γ̂(τ)} for all y ∈ X_{k} and τ ∈ T_{k}.

By (6b), γ̂: X → Ĵ is well defined. In addition,

(8) γ̂ is constant on each X_{k}^{τ} with τ ∈ T_{k}. Hence, by (6b), γ̂ is continuous.

Taking τ = 1 in (7), gives γ̂(y) = γ̂(y_{k}) for all y ∈ X_{k}. Hence, by (7),

(9)  γ̂(y^{τ}) = γ̂(y)^{γ̂(τ)} for all y ∈ X_{k} and τ ∈ T_{k}.

We claim that

(10) γ̂(G_{x}) ≤ B̂_{γ̂(x)} for every x ∈ X.

Indeed, x = y^{τ}, where y ∈ X_{k}, τ ∈ T_{k}. By (6a), y ∈ V_{y_{k}}, that is, G_{y} ≤ G_{y_{k}}K, so γ̂(G_{y}) ≤ γ̂(G_{y_{k}}) ≤ B̂_{γ̂(y_{k})} = B̂_{γ̂(y)}. By (9), γ̂(y)^{γ̂(τ)} = γ̂(x). Hence, γ̂(G_{x}) = γ̂(G_{y}^{τ}) =

γ̂(G_{y})^{γ̂(τ)} ≤ B̂_{γ̂(y)}^{γ̂(τ)} = B̂_{γ̂(y)^{γ̂(τ)}} = B̂_{γ̂(x)}, as claimed.

We know that α̂ ∘ γ̂ = φ̂ on G. But we do not know that α̂ ∘ γ̂ = φ̂ on X. Therefore, we define γ = β ∘ γ̂: G → B and γ = β ∘ γ̂: X → J and prove directly that γ: G → A is a morphism which solves embedding problem (3).

Part C: The morphism γ: G → B. An application of β on (8), (9), and (10) implies: (11a) γ is constant on each X_{k}^{τ} with τ ∈ T_{k}, so, by (6b), γ: X → J is continuous. (11b) γ(y^{τ}) = γ(y)^{γ(τ)} for all y ∈ X_{k} and τ ∈ T_{k} (11c) γ(G_{x}) ≤ B_{γ(x)} for every x ∈ X.

Claim C1: α ∘ γ = φ. That α ∘ γ = φ on G follows from the equality α̂ ∘ γ̂ = φ̂ on G. Consider therefore x ∈ X. Since φ̂: G → Â is an epimorphism, there is y ∈ X such that φ̂(y) = α̂(γ̂(x)) and φ̂(G_{y}) = Â_{α̂(γ̂(x))}. By (10),

φ̂(G_{x}) = α̂(γ̂(G_{x})) ≤ α̂(B̂_{γ̂(x)}) = Â_{α̂(γ̂(x))} = φ̂(G_{y}).

By (4), φ(x) = φ(y). In addition,

φ(y) = φ̄(φ̂(y)) = φ̄(α̂(γ̂(x))) = α(β(γ̂(x))) = α(γ(x)).

Hence, φ(x) = α(γ(x)), as claimed.

Claim C2: γ(G_{y_{k}}) is contained in the stabilizer S_{γ(y_{k})} of γ(y_{k}) in B. Let j = γ(y_{k}). By Claim C1, α(j) = φ(y_{k}). By [HJPa, Remark 2.1], G_{y_{k}} = S_{y_{k}}. Hence,

(12)  α(γ(G_{y_{k}})) = φ(G_{y_{k}}) = φ(S_{y_{k}}) ≤ S_{φ(y_{k})} = S_{α(j)}.

By (11c), γ(G_{y_{k}}) ≤ B_{j}. Since α: B → A is a cover, α: B_{j} → A_{α(j)} is an isomorphism that maps S_{j} onto S_{α(j)} [HJPa, Lemma 2.2]. Therefore, by (12), γ(G_{y_{k}}) ≤ S_{γ(y_{k})}.

Claim C3: γ preserves the action. We prove first that γ(y^{σ}) = γ(y)^{γ(σ)} for all y ∈ X_{k} and σ ∈ G. To this end we use (6b) to write σ = λτ with λ ∈ G_{y_{k}}K and τ ∈ T_{k}. Then γ(λ) ∈ γ(G_{y_{k}}K) = γ(G_{y_{k}}). Hence, by Claim C2, γ(y_{k})^{γ(λ)} = γ(y_{k}). Whence, by (11a), γ(y)^{γ(λ)} = γ(y). By (6a), y^{λ} ∈ X_{k}^{λ} = X_{k}. Hence, by (11a), γ(y^{λ}) = γ(y), and by (11b), γ((y^{λ})^{τ}) = γ(y^{λ})^{γ(τ)}. Therefore, γ(y^{σ}) = γ((y^{λ})^{τ}) = γ(y^{λ})^{γ(τ)} = γ(y)^{γ(τ)} = γ(y)^{γ(λ)γ(τ)} = γ(y)^{γ(σ)}.

Now consider x ∈ X_{k}^{τ′} with τ′ ∈ T_{k}. Write x = y^{τ′} with y ∈ X_{k}. Let g ∈ G. By the preceding paragraph, γ(x^{g}) = γ(y^{τ′g}) = γ(y)^{γ(τ′g)} = γ(y)^{γ(τ′)γ(g)} = γ(y^{τ′})^{γ(g)} = γ(x)^{γ(g)}, as claimed.

Thus, γ is a solution of embedding problem (3).

Let G be a profinite group and G an étale profinite G-invariant subset of Subgr(G). Suppose N_{G}(Γ) = Γ for each Γ ∈ G. Choose a homeomorphic copy X of G and a homeomorphism x ↦ G_{x} of X onto G. The action of G on G induces an action on X making G = (G, X, G_{x})_{x∈X} a proper group structure. In this case we also refer to (G, G) as a proper group structure. We call (G, G) projective if G is projective.

Proposition 5.4: Let G be a profinite group and G an étale compact G-invariant subset of Subgr(G). Suppose 1 ∉ StrictClosure(G) and G is strongly G-projective. Then (G, G_{max}) is a proper projective group structure.

Proof: By Lemma 4.6, N_{G}(Γ) = Γ for each Γ ∈ G_{max} and G is strongly G_{max}-projective. By Proposition 4.8, G_{max} is étale profinite. It follows, (G, G_{max}) is a proper group structure. By Proposition 5.3, (G, G_{max}) is projective.

Remark 5.5: Relatively projective groups. Let G and G be as in Proposition 5.4. Then G_{max} is étale profinite. Let Γ_{1}, Γ_{2} be distinct groups in G_{max}. By Lemma 4.5, Γ_{1}∩Γ_{2} = 1. Choose étale open-closed neighborhoods U_{1} and U_{2} of Γ_{1} and Γ_{2} in G_{max}, respectively, with G = U_{1} ⋃ · U_{2}. By [HJPa, Lemma 2.3], the union of all Γ in U_{i} is a closed subset of G. Thus, G is separated in the sense of [Har, Def. 3.1]. In addition, G is strongly G_{max}- projective. Consequently, G is projective relative to G in the sense of [Har, Def. 4.2].

We interpret the notions of a “morphism” and a “cover” of proper group structures in terms of the pairs (G, G): Let (H, H) and (G, G) be proper group structures. Then a morphism φ: (H, H) → (G, G) is just a homomorphism φ: H → G which maps H into Con(G). In other words, for each Δ ∈ H there is Γ ∈ G with φ(Δ) ≤ Γ.

The morphism φ is a cover if (13a) φ(H) = G, φ(H) = G, (13b) φ is injective on each Δ ∈ H, and (13c) if Δ, Δ′ ∈ H and φ(Δ) = Δ′, then there exists κ ∈ Ker(φ) with Δ^{κ} = Δ′.

A sub-group-structure of (H, H) is a proper group structure (H_{0}, H_{0}) with H_{0} ≤ H and H_{0} ⊆ H. Specializing [HJPa, Cor. 4.3] to proper group structures gives the following result:

Proposition 5.6: Let φ: (H, H) → (G, G) be a cover of proper group structures. Suppose (G, G) is projective. Then (H, H) has a sub-group-structure (H_{0}, H_{0}) which φ maps isomorphically onto (G, G).

6. Big Quotients

Let G be a profinite group and G a subset of Subgr(G). We have already mentioned in Section 4 that if G is strongly G-projective, then G is G-projective. We show in this section that the converse is also true if G there are only finitely many isomorphism types of groups in G and they have a “system of big quotients”.

Let C be a finite set of finitely generated profinite groups. Each profinite group Δ which is isomorphic to a group in C is of type C. A set G of profinite groups is said to be of type C if each H ∈ G is of type C.

Let G be a profinite group and G a subset of Subgr(G). For each Γ ∈ C let G_{Γ} = {H ∈ G | H ≅ Γ}. We prove an analog of Lemma 4.3 for G-projective groups:

Lemma 6.1: Let G be a profinite group and G a subset of Subgr(G) of type C. Suppose G is G-projective and

(1)  (φ: G → A, α: B → A)

is a G-embedding problem with A finite and rank(B) ≤ ℵ_{0}. Then (1) is solvable.

Proof: There exists a descending sequence Ker(α) = N_{0} ≥ N_{1} ≥ N_{2} ≥ ⋯ of open normal subgroups of B with trivial intersection. Identify A with B/N_{0} and α with the quotient map B → B/N_{0}. Let φ_{0} = φ and α_{0} = α. For each i and j with j ≥ i ≥ 0 let α_{i}: B → B/N_{i} and α_{ji}: B/N_{j} → B/N_{i} be the quotient maps.

Claim: Let i ≥ 0 and let φ_{i}: G → B/N_{i} be a homomorphism such that (φ_{i}: G → B/N_{i}, α_{i}: B → B/N_{i}) is a G-embedding problem for G. Then there is a homomorphism φ_{i+1}: G → B/N_{i+1} such that α_{i+1,i} ∘ φ_{i+1} = φ_{i} and (φ_{i+1}: G → B/N_{i+1}, α_{i+1}: B → B/N_{i+1}) is a G-embedding problem for G.

Once the claim has been proved, we may inductively construct for each i ≥ 0 a homomorphism φ_{i+1}: G → B/N_{i+1} with α_{i+1,i} ∘ φ_{i+1} = φ_{i}. The maps φ_{i} define a γ ∈ Hom(G, B) with α ∘ γ = φ.

Without loss we prove the claim for i = 0. To this end note that for each j, (φ: G → A, α_{j0}: B/N_{j} → A) is a finite G-embedding problem of G. Indeed, given Γ ∈ G, there is a homomorphism γ′: Γ → B with α ∘ γ′ = φ|_{Γ}. Thus, α_{j0} ∘ (α_{j} ∘ γ′) = φ|_{Γ}, as desired.

For each β ∈ Hom(G, B/N_{j}) let

β ∘ ∏_{Γ∈C} Hom(Γ, G) = {(β ∘ ψ_{Γ})_{Γ∈C} | ψ_{Γ} ∈ Hom(Γ, G) for each Γ ∈ C}.

This is a subset of ∏_{Γ∈C} Hom(Γ, B/N_{j}). Since C is finite, each Γ ∈ C is finitely generated, and B/N_{j} is finite, ∏_{Γ∈C} Hom(Γ, B/N_{j}) is finite. Hence, the collection of subsets

H_{j} = {β ∘ ∏_{Γ∈C} Hom(Γ, G) | β ∈ Hom(G, B/N_{j}), α_{j0} ∘ β = φ}

of ∏_{Γ∈C} Hom(Γ, B/N_{j}) is finite. Since G is G-projective, H_{j} is nonempty.

The map β ∘∏_{Γ∈C} Hom(Γ, G) ↦ α_{j+1,j} ∘β ∘∏_{Γ∈C} Hom(Γ, G) maps H_{j+1} into H_{j}. Hence, \underleftarrow{lim} H_{j} ≠ ∅. Thus, there are homomorphisms β_{j}: G → B/N_{j} with α_{j0} ∘ β_{j} = φ

and

(2)  α_{j+1,j} ∘ β_{j+1} ∘ ∏_{Γ∈C} Hom(Γ, G) = β_{j} ∘ ∏_{Γ∈C} Hom(Γ, G), j = 0, 1, 2, ….

In particular, α_{1,0} ∘ β_{1} = φ.

We prove that (β_{1}: G → B/N_{1}, α_{1}: B → B/N_{1}) is a G-embedding problem for G. To this end consider Γ ∈ C and H ∈ G_{Γ}. Then H ≅ Γ. Hence, by (2),

(3)  α_{j+1,j} ∘ β_{j+1} ∘ Hom(H, G) = β_{j} ∘ Hom(H, G), j = 0, 1, 2, …, .

Use (3) to inductively construct homomorphisms η_{j}: H → B/N_{j}, j = 1, 2, … with η_{1} = β_{1}|_{H} and α_{j+1,j} ∘ η_{j+1} = η_{j}. The η_{j}’s define a homomorphism η: H → B with α_{1} ∘ η = β_{1}, as needed. This concludes the proof of the claim.

Lemma 6.2: Let C be a finite set of finitely generated profinite groups, G a profinite group, and G a subset of Subgr(G) of type C. Consider a finite G-embedding problem with local data for G

(4)  (φ: G → A, α: B → A, B).

Then there are (5a) a positive integer e, (5b) a finite set {Δ_{λ} | λ ∈ Λ} of groups of type C, and (5c) an epimorphism β: B^{∗} = F̂_{e} ∗ ∏ ∗_{λ∈Λ} Δ_{λ} → B such that

(6)  (φ: G → A, α ∘ β: B^{∗} → A)

is a G-embedding problem for G with A finite, rank(B^{∗}) ≤ ℵ_{0}, and β(Δ_{λ}) ∈ B for each λ ∈ Λ.

Proof: The proof has two parts.

Part A: Free product. For each Γ ∈ C let Λ_{Γ} be the set of all homomorphisms λ: Γ → B satisfying

(7) λ(Γ) ∈ B and there is an embedding ε: Γ → G such that α ∘ λ = φ ∘ ε. Since Γ is finitely generated and B is finite, Λ_{Γ} is a finite set. For each λ ∈ Λ_{Γ}

choose an isomorphic copy Δ_{λ} of Γ and an isomorphism δ_{λ}: Δ_{λ} → Γ. Let Λ = ⋃ · _{Γ∈C} Λ_{Γ}. Then {Δ_{λ} | λ ∈ Λ} is a finite set of groups of type C. Put e = rank(B). Choose an epimorphism β_{e}: F̂_{e} → B. Then consider the free product B^{∗} = F̂_{e} ∗ ∏ ∗_{λ∈Λ} Δ_{λ}. Let β: B^{∗} → B be the unique epimorphism whose restriction to F̂_{e} is β_{e} and to Δ_{λ} is λ ∘ δ_{λ}. By (7), β(Δ_{λ}) ∈ B for all λ ∈ Λ.

Part B: G-embedding problem. Let Γ ∈ C and H ∈ G_{Γ}. Then, there is an isomorphism θ: Γ → H. The definition of embedding problems with local data (Section 4) gives a homomorphism η of H into B such that η(H) ∈ B and α ∘ η = φ ∘ ι, where ι is the inclusion H → G. Put λ = η ∘ θ and ε = ι ∘ θ. Then α ∘ λ = φ ∘ ε, so λ ∈ Λ_{Γ}. Thus, δ_{λ}^{−1} ∘ θ^{−1} maps H onto the subgroup Δ_{λ} of B^{∗}. Furthermore, α ∘ β ∘ (δ_{λ}^{−1} ∘ θ^{−1}) = α ∘ (λ ∘ δ_{λ}) ∘ (δ_{λ}^{−1} ∘ θ^{−1}) = α ∘ λ ∘ θ^{−1} = α ∘ η ∘ θ ∘ θ^{−1} = φ ∘ ι. Therefore, (6) is a G-embedding problem for G.

Lemma 6.3: Let G be a profinite group and G a subset of Subgr(G) of type C. For each Γ ∈ C let Γ̄ be a finite quotient of Γ. Suppose G_{Γ} is strictly closed in Subgr(G). Then, G has an open normal subgroup N satisfying this: For each Γ ∈ C and each H ∈ G_{Γ} the group Γ̄ is a quotient of H/H ∩ N = HN/N.

Proof: Let Γ ∈ C and H ∈ G_{Γ}. Then H ≅ Γ, so H has an open normal subgroup M_{H} with H/M_{H} ≅ Γ̄. Choose N_{H} ∈ OpenNormal(G) with H ∩ N_{H} ≤ M_{H}. Let U_{H} = {H′ ∈ G_{Γ} | H′N_{H} = HN_{H}}. Then U_{H} is a strictly open neighborhood of H in G_{Γ}. By assumption, G_{Γ} is strictly compact. Hence, there are H_{Γ,1}, …, H_{Γ,m(Γ)} ∈ G_{Γ}

with G_{Γ} = ⋃_{i=1}^{m(Γ)} U_{H_{Γ,i}}.

Let N = ⋂_{Γ∈C} ⋂_{i=1}^{m(Γ)} N_{H_{Γ,i}}. Consider Γ ∈ C and H ∈ G_{Γ}. Then there is i with HN_{H_{Γ,i}} = H_{Γ,i}N_{H_{Γ,i}}. By construction, N ≤ N_{H_{Γ,i}}. This gives a sequence H/H ∩ N ⟶ H/H ∩ N_{H_{Γ,i}} ≅ HN_{H_{Γ,i}}/N_{H_{Γ,i}} = H_{Γ,i}N_{H_{Γ,i}}/N_{H_{Γ,i}} ≅ H_{Γ,i}/H_{Γ,i} ∩ N_{H_{Γ,i}} ⟶ H_{Γ,i}/M_{H_{Γ,i}} ≅ Γ̄ where the arrows are epimorphisms. Therefore, Γ̄ is a quotient of H/H ∩ N.

Definition 6.4: Big quotients. For each Γ ∈ C let Γ̄ be a finite quotient of Γ. We say {Γ̄ | Γ ∈ C} is a system of big quotients for C if has the following property: Let e be a nonnegative integer, J a finite set, and for each j ∈ J let Δ_{j} be a profinite group of type C. Consider the free product B^{∗} = F̂_{e} ∗ ∏ ∗_{j∈J} Δ_{j}. Let Γ ∈ C and let Δ be a closed

subgroup of B^{∗} with epimorphisms Γ ⟶^{γ} Δ → Γ̄. Then Δ is conjugate to a closed subgroup of some Δ_{j} and γ is an isomorphism.

Proposition 6.5: Let C be a finite set of finitely generated groups, G a profinite group, and G a G-invariant subset of Subgr(G) of type C. Suppose C has a system of finite big quotients and G_{Γ} is strictly closed in Subgr(G) for each Γ ∈ C, and G is G-projective. Then:

(a) G is strongly G-projective.

(b) There is a homomorphism δ: G → ∏ ∗_{Γ∈C} Γ which maps each H ∈ G injectively into a conjugate of some Γ ∈ C.

(c) Suppose in addition, 1 ∉ C. Then (G, G_{max}) is a proper projective group structure.

Proof of (a): By assumption, G = ⋃_{Γ∈C} G_{Γ} is strictly closed. By [HJPa, Remark 1.2], G is étale compact. It remains to solve a finite G-embedding problem for G with local data (4). Let {Γ̄ | Γ ∈ C} be a system of finite big quotients for C.

Part A: Γ̄ is a quotient of φ(H) for each Γ ∈ C and each H ∈ G. Lemma 6.3 gives N ∈ OpenNormal(G) such that Γ̄ is a quotient of H/H ∩ N for all Γ ∈ C and H ∈ G_{Γ}. We may assume N ≤ Ker(φ), otherwise replace N with N ∩ Ker(φ). Put A′ = G/N. Let φ′: G → A′ be the quotient map and φ̄: A′ → A the map induced by φ. Then Γ̄ is a quotient of φ′(H) for all Γ ∈ C and H ∈ G_{Γ}. Also, φ = φ̄ ∘ φ′. Put B′ = B ×_{A} A′ and let α′: B′ → A′ and β: B′ → B be the canonical projections.

Put B′_{0} = {B_{0} ×_{A} φ′(H) | B_{0} ∈ B, H ∈ G, α(B_{0}) = φ(H)} and B′ = Con(B′_{0}). Then φ′(G) ⊆ α′(B′). By definition, α is injective on each B_{0} ∈ B. Therefore, α′ is injective on each B′_{0} ∈ B′_{0}, hence on each B′_{0} ∈ B′. Thus,

(8)  (φ′: G → A′, α′: B′ → A′, B′)

is a finite G-embedding problem with local data for G.

Since β(B′) ⊆ B, any solution γ′ of (8) gives rise to a solution β ∘ γ′ of embedding problem (4). Thus, replacing (4) with (8), if necessary, we may assume Γ̄ is a quotient of φ(H) for each Γ ∈ C and each H ∈ G.

Part B: Solving embedding problem (6). Lemma 6.2 gives a G-embedding problem (6) for G with rank(B^{∗}) ≤ ℵ_{0}, and β(Δ_{λ}) ∈ B for each λ ∈ Λ. Since G is G-projective, Lemma 6.1 gives a homomorphism γ^{∗}: G → B^{∗} with α ∘ β ∘ γ^{∗} = φ. We claim that β ∘ γ^{∗} solves (4).

Let Γ ∈ C and H ∈ G_{Γ}. Put Δ = γ^{∗}(H). Then Δ is a subgroup of B^{∗} as well as a quotient of Γ. Moreover, α(β(Δ)) = φ(H). Therefore, by Part A, Γ̄ is a quotient of Δ. By the definition of big quotients, Δ is conjugate to a closed subgroup of Δ_{λ} for some λ ∈ Λ and γ^{∗} is injective on H. Since β(Δ_{λ}) ∈ B, we have β ∘ γ^{∗}(H) ∈ B.

Proof of (b): The proof of (a), say, starting with embedding problem (4) with B = A = 1, gives a homomorphism γ^{∗}: G → F̂_{e}∗∏ ∗_{λ∈Λ} Δ_{λ} which maps each H ∈ G isomorphically into a conjugate of some Δ_{λ}. Define a homomorphism δ^{∗}: F̂_{e} ∗ ∏ ∗_{λ∈Λ} Δ_{λ} → ∏ ∗_{Γ∈C} Γ which is the trivial map on F̂_{e} and maps each Δ_{λ} ∈ G_{Γ} isomorphically onto Γ. Then δ = δ^{∗} ∘ γ^{∗} is a homomorphism of G into ∏ ∗_{Γ∈C} Γ which maps each H ∈ G injectively into a conjugate of some Γ ∈ C.

Proof of (c): Since Γ ≠ 1 for each Γ ∈ C and G is strictly closed, 1 ∉ StrictClosure(G). By (a), G is strongly G-projective. Therefore, by Proposition 5.4, (G, G_{max}) is a proper projective group structure.

7. P-adically Closed Fields

We prove in the next section that any finite family of absolute Galois groups of Padically closed fields has a system of big quotients. This section gives the necessary prerequisites for the proof.

Let p be a prime number. Denote the algebraic closure of ℚ in ℚ_{p} by ℚ_{p,abs}. It is well defined up to an isomorphism.

Let (L, v) be a valued field. Call (L, v) P-adic if there is a prime number p satisfying these conditions: (1a) The residue field L̄_{v} is finite, say with q = p^{f} elements. (1b) There is π ∈ L with a smallest positive value v(π) in v(L^{×}). Call π a prime

element of (L, v). (1c) There is a positive integer e with v(p) = ev(π).

Refer to (p, e, f) as the type of (L, v) and to p as the residue characteristic of (L, v). We say (L, v) is P-adically closed if (L, v) admits no finite proper P-adic extension of the same type. Refer to a field L as P-adically closed if L admits a valuation v with (L, v) P-adically closed.

Remark 7.1: Comparison with former definitions. Prestel and Roquette [PrR] use “padically closed” instead of “P-adically closed of residue characteristic p”. The same expression, “p-adically closed”, is used in [HaJ2] for “P-adically closed field of type (p, 1, 1)”.

The proposition below summarizes well known facts about P-adically closed fields (see also [Pop1, Sec. 1]). We use F ≡ F′ to denote elementary equivalence between fields and (F, v) ≡ (F′, v′) to denote elementary equivalence between valued fields.

Proposition 7.2:

(a) ℚ_{p} is P-adically closed of type (p, 1, 1).

(b) Every P-adically closed valued field (L, v) is Henselian of characteristic 0.

(c) A field L is P -adically closed for at most one P -adic valuation.

(d) Suppose (K, v) is a P-adic field. Then (K, v) has a P-adically closed algebraic extension (L, w) of the same type. Call (L, w) a P-adic closure of (K, v). If (K, v) is discrete, then (L, w) is uniquely determined up to a K-isomorphism.

(e) In the notation of (d), L is minimal among all P -adically closed extensions of K.

(f) Let K be a subfield of a P-adically closed field L. Suppose K is algebraically closed in L. Then, K is a P-adically closed field of the same type as L and K ≡ L.

Moreover, the restriction of the P-adic valuation of L to K is the P-adic valuation of K.

(g) A P-adic field (L, v) is P-adically closed if and only if (L, v) is Henselian and v(L^{×})/nv(L^{×}) ≅ ℤ/nℤ for every positive integer n.

(h) Suppose a field L is elementarily equivalent (in the language of fields) to a P-adically closed field F. Then L is a P-adically closed field of the same type as F. Moreover, let v (resp. w) be the P-adic valuation of L (resp. F). Then (L, v) ≡ (F, w).

(i) Every finite extension of a P-adically closed field is a P-adically closed field of the same residue characteristic.

(j) Every P-adically closed field of residue characteristic p is elementarily equivalent to a finite extension of ℚ_{p,abs} and also to a finite extension of ℚ_{p}.

(k) Let L be a P-adically closed field. Then Gal(L) is a finitely generated prosolvable group.

(l) Let L be a P-adically closed field and L_{0} = L ∩ ℚ̃. Then res: Gal(L) → Gal(L_{0}) is an isomorphism.

(m) Let F be a P-adically closed field and F′ an arbitrary field. Suppose F′ ≡ F. Then Gal(F′) ≅ Gal(F).

(n) Let F be a P -adically closed field and F′ an arbitrary field. Suppose Gal(F′) ≅ Gal(F). Then F′ is a P -adically closed field of the same type as F.

Proof of (a): Let K be a finite proper extension of ℚ_{p}. Then [K : ℚ_{p}] = ef where e is the ramification index and f is the residue degree [CaF, p. 19, Prop. 3]. In particular, e = v(p), where v is the unique normalized p-adic valuation of K. Also, the residue fields of ℚ_{p} and K are 𝔽_{p} and 𝔽_{p^{f}}, respectively. Hence, e > 1 or f > 1. This proves (a).

Proof of (b): That (L, v) is Henselian is stated in [PrR, p. 34, Thm. 3.1]. By (1a), p = char(L̄_{v}) is a prime number. Hence, either char(L) = 0 or char(L) = p. By (1b) and (1c), v(p) ≠ 0. Therefore, char(L) = 0.

Proof of (c): Suppose v and v′ are P-adic valuations of L. By (b), both are Henselian. Since their residue fields are not separably closed (by (1a)), F.K. Schmidt - Engler [Jar2, Prop. 13.4] implies v is equivalent to v′.

Proof of (d): See [PrR, p. 37, Thm. 3.2].

Proof of (e): Let (L_{0}, w_{0}) be a P-adically closed field with K ⊆ L_{0} ⊆ L. Denote the unique P -adically closed valuation of L by w. By (b), both (L, w) and (L_{0}, w_{0}) are Henselian. Extend w_{0} to a valuation w_{1} of L. Then (L, w_{1}) is Henselian.

Assume w_{1} is inequivalent to w. By (1a), both L̄_{w} and L̄_{w_{1}} are algebraic extensions of finite fields. Hence, none of them is the residue field of a nontrivial valuation of the other. This means, w and w_{1} are incomparable. Hence, by F.K. Schmidt - Engler [Jar, Prop. 13.4], L̄_{w} is separably closed , in contradiction to (1a). Therefore, w and w_{1} are equivalent.

Thus, (L, w) extends (L_{0}, w_{0}) and (L_{0}, w_{0}) extends (K, v). Since (L, w) and (K, v) have the same type, also (L_{0}, w_{0}) has the same type. In particular, the residue characteristic of (L_{0}, w_{0}) is p. Since (L_{0}, w_{0}) is P-adically closed, (L_{0}, w_{0}) = (L, w), as contended.

Proof of (f): [PrR, p. 38, Thm. 3.4] says K is P-adically closed of the same type as L. Moreover, the P -adic valuation of K is the restriction of the P-adic valuation of L. By [PrR, p. 86, Thm. 5.1], K ≡ L.

Proof of (g): See [PrR, p. 34, Thm. 3.1].

Proof of (h): Denote the P-adic valuation of F by v. Let (p, e, f) be the type of (F, v) and π a prime element F. Consider the Kochen operator

γ(X) = \frac{1}{π} \frac{X^{q} − X}{(X^{q} − X)^{2} − 1},

with q = p^{f}. Put γ(F) = {γ(x) | x ∈ F and x^{q} − x ≠ ±1}. By [JaR, Lemma 4.1(iii)], γ(F) is the valuation ring of v.

Since L ≡ F, γ(L) is a valuation ring of L. Denote the corresponding valuation by w. Then (L, w) ≡ (F, v). Since (F, v) satisfies (1), so does (L, w). Thus, (L, w) is P-adic.

Finally note: The conditions of (g) on a P-adic field to be P-adically closed are elementary in the language of valued fields. Consequently, (L, w) is P-adically closed.

Proof of (i): Let (L, v) be a P-adically closed field and L′ a finite extension of L. Since L is Henselian, v uniquely extends to a valuation v of L′ and (L′, v) is Henselian. Since both [L̄′_{v} : L̄_{v}] and (v((L′)^{×}) : v(L^{×})) are finite, (L′, v) is a P -adic valued field and v((L′)^{×})/nv((L′)^{×})) ≅ ℤ/nℤ. Consequently, L′ is P-adically closed.

Proof of (j): Let (L, v) be a P-adically closed field of residue characteristic p. By (b), char(L) = 0. Put L_{0} = L ∩ ℚ̃. By (f), L_{0} is a P-adically closed field of the same type as L and L_{0} ≡ L.

Let v_{0} be the P -adic valuation of L_{0}. By (b), (L_{0}, v_{0}) is Henselian. Moreover, v_{0}|_{ℚ} is the p-adic valuation v_{p} of ℚ. Hence, ℚ_{p,abs} ⊆ L_{0}. The relation [K : ℚ_{p,abs}] = ef for finite extensions K/ℚ_{p,abs} and the finiteness of the type of L_{0} imply [L_{0} : ℚ_{p,abs}] < ∞.

It follows that F = L_{0}ℚ_{p} is a finite extension of ℚ_{p} with F ∩ ℚ̃ = L_{0}. By (i), F is P-adically closed. By (f), F ≡ L_{0}. Consequently, F ≡ L.

Proof of (k) and (l): Let L_{0} and F be as in the proof of (j). By [Jan, Satz 3.6], Gal(F) is finitely generated (see also [JRi2, p. 2]). By [CaF, p. 31, Cor. 1], Gal(F) is prosolvable. By Krasner’s Lemma, ℚ̃ℚ_{p} = ℚ̃_{p}, so res: Gal(F) → Gal(L_{0}) is an isomorphism and Gal(L_{0}) is finitely generated. Since L ≡ L_{0}, every finite quotient G of Gal(L) is a finite quotient of Gal(L_{0}) (as the proof of [FrJ, Prop. 18.12] shows). It follows from [FrJ, Props. 15.3 and 15.4] that the epimorphism res: Gal(L) → Gal(L_{0}) is an isomorphism. Consequently, Gal(L) is prosolvable and finitely generated.

Proof of (m): By (h), F′ is a P-adically closed field. Let F_{0} = F ∩ ℚ̃ and F′_{0} = F′ ∩ ℚ̃. Then F_{0} ≡ F′_{0}. Hence, F_{0} ≅ F′_{0} [FrJ, Lemma 18.19]. By (l), Gal(F) ≅ Gal(F_{0}) and Gal(F′) ≅ Gal(F′_{0}). Therefore, Gal(F) ≅ Gal(F′).

Proof of (n): Efrat [Efr, Thm. A] (in the case p ≠ 2) and Koenigsmann [Koe, Thm. 4.1] (in general) construct a Henselian valuation v′ of F′ with char(F̄′_{v′}) ≠ 0. It follows from [Pop1, E9] that F′ is P-adically closed. Moreover, if F is a finite extension of ℚ_{p}, then so is F′.

Lemma 7.3: For each prime number p the group Gal(ℚ_{p}) is torsion free.

Proof: For p ≠ 2 there is x ∈ ℚ_{p} with x^{2} +p−1 = 0 (Hensel’s lemma). For p = 2 there is x ∈ ℚ_{2} with x^{2} + 7 = 0 (use Hensel-Rychlik). In both cases a sum of nonzero squares in ℚ_{p} is 0. Hence, ℚ_{p} is not formally real. Therefore, by Artin-Schreier, Gal(ℚ_{p}) is torsion free.

We summarize some well known facts about real closed fields and algebraically closed fields of characteristic 0.

Remark 7.4: Algebraically closed and real closed fields. Let 𝔽 be a finite extension of ℝ. Then either 𝔽 = ℝ or 𝔽 = ℂ. Suppose F ≡ 𝔽. If 𝔽 = ℝ, then F is real closed and Gal(F) = ℤ/2ℤ. If 𝔽 = ℂ, then F is algebraically closed and Gal(F) is trivial. Now let K be a subfield of F. Then res: Gal(F) → Gal(F ∩ K̃) is an isomorphism and F ≡ F ∩ K̃ [Pre, p. 53, Cor. 5.6 and p. 51, Cor. 5.3]. Conversely, if F′ extends F and F′ ≡ F, then F′ ∩ F̃ = F.

8. Construction of Big Quotients for Classical Groups

We say that a field 𝔽 is classical local of characteristic 0 if 𝔽 is either ℝ, ℂ, or a finite extension of ℚ_{p} for some p. A profinite group G is classical local of characteristic 0 if G is isomorphic to the absolute Galois group of a classical local field of characteristic 0.

Let F be a finite set of classical local fields of characteristic 0. Put

C = {Gal(𝔽) | 𝔽 ∈ F}.

We have already mentioned that each Γ ∈ C is finitely generated and prosolvable (Proposition 7.2(k)). We use the next result together with Lemma 7.3 to equip C with a system of big quotients.

Notation 8.1: Let p be a prime number and G a profinite group. Denote the maximal pro-p quotient of G by G(p). Let G_{p} be a p-Sylow subgroup of G.

Proposition 8.2: Let p, l be prime numbers and 𝔽 be a finite extension of ℚ_{p}. Then 𝔽 has a finite Galois extension 𝔽′ with the following properties:

(a) Let Δ be a quotient of Gal(𝔽) which has Gal(𝔽′/𝔽) as a quotient. Then, Δ_{p} is not a free pro-p group and Δ_{l} is not a free pro-l group.

(b) Let p′ be a prime number, L an algebraic extension of ℚ_{p′}, and γ: Gal(𝔽) → Gal(L) an epimorphism. Suppose there is an epimorphism β: Gal(L) → Gal(𝔽′/𝔽). Then p = p′, γ is an isomorphism, and [𝔽 : ℚ_{p}] = [L : ℚ_{p}].

Proof: By Proposition 7.2(k), Gal(𝔽) is finitely generated.

Construction of 𝔽′: Denote the compositum of all extensions of 𝔽 of degree at most max(p − 1, l − 1) by E_{0}. In particular, E_{0} contains the roots of unity ζ_{p} and ζ_{l} of order p and l, respectively.

By [HaJ2, Lemma 11.1], E_{0} has a finite extension E_{1} with this property:

(1) rank(Gal(E′_{1}/E_{0})(p)) = rank(Gal(E_{0})(p)) for every Galois extension E′_{1} of E_{0} which contains E_{1}. Since ζ_{p} ∈ E_{0}, Gal(E_{0})(p) is not a free pro-p group [Koc, p. 96, Satz 10.3]. By

[HaJ2, Lemma 11.2], E_{0} has a proper finite p-extension E_{2,p} with this property: (2a) For every Galois extension E′_{2} of E_{0} containing E_{2,p}, the group Gal(E′_{2}/E_{0}) is not

a free pro-p group. Similarly, E_{0} has a proper finite l-extension E_{2,l} satisfying this: (2b) For every Galois extension E′_{2} of E_{0} containing E_{2,l}, the group Gal(E′_{2}/E_{0}) is not

a free pro-l group. Put E_{2} = E_{2,p}E_{2,l}.

Since Gal(ℚ_{p}) is finitely generated, ℚ_{p} has only finitely many extensions of degree [𝔽 : ℚ_{p}]. Let L_{1}, …, L_{k} be all extensions of ℚ_{p} satisfying this:

(3) [L_{j} : ℚ_{p}] = [𝔽 : ℚ_{p}], Gal(L_{j}) is a quotient of Gal(𝔽), but Gal(L_{j}) ≅ G̸al(𝔽), j = 1, …, k (k may be 0). For each j choose a finite Galois extension F_{j} of 𝔽 such that Gal(F_{j}/𝔽) is not a

quotient of Gal(L_{j}) [FrJ, Prop. 15.4]. Let 𝔽′ be the compositum of all extensions of 𝔽 of degree at most m = max([E_{i} : 𝔽], [F_{j} : 𝔽])_{i=1,2; j=1,...,k}. Then 𝔽′ is a finite Galois extension of 𝔽 which contains E_{1}, E_{2}, F_{1}, …, F_{k}.

Proof of (a): Let Δ be as in (a). Then Δ ≅ Gal(M/𝔽) for some Galois extension M of 𝔽. Since Gal(𝔽′/𝔽) is a quotient of Δ, there is a Galois extension 𝔽′′ of 𝔽 in M with Gal(𝔽′′/𝔽) ≅ Gal(𝔽′/𝔽). In particular, 𝔽′′ is the compositum of extensions of 𝔽 of degree at most m, so 𝔽′′ ⊆ 𝔽′. Since [𝔽′′ : 𝔽] = [𝔽′ : 𝔽], we have 𝔽′ = 𝔽′′ ⊆ M.

Assume Gal(M/𝔽)_{p} is a free pro-p group. Then Gal(M/E_{0})_{p} is also a free pro-p group [FrJ, Cor. 20.38], so cd_{p}Gal(M/E_{0}) ≤ 1 [Rib, p. 235, Thm. 6.5]. Therefore, by [Rib, p, 255, Thm. 3.2], Gal(M/E_{0})(p) is pro-p free. This contradiction to (2a) proves that Gal(M/𝔽)_{p} is not a free pro-p group. Similarly, Gal(M/𝔽)_{l} is not a free pro-l group.

Proof of (b): Let p′, L, γ, and β be as in (b). Denote the fixed field of Ker(γ) (resp. Ker(β ∘ γ)) in ℚ̃_{p} by N (resp. F′). Then F′ is a Galois extension of 𝔽 in N satisfying Gal(F′/𝔽) ≅ Gal(𝔽′/𝔽). In particular, F′ is a compositum of extensions of 𝔽 of degree at most m. Hence, F′ ⊆ 𝔽′. Since [F′ : 𝔽] = [𝔽′ : 𝔽], we have F′ = 𝔽′.

By construction, E_{0} ⊂ E_{2,p} ⊆ 𝔽′ ⊆ N and E_{2,p}/E_{0} is a proper p-extension, so p divides [N : E_{0}]. Let E_{0}^{(p)} be the maximal pro-p extension of E_{0}. Then Gal(N ∩ E_{0}^{(p)}/E_{0}) is the maximal pro-p quotient of Gal(N/E_{0}). Also, E_{2,p} ⊆ N ∩ E_{0}^{(p)}. Hence, by (2a), Gal(N ∩ E_{0}^{(p)}/E_{0}) is not a free pro-p group. It follows from [Rib, p. 255] that cd_{p}Gal(N/E_{0}) > 1.

Let L_{0} be the fixed field of γ(Gal(E_{0})) in ℚ̃_{p′}. Then Gal(N/E_{0}) ≅ Gal(L_{0}). Hence, by the preceding paragraph, cd_{p}Gal(L_{0}) > 1. This implies, p^{∞} ∤ [L_{0} : ℚ_{p′}] [Rib, p. 291–292]. Also, L_{0} is the compositum of all extensions of L of degree at most max(p − 1, l − 1). In particular, ζ_{p} ∈ L_{0}. By (1) applied to N instead of to E′_{1} and by [Neu, Satz 4]

(4) rank(Gal(L_{0})(p)) = rank(Gal(N/E_{0})(p)) = rank(Gal(E_{0})(p)) = 2 + [E_{0} : ℚ_{p}].

In particular, rank(Gal(L_{0})(p)) ≥ 3. Hence, p′ = p and rank(Gal(L_{0})(p)) = 2+[L_{0} : ℚ_{p}] [Neu, Satz 4]. It follows from (4) that [E_{0} : ℚ_{p}] = [L_{0} : ℚ_{p}]. Since [E_{0} : 𝔽] = [L_{0} : L], this implies [𝔽 : ℚ_{p}] = [L : ℚ_{p}].

Finally assume Gal(L) ≅ G̸al(𝔽). By assumption, Gal(L) is a quotient of Gal(𝔽). Hence, L = L_{j} with 1 ≤ j ≤ m. By construction, Gal(F_{j}/𝔽) is not a quotient of Gal(L). Since 𝔽 ⊆ F_{j} ⊆ 𝔽′, this implies Gal(𝔽′/𝔽) is not a quotient of Gal(L), in contradiction to our assumption. Thus, Gal(𝔽) ≅ Gal(L). Consequently, by [FrJ, Prop. 15.3], γ is an isomorphism.

The following result gives sufficient conditions for a finite set C of finitely generated profinite groups to have a system of big quotients (Definition 6.4):

Proposition 8.3: Let C be a finite set of finitely generated profinite groups. Suppose each Γ ∈ C is finite or prosolvable. For each infinite Γ ∈ C let Γ̄ be a finite quotient of Γ and for each finite Γ ∈ C let Γ̄ = Γ. Suppose there exists a prime number l such that for every infinite Γ ∈ C and every profinite group Δ with epimorphisms Γ ⟶^{γ} Δ ⟶ Γ̄ the following holds: (5a) Γ_{l} is torsion free. (5b) Δ_{l} is not a free pro-l group. (5c) There is a prime number p ≠ l such that Δ_{p} is not a free pro-p group. (5d) If Δ is isomorphic to a subgroup of some Γ′ ∈ C, then γ is an isomorphism.

Then {Γ̄ | Γ ∈ C} is a system of big quotients for C.

Proof: Let {Δ_{j} | j ∈ J} be a finite collection of profinite groups of type C and let e be a positive integer. Put B^{∗} = F̂_{e} ∗ ∏ ∗_{j∈J} Δ_{j}. Let Γ be a group in C and Δ a closed

subgroup of B^{∗} with epimorphisms Γ ⟶^{γ} Δ → Γ̄.

First suppose Δ is finite. By [HeR, p. 160, Thm. 1], Δ is conjugate to a closed subgroup of some Δ_{j}. By (5d), γ is an isomorphism.

Now suppose Δ is prosolvable. Let l be a prime number as in the proposition. By the first paragraph and (5a) (applied to Δ_{j} instead of to Γ), no element of B^{∗} has order l. In particular, Δ_{l} is torsion free. By [Pop3, Thm. 2(2)], Δ is conjugate to a subgroup of some Δ_{j}. Again, by (5d), γ is an isomorphism.

Lemma 8.4: Let F be a finite set of classical local fields of characteristic 0. Put C = {Gal(𝔽) | 𝔽 ∈ F}. Then C has a system of big quotients.

Proof: Let S be the set of all residue characteristics of 𝔽 ∈ F. Choose a prime number l not in S ∪ {2}. By Proposition 7.2(k) and Remark 7.4, each Γ ∈ C is finitely generated and prosolvable. Moreover, Γ_{l} is torsion free (Lemma 7.3). Omit ℂ from F, if necessary, to assume 1 ∉ C. For each Γ ∈ C choose 𝔽 ∈ F with Γ ≅ Gal(𝔽). If Γ is finite (i.e. 𝔽 = ℝ) choose Γ̄ = Γ. If Γ is infinite and 𝔽 is a finite extension of ℚ_{p}, let Γ̄ = Gal(𝔽′/𝔽), where 𝔽′ is the finite extension of 𝔽 given by Proposition 8.2. We apply Lemma 7.3 to prove that {Γ̄ | Γ ∈ C} is a system of big quotients for C in the sense of Definition 6.4:

Since each Γ ∈ C is an absolute Galois group, Γ_{l} is torsion free. Let 𝔽 ∈ F, Γ = Gal(𝔽), Δ be a profinite group, and Γ ⟶^{γ} Δ → Γ̄ epimorphisms. Let p be the residue characteristic of 𝔽. By Proposition 8.2(a), Δ_{p} is not a free pro-p group and Δ_{l} is not a free pro-l group. Finally, suppose Δ is isomorphic to a subgroup of some Γ′ ∈ C. Identify Δ with Gal(L) where L is an algebraic extension of ℚ_{p′}. By Proposition 8.2, γ is an isomorphism. Thus, all parts of Condition (4) holds. By Proposition 8.3, C has a system of big quotients.

9. Spaces of Classically Local Fields

Let F be a finite set of classical local fields of characteristic 0 and let K be a field. For each 𝔽 ∈ F let AlgExt(K, 𝔽) be the set of all algebraic extensions of K which are elementarily equivalent to 𝔽. Put AlgExt(K, F) = ⋃_{𝔽∈F} AlgExt(K, 𝔽). Call a field K pseudo-F-closed (abbreviated PFC) if it is pseudo-AlgExt(K, F)-closed; that is, V (K) ≠ ∅ for each smooth absolutely irreducible variety V satisfying V (F) ≠ ∅ for all F ∈ AlgExt(K, F).

We call a profinite group G (strongly) F-projective, if G is (strongly) Gprojective, where G = ⋃_{𝔽∈F}{H ∈ Subgr(G) | H ≅ Gal(𝔽)}.

Our first main result is that “K is PFC” implies “Gal(K) is strongly F-projective”. The only still missing ingredient of the proof is the strict closedness of AlgExt(K, 𝔽).

Lemma 9.1: Let K be a field and 𝔽 a finite extension of ℚ_{p} or of ℝ. Then AlgExt(K, 𝔽) is strictly closed in AlgExt(K).

Proof: We prove the theorem in the case where 𝔽 is a finite extension of ℚ_{p}. The same proof applies to the case where 𝔽 is ℝ or ℂ. We only have to replace the references to Proposition 7.2 by references to Remark 7.4. By Proposition 7.2(f) and Remark 7.4, 𝔽 ≡ 𝔽 ∩ ℚ̃. Hence, we may replace 𝔽 by 𝔽 ∩ ℚ̃.

So, assume without loss, 𝔽 is a finite extension of ℚ_{p,alg}. By [FrJ, Lemma 18.19], AlgExt(ℚ, 𝔽) = {𝔽^{σ} | σ ∈ Gal(ℚ)}. Thus, AlgExt(ℚ, 𝔽) is the image of the strictly continuous map Gal(ℚ) → AlgExt(ℚ) given by σ ↦ 𝔽^{σ}. Since both spaces are profinite, AlgExt(ℚ, 𝔽) is strictly closed in AlgExt(ℚ).

Now put X = AlgExt(K, 𝔽). Consider F ∈ StrictClosure(X). Then, for every finite Galois extension N of K, W_{N} = {E ∈ X | E ∩ N = F ∩ N} ≠ ∅. By Proposition 7.2(m), Gal(E) ≅ Gal(𝔽) for each E ∈ X. Therefore,

(1) every finite quotient of Gal(F) is a finite quotient of Gal(𝔽).

Conversely, let F_{0} = F ∩ ℚ̃. Observe that the map φ: AlgExt(K) → AlgExt(ℚ) given by L ↦ L ∩ ℚ̃ is strictly continuous. It maps AlgExt(K, 𝔽) into AlgExt(ℚ, 𝔽) (by Proposition 7.2(f)). Hence, F_{0} = φ(F) ∈ StrictClosure(AlgExt(ℚ, 𝔽)). By the second paragraph of the proof, F_{0} ∈ AlgExt(ℚ, 𝔽). Thus, F_{0} ≡ 𝔽. Hence, by Proposition 7.2(m), Gal(F_{0}) ≅ Gal(𝔽), so Gal(𝔽) is an image of Gal(F). In particular, every finite quotient of Gal(𝔽) is a finite quotient of Gal(F). Combining with (1), we conclude that Gal(F) and Gal(𝔽) have the same finite quotients. By Proposition 7.2(k), Gal(𝔽) is finitely generated. Hence, by [FrJ, Prop. 15.4], Gal(F) ≅ Gal(𝔽). It follows that Gal(F) is finitely generated and isomorphic to Gal(F_{0}). Since res: Gal(F) → Gal(F_{0}) is surjective, it is bijective [FrJ, Prop. 15.3].

Next observe that the intersection of finitely many sets W_{N} contains a set of this form. Hence the intersection is nonempty. Therefore, there is an ultrafilter D of X which contains each W_{N}. Put F^{∗} = ∏_{E∈X} E/D. By the fundamental property of ultraproducts, F^{∗} ≡ 𝔽 [FrJ, Cor. 6.12].

Embed F in F^{∗} by mapping each x ∈ F onto the element (x_{E})/D where x_{E} is x if x ∈ E and x_{E} = 0 otherwise. Put F_{0}^{∗} = F^{∗} ∩ ℚ̃. Then, by Proposition 7.2(f), F_{0} ≡ 𝔽 ≡ F^{∗} ≡ F_{0}^{∗}. Also, F_{0} ⊆ F_{0}^{∗}. Let x ∈ F_{0}^{∗}. Put f = irr(x, ℚ). Since F_{0} ≡ F_{0}^{∗}, the number of roots of f in F_{0} is equal to the number of roots of f in F_{0}^{∗}. Hence, x ∈ F_{0}. Therefore, F_{0} = F_{0}^{∗}. By Proposition 7.2(l), res: Gal(F^{∗}) → Gal(F_{0}) is an isomorphism. Hence, so is res: Gal(F^{∗} ∩ K̃) → Gal(F_{0}). Since F ⊆ F^{∗} ∩ K̃ and res: Gal(F) → Gal(F_{0}) is an isomorphism, we have F = F^{∗} ∩ K̃. Again, by Proposition 7.2(f), F ≡ F^{∗}. Consequently, F ∈ X, as desired.

In order to formulate the first main result of this work we have to impose a certain restriction on F.

Remark 9.2: Isomorphism of Galois groups of P -adic fields. We say F is closed under Galois isomorphism if for all classical local fields 𝔽, 𝔽′ the following holds:

(1) 𝔽 ∈ F and Gal(𝔽) ≅ Gal(𝔽′) implies 𝔽′ ∈ F.

Actually, by Remark 7.4, it suffices to impose Condition (2) only for a finite extension 𝔽 of ℚ_{p} and a finite extension 𝔽′ of ℚ_{p′}. By Proposition 8.2, Gal(𝔽) ≅ Gal(𝔽′) implies p = p′ and [𝔽′ : ℚ_{p}] = [𝔽 : ℚ_{p}]. So, for each 𝔽 ∈ F there are only finitely many fields 𝔽′ with Gal(𝔽′) ≅ Gal(𝔽).

Section 2 of [JRi1] gives for each p examples of nonisomorphic extensions F and F′ of ℚ_{p} with Gal(F) ≅ Gal(F′). Indeed, [JRi1, p. 2, Thm.] and [Rit, p. 281, Thm.] prove for arbitrary finite extensions F, F′ of ℚ_{p} (if p = 2, the theorem assumes \sqrt{−1} ∈ F) that Gal(F) ≅ Gal(F′) if and only if [F : ℚ_{p}] = [F′ : ℚ_{p}] and F ∩ ℚ_{p,ab} = F′ ∩ ℚ_{p,ab}. Here ℚ_{p,ab} is the maximal abelian extension of ℚ_{p}.

Finally consider classical local fields F and F′ of characteristic 0. Suppose F is elementarily equivalent to F′. Then F is isomorphic to F′. Indeed, we may assume F is a finite extension of ℚ_{p} and F′ is a finite extension of ℚ_{p′}. By 7.2(h), p = p′. Let F_{0} = F ∩ ℚ̃ and F′_{0} = F′ ∩ ℚ̃. By 7.2(f), F_{0} ≡ F and F′_{0} ≡ F′. Hence, by [FrJ, Lemma 18.19], F_{0} ≅ F′_{0}. We may therefore assume F_{0} = F′_{0} and F_{0} is a finite extension of ℚ_{p,alg}. But then the isomorphism res: Gal(ℚ_{p}) → Gal(ℚ_{p,alg}) (see 7.2(l)) maps both Gal(F) and Gal(F′) onto Gal(F_{0}). Consequently F = F′.

Lemma 9.3: Let F be a finite set of classical local fields of characteristic 0. Suppose F is closed under Galois isomorphism. Then for every field K we have

(3) ⋃_{𝔽∈F}{F ∈ AlgExt(K) | F ≡ 𝔽} = ⋃_{𝔽∈F}{F ∈ AlgExt(K) | Gal(F) ≅ Gal(𝔽)}.

Proof: By Proposition 7.2(m), the left hand side of (3) is contained in its right hand side. Conversely, let F ∈ AlgExt(K) and 𝔽 ∈ F be fields with Gal(F) ≅ Gal(𝔽). If 𝔽 is real closed, then so is F and F ≡ 𝔽 (Remark 7.4). Otherwise, 𝔽 is a finite extension of ℚ_{p} for some p. By Proposition 7.2(n), F is elementarily equivalent to a finite extension 𝔽′ of ℚ_{p}. Hence, by Proposition 7.2(m), Gal(𝔽′) ≅ Gal(F) ≅ Gal(𝔽). Since F is closed under Galois isomorphism, 𝔽′ ∈ F. Consequently, F belongs to the left hand side of (3).

Theorem 9.4: Let F be a finite set of classical local fields of characteristic 0 not containing ℂ which is closed under Galois isomorphism. Let K be a PFC field. Put

G = ⋃_{𝔽∈F}{Gal(F) | F ∈ AlgExt(K) and Gal(F) ≅ Gal(𝔽)}.

Then Gal(K) is strongly F-projective and (Gal(K), G_{max}) is a proper projective group structure.

Proof: Let C = {Gal(𝔽) | 𝔽 ∈ F}. For each Γ ∈ C let

G_{Γ} = ⋃_{{𝔽∈F \atop Gal(𝔽)≅Γ}}\{Gal(F) | F ∈ AlgExt(K, 𝔽)\}.

By Lemma 9.1, G_{Γ} is strictly closed in Subgr(Gal(K). By Lemma 9.3, G = ⋃_{Γ∈C} G_{Γ}, so G is strictly closed in Subgr(Gal(K)). Hence, by [HJPa, Remark 1.2], G is étale compact. By Proposition 3.1, Gal(K) is G-projective. By Lemma 8.4, C has a system of big quotients. It follows from Proposition 6.5 that Gal(K) is strongly G-projective and (Gal(K), G_{max}) is a proper projective group structure.

10. Realization of Strongly Projective Groups as Absolute Galois Groups

The second main result of this work is a converse to Theorem 9.4. We consider again a finite set F of classical local fields of characteristic 0 not containing ℂ. We prove that each F-projective group G which satisfies the group theoretic analog of Lemma 9.1 is isomorphic to Gal(K) for some PFC field K. Moreover, we construct K equipped with a “field-valuation structure” satisfying the “block approximation condition”. We recall the definition of these concepts from [HJPa]:

A field structure is a data K = (K, X, K_{x})_{x∈X} where K is a field, X is a profinite space with a continuous action of Gal(K) on X, and for each x ∈ X, K_{x} is separable algebraic extension of K satisfying the following conditions: (1a) For each finite separable extension L of K the set X_{L} = {x ∈ X | L ⊆ K_{x}} is

open. (1b) K_{x^{σ}} = K_{x}^{σ} for all x ∈ X and σ ∈ Gal(K). (1c) {σ ∈ Gal(K) | x^{σ} = x} ⊆ Gal(K_{x}). Thus, Gal(K) = (Gal(K), X, Gal(K_{x}))_{x∈X} is a group structure called the absolute Galois structure associated with K [HJPa, Section 6].

Denote the set of all valuations, including the trivial one, of a field L by Val(L). A subbasis for the patch topology of Val(L) consists of all sets

Val_{a}(K) = {v ∈ Val(K) | v(a) > 0}, Val′_{a}(K) = {v ∈ Val(K) | v(a) ≥ 0}

with a ∈ K.

A field-valuation structure is a structure K = (K, X, K_{x}, v_{x})_{x∈X} satisfying the following conditions: (2a) (K, X, K_{x})_{x∈x} is a field structure. (2b) v_{x} is a valuation of K_{x} satisfying v_{x^{σ}} = v_{x}^{σ} for all x ∈ X and σ ∈ Gal(K). Here

v_{x}^{σ}(u^{σ}) = v_{x}(u) for each u ∈ K_{x}. (2c) For each finite separable extension L of K define a map ν_{L}: X_{L} → Val(L) by

ν_{L}(x) = v_{x}|_{L}. Then ν_{L} is continuous. We call K Henselian if (K_{x}, v_{x}) is Henselian for each x ∈ X.

The absolute Galois structure associated with K is the same associated with the underlying field structure, namely Gal(K) = (Gal(K), X, Gal(K_{x}))_{x∈X}. We call K proper if Gal(K) is proper.

Definition 10.1: Block approximation condition. A block approximation problem for a field-valuation structure K = (K, X, K_{x}, v_{x})_{x∈X} is a data (V, X_{i}, L_{i}, a_{i}, c_{i})_{i∈I_{0}} satisfying this: (3a) (Gal(L_{i}), X_{i})_{i∈I_{0}} is a special partition of Gal(K): For each i ∈ I_{0} the set X_{i} is

open-closed in X, for all x ∈ X_{i} we have L_{i} ⊆ K_{x}, Gal(L_{i}) = {σ ∈ Gal(K) | X_{i}^{σ} = X_{i}}, and X = ⋃ · _{i∈I_{0}} ⋃ · _{ρ∈R_{i}} X_{i}^{ρ}, where R_{i} is any subset of Gal(K) satisfying

Gal(K) = ⋃ · _{ρ∈R_{i}} Gal(L_{i})ρ_{i}.

(3b) V is a smooth affine variety over K. (3c) a_{i} ∈ V (L_{i}). (3d) c_{i} ∈ K^{×}.

A solution of the problem is a point a ∈ V (K) with v_{x}(a − a_{i}) > v_{x}(c_{i}) for all i ∈ I_{0} and x ∈ X_{i}. We say K satisfies the block approximation condition if each block approximation problem for K is solvable.

The block approximation condition has several interesting consequences:

Proposition 10.2 ([HJPa, Proposition 12.3]): Let K = (K, X, K_{x}, v_{x})_{x∈X} be a fieldvaluation structure satisfying the block approximation condition.

(a) Put K = {K_{x} | x ∈ X}. Then K is PKC.

(b) Suppose x_{1}, …, x_{n} ∈ X lie in distinct Gal(K)-orbits. Then v_{x_{1}}|_{K}, … v_{x_{n}}|_{K} satisfies the weak approximation theorem.

(c) Suppose x, y ∈ X lie in distinct Gal(K)-orbits. Then v_{x}|_{K} and v_{y}|_{K} are independent.

(d) Suppose X has more than one Gal(K)-orbit. Then the trivial valuation is not in ν_{K}(X).

(e) For each x ∈ X, K is v_{x}-dense in K_{x}; and

(f) (K_{x}, v_{x}) is a Henselian closure of (K, v_{x}|_{K}).

Theorem 10.3: Let F be a finite set of classical local fields of characteristic 0 and G an F-projective group. Let

C = {Gal(𝔽) | 𝔽 ∈ F} and G = Subgr(G, C) = ⋃_{Γ∈C} Subgr(G, Γ).

Suppose: (5a) ℂ ∉ F. (5b) F is closed under Galois isomorphism. (5c) Subgr(G, Γ) is strictly closed in Subgr(G) for each Γ ∈ C. Then there is a proper field-valuation structure K = (K, X, K_{x}, v_{x})_{x∈X} such that: (6a) K satisfies the block approximation condition. (6b) There is an isomorphism φ: (G, G_{max}) → Gal(K); in particular G ≅ Gal(K). (6c) {K_{x} | x ∈ X} = AlgExt(K, F)_{min}. (6d) K is PFC.

Proof: By Lemma 8.4, C has a system of finite big quotients. By Proposition 7.2(k), each Γ ∈ C is finitely generated and prosolvable. Finally, by assumption, G is Gprojective. Hence, by Proposition 6.5, G = (G, G_{max}) is a proper projective group structure. Moreover, Proposition 6.5 gives a homomorphism δ: G → ∏ ∗_{Γ∈C} Γ which maps each H ∈ G injectively into a conjugate of some Γ ∈ C. By assumption, each Γ ∈ C is the absolute Galois group of a Henselian algebraic extension of ℚ or a real closure of ℚ. Therefore, by [Gey, Thm. 10.1], we may identify ∏ ∗_{Γ∈C} Γ with Gal(D) for some algebraic extension field D of ℚ. Let E be the fixed field of δ(G) in ℚ̃. Then δ: G → Gal(E) is an epimorphism of profinite groups which extends to a cover δ: G → Gal(E) of group structures, with E being a field structure whose underlying field is E. Indeed, E is the associated field structure to the quotient structure (G, G_{min})/Ker(δ) [HJPa, Example 2.5]. Note that Gal(E) need not be proper.

Put X = G_{max}. By [HJPa, Thm. 15.4], there is a proper Henselian field-valuation structure K = (K, X, K_{x}, v_{x})_{x∈X} which satisfies the block approximation condition and there is an isomorphism φ: G → Gal(K) such that (7a) E = K ⋂ ℚ̃, E_{δ(x)} = K_{x} ∩ ℚ̃, and v_{x} is trivial on E_{δ(x)} for all x ∈ X. (7b) res_{K̃/Ẽ} ∘ φ = δ.

By Lemma 9.3, F ∈ AlgExt(K, F) if and only if Gal(F) ≅ Gal(𝔽) for some 𝔽 ∈ F. Therefore the isomorphism φ: G → Gal(K) establishes, via Galois correspondence, a bijection of Subgr(G, C) onto AlgExt(K, F) which maps Subgr(G, C)_{max} onto AlgExt(K, F)_{min}. This proves (6c)

By Proposition 10.2(a), K is pseudo-{K_{x} | x ∈ X}-closed. Therefore, by (6c), K is PFC.

Problem 10.4: Is it possible to remove the condition “F is closed under Galois isomorphism” from Theorem 10.3?

References [Ax] J. Ax, A metamathematical approach to some problems in number theory, AMS Proc.

Symp. Pure Math. XX (1971), 161–190.

[Bre] G. E. Bredon, Introduction to Compact Transformation Groups, Pure and applied Mathematics 46, Academic Press, New Your, 1972.

[CaF] J.W.S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, London, 1967.

[Efr] I. Efrat, A Galois-theoretic characterization of p-adically closed fields, Israel Journal of Mathematics 91 (1995), 273-284.

[FrJ] M. D. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik (3) 11, Springer-Verlag, Heidelberg, 1986.

[Gey] W.-D. Geyer, Galois groups of intersections of local fields, Israel Journal of Mathematics 30 (1978), 382–396.

[Har] D. Haran, On closed subgroups of free products of profinite groups, Proceedings of the London Mathematical Society 55 (1987), 266–289.

[HeR] W. Herfort and L. Ribes, Torsion elements and centralizers in free products of profinite groups, Journal für die reine und angewandte Mathematik 358 (1985), 155-161.

[HaJ1] D. Haran and M. Jarden, The absolute Galois group of a pseudo real closed field, Annali della Scuola Normale Superiore — Pisa, Serie IV, 12 (1985), 449–489.

[HaJ2] D. Haran and M. Jarden, The absolute Galois group of a pseudo p-adically closed field, Journal für die reine und angewandte Mathematik 383 (1988), 147–206.

[HJPa] D. Haran, M. Jarden, and F. Pop, Projective group structures as absolute Galois structures with block approximation, http://www.math.uiuc.edu/Algebraic-Number-Theory/0367, September, 2002

[Jan] U. Jannsen, Über Galoisgruppen lokaler Körper, Inventiones mathematicae 70 (1982), 53–69.

[Jar1] M. Jarden, Algebraic realization of p-adically projective groups, Compositio Mathematica 79 (1991), 21–62.

[Jar2] M. Jarden, Intersection of local algebraic extensions of a Hilbertian field (A. Barlotti et al., eds), NATO ASI Series C 333 343–405, Kluwer, Dordrecht, 1991

[JaR] M. Jarden and Peter Roquette, The Nullstellensatz over p-adically closed fields, Journal of the Mathematical Society of Japan 32 (1980), 425–460.

[JRi1] M. Jarden and J. Ritter, On the characterization of local fields by their absolute Galois groups, Journal of Number Theory 11 (1979), 1–13.

[JRi2] M. Jarden and J. Ritter, On the Frattini subgroup of the absolute Galois group of a local field, Israel Journal of Mathematics 74 (1991), 81–90.

[Koc] H. Koch, Galoissche Theorie der p-Erweiterungen, Mathematische Monographien 10, Berlin 1970.

[Koe] J. Koenigsmann, From p-rigid elements to valuations (with a Galois-characterisation of p-adic fields), Journal für die reine und angewandte Mathematik 465 (1995), 165– 182.

[Koe2] J. Koenigsmann, Relatively projective groups as absolute Galois groups, Israel Journal of Mathematics 127 (2002), 93–129.

[Neu] J. Neukirch, Kennzeichnung der p-adischen und der endlichen algebraischen Zahlkörper, Inventiones mathematicae 6 (1969), 296-314.

[Pop1] F. Pop, Galoissche Kennzeichnung p-adisch abgeschlossener Körper, Journal für die reine und angewandte Mathematik 392 (1988), 145–175.

[Pop2] F. Pop, Classically projective groups, preprint, Heidelberg, 1990.

[Pop3] F. Pop, On prosolvable subgroups of profinite free products and some applications, manuscripta mathematica 86 (1995), 125–135.

[Pre] A. Prestel, Lectures on Formally Real Fields, Lecture Notes in Mathematics 1093, Springer, Berlin, 1984.

[PrR] A. Prestel and P. Roquette, Formally p-adic Fields, Lecture Notes in Mathematics 1050, Springer, Berlin, 1984.

[Rib] L. Ribes, Introduction to Profinite Groups and Galois Cohomology, Queen’s papers in Pure and Applied Mathematics 24, Queen’s University, Kingston, 1970.

[RiZ] L. Ribes and P. Zalesskii, Profinite Groups, Ergebnisse der Mathemtik III 40, Springer, Berlin, 2000.

15 November, 2003