Manuscript, variant of Oct, 2002; 1–47

The birational anabelian conjecture — r e v i s i t e d —

By Florian Pop*

Introduction

Let ℓ be a fixed rational prime number. For every field K of positive characteristic p ≠ ℓ, let K^{cyclc} be a maximal cyclotomic extension of K. Thus K^{cyclc} = K \overline{𝔽}_{p}, where \overline{𝔽}_{p} is an algebraic closure of the prime field 𝔽_{p} of K. Further let K^{ℓ} be a maximal Galois pro-ℓ extension of K^{cycl}. By general Galois theory, it is clear that K^{ℓ}|K is a Galois extension. We denote by G_{K}^{ℓ} = Gal(K^{ℓ}|K) its Galois group.

The aim of this paper is to prove the following cyclotomic by pro-ℓ version of Grothendieck’s birational anabelian conjecture in positive characteristic:^{1}

Theorem. Let F be the category of all finitely generated fields of positive characteristic having absolute transcendence degree td(K) > 1. There exists a group theoretic recipe by which we can recover every field K ∈ F from G_{K}^{ℓ}, up to a pure inseparable extensions. This recipe is invariant under profinite group isomorphisms. In particular, if K and L are in F, there exists a canonical bijection

Isom^{i} (L^{i}, K^{i}) ⟶ Out (G_{K}^{ℓ}, G_{L}^{ℓ}) ,

where ( )^{i} denotes pure inseparable closure, and Isom^{i} means up to Frobenius twists, and Out denotes outer isomorphisms of profinite groups.

Equivalently, if Φ : G_{K}^{ℓ} → G_{L}^{ℓ} is an isomorphism of profinite groups, then up to a Frobenius twist, there exists a unique field isomorphism ϕ : L^{i}^{ℓ} → K^{i}^{ℓ} such that Φ(g) = ϕ^{−1} g ϕ for all g ∈ G_{K}^{ℓ}. In particular, ϕ(L^{i}) = K^{i}.

Remarks.

1) The Theorem above implies corresponding “full Galois assertions”, i.e., the corresponding assertions for the full Galois group G_{K} in stead of the quotient G_{K}^{ℓ}. In particular the Galois characterization of finitely generated fields of Kronecker dimension > 1 (in positive characteristic), see [P2], [P3], follows from the Main Theorem above. Thus the above result generalizes in a nontrivial way the celebrated result by Neukirch, Ikeda, Iwasawa, Uchida concerning the Galois characterization global fields.

* This work was partially written during my visit to the IAS Princeton in 1994 and 1996. I would like to thank Pierre Deligne for several fruitful discussions and suggestions.

1 A similar result is true as well if we include function fields of characteristic zero. This will be shown in a subsequent paper.

2) The proof goes by induction on d = td(K) > 1, but information from the pro-ℓ Galois theory of global function fields will be used in an essential way. We remark the a corresponding cyclic by pro-ℓ result for global function fields would bring a major simplification of the proof of the general case. Unfortunately, in spite of several efforts in the last years, the case of global function fields is still open. Second, we strongly believe that a stronger assertion holds, namely that the above recipe for detecting the field structure from G_{K}^{ℓ} is actually invariant under open homomorphisms of profinite groups. Thus in fact a “cyclic by pro-ℓ Hom-form” of the birational anabelian conjecture should hold. See also the comments at point 3) below. Finally, we remark that in stead of working with the full cyclic by pro-ℓ Galois group, one could work as well with truncations of the pro-ℓ part. This is doable, but the resulting assertions are quite technical, and at the moment maybe too complicated in order to be interesting...

3) In characteristic zero, one should mention here the very strong results by Mochizuki [M1], [M2], concerning anabelian schemes over sub-p-adic fields k. Among other things, he shows the following: Given regular function fields K|k and L|k, every open G_{k}-homomorphism G_{K} → G_{L} arises canonically from a kembedding of fields L → K. This shows that one could expect better functorial properties of the “recipe” for detecting K from G_{K}^{ℓ}. Nevertheless, there are some essential differences between results proved earlier and ours in the present manuscript: First, in Mochizuki’s approach, one deals with a relative full Galois situation, i.e., G_{k}-morphisms of the corresponding full Galois groups G_{K} and G_{L}. Second, comparing with previous results in positive characteristic, e.g., by Tamagawa and/or by the author, in the present situation we are missing a very essential previous ingredient, namely the full Frobenius: It is not possible to detect it at a pro-ℓ level.

Rough idea of Proof

Let 𝔎|𝔨 be an extension of fields of characteristic ≠ ℓ. We consider Galois extensions 𝔎′|𝔎 with the following properties:

i) 𝔎′ contains a separable closure 𝔨′ of 𝔨. Set 𝔎_{0} = 𝔎 𝔨′ in 𝔎′.

ii) 𝔎′|𝔎_{0} is a pro-ℓ extension.

iii) There exists a Galois sub-extension 𝔎_{1} of 𝔎′|𝔎 containing 𝔨′ such that 𝔎′|𝔎_{1} is a maximal pro-ℓ Abelian extension of 𝔎_{1}.

October 10, 2002  PRO-ℓ BIRATIONAL ANABELIAN GEOMETRY  3

We remark that the extension K^{ℓ}|K as introduced before the Main Theorem satisfies the conditions i), ii), iii). Thus working with K′|K is a little bit more general than working with K^{ℓ}|K, and offers more flexibility.

To fix notations, let G′_{𝔎} = Gal(𝔎′|𝔎), and G′_{𝔨} = Aut(𝔨′|𝔨), and let denote pr : G′_{𝔎} → G′_{𝔨} the canonical projection.

Let us fix an isomorphism ı : 𝕋_{ℓ} → ℤ_{ℓ}(1). By Kummer Theory there is a functorial isomorphism

(∗)  δ̂_{𝔎} : 𝔎̂ ⟶ H^{1}(G′_{𝔎} , 𝕋_{ℓ}) ⟶^{ı} H^{1}(𝔎, ℤ_{ℓ}(1)) ,

where 𝔎̂ denotes the ℓ-adic completion of the multiplicative group 𝔎^{×} of 𝔎. Suppose that 𝔎|𝔨 is a regular field extension, thus in particular, the canonical projection pr : G′_{𝔎} → G′_{𝔨} is surjective. By the functoriality of δ̂, we get a commutative diagram of the form

𝔨̂  ⟶  𝔎̂
↓δ̂_{𝔨}  ↓δ̂_{𝔎}
H^{1}(𝔨, ℤ_{ℓ}(1)) ⟶ H^{1}(𝔎, ℤ_{ℓ}(1))

Setting Ĥ_{𝔎/ 𝔨} = 𝔎̂/𝔨̂, and H^{1}(𝔎/𝔨, ℤ_{ℓ}(1)) = H^{1}(𝔎, ℤ_{ℓ}(1))/H^{1}(𝔨, ℤ_{ℓ}(1)) we have a canonical isomorphism δ̂_{𝔎/ 𝔨} : Ĥ_{𝔎/ 𝔨} → H^{1}(𝔎/𝔨, ℤ_{ℓ}(1)).

Now suppose that 𝔎|𝔨 is a regular function field. Then 𝔎^{×}/𝔨^{×} is a free Abelian group, and the ℓ-adic completion homomorphism is an embedding:

ȷ_{𝔎/ 𝔨} : 𝔎^{×}/𝔨^{×} ↪ Ĥ_{𝔎/ 𝔨} ⟶^{δ̂_{𝔎 / 𝔨}}H^{1}(𝔎, ℤ_{ℓ}(1))/H^{1}(𝔨, ℤ_{ℓ}(1)) .

Now let us denote P(𝔎) = 𝔎^{×}/𝔨^{×}, and view P(𝔎) as the projectivization of the (infinite) dimensional 𝔨-vector space (𝔎, +). Suppose that we have a Galois theoretic recipe in order to detect: First the image of P(𝔎) inside the Galois theoretically “known” Ĥ_{𝔎/ 𝔨}. Second, the projective lines in P(𝔎). Remark that the multiplication l_{x} by any non-zero element x ∈ 𝔎 defines an “automorphisms” of P(𝔎) which respects “co-lineations”. This automorphism is noting but the translation l_{ȷ_{𝔎 / 𝔨} (x)} in P(𝔎), this time viewed again as the multiplicative group P(𝔎) = 𝔎^{×}/𝔨^{×}. Now using the Fundamental Theorem of projective geometry, see e.g. Artin [A], it follows that the additive structure of 𝔎 can be deduced from the knowledge of all the projective lines in P(𝔎). And finally, since the “multiplications” l_{ȷ_{𝔎 / 𝔨} (x)} do respect this structure, we finally deduce from this the field structure of 𝔎, as well the field extension 𝔎|𝔨.

Now coming back to the case of a finitely generated field of positive characteristic K, let k is its field of constants, i.e., the field of absolute algebraic elements in K. Then k is a finite field, and K is a regular function field over k. In order to use the idea of the proof above, we will give Galois theoretic recipes in order to detect:

I) The projection pr : G_{K}^{ℓ} → G_{k} and the ℓ-Frobenius of k.

Here, by the ℓ-Frobenius of k we mean the Frobenius of the Galois group of the ℓ^{∞}-cyclotomic extension of k. In particular, from this we can deduce the ℓ-adic completions of K^{×} and of k^{×}. Thus finally Ĥ_{K/k}.

II) K_{(ℓ)} := K^{×} ⊗ ℤ_{(ℓ)} inside K̂.

This almost answers the question about detecting P(K): We know namely its ℤ_{(ℓ)}-version, but not P(K) itself.

III) P(K) and its projective lines.

Organization of the paper

Part I: Here we put together the necessary tools for the proof as follows:

a) First, a “cyclotomic by pro-ℓ” form of the Local theory from [P1]. It relies on results by Ware [W], Koenigsmann [Ko], see also Bogomolov– Tschinkel [B–T], Efrat [Ef], Engeler–Koenigsmann [E–K]. Alternatively, one could develop the ideas from [P1] in this new (simpler) context. The aim of this local theory is to recover in a functorial way from G_{K}^{ℓ} the set 𝔇_{K} of all the Zariski prime divisors of K, where K is any finitely generated field of positive characteristic with td(K) > 0. In particular, if td(K) = 1, the set 𝔇_{K} is in a natural bijection with the set of closed points of the (unique) complete normal model X of the global function field K. Nevertheless, if td(K) > 1, then 𝔇_{K} is “huge”, and much bigger than the set of Zariski prime divisors 𝔇_{X} defined by the Weil prime divisors of any model X of K.

b) As a consequence of a) above, we show that given a finitely generated field K of positive characteristic with td(K) > 0, say with constant field k, the following invariants of K|k are group theoretically encoded in G_{K}^{ℓ}: First, the projection G_{K}^{ℓ} → G_{k}. Second, the cardinality q of k, and the ℓ-Frobenius of k. Thus finally, the ℓ-cyclotomic character χ : G_{K}^{ℓ} → ℤ_{ℓ}(1) is group theoretically encoded in G_{K}^{ℓ}. In particular, if td(K) = 1, the degree d_{x} of every closed point x ∈ X of the complete normal model X of K is encoded in G_{K}^{ℓ}.

c) Third, a more technical result by which we recover the “geometric sets of prime divisors” of any finitely generated field K of positive characteristic. By definition, a geometric set of prime divisors of K is the set of Zariski prime divisors 𝔇_{X} ⊂ 𝔇_{K} defined by the Weil prime divisors of some quasiprojective normal model X of K. This result itself relies on de Jong’s theory of alterations [J].

Part II: Following a suggestion by Deligne [D2], we first develop a kind of “abstract non-sense” which we call abstract pre-Galois formations, reminding one in some sense of the abstract class field theory. The aim of this theory is to lay an axiomatic strategy for the proof of the main result.

The main example of pre-Galois formations are the concrete ones, which arise from geometry and arithmetic.

Elaborating on this, we next define abstract Galois formations, which come closer to approximating the concrete Galois formations. A main results here is Theorem 4.12 of Section 4), which shows that for a concrete Galois formation based on a finitely generated field 𝔎 with char(𝔎) > 0, the concrete field formation 𝔎_{(ℓ)} defining it is actually Galois theoretically encoded in Gℓ_{𝔎}.

We finally mention Theorem 5.12, which asserts that under certain hypotheses, a Galois formation G can carry at most one structure of a concrete Galois formation. This result is the last key point in proving the main Theorem above.

PART I

1. Local theory (revisited)

In this section we will recall the main facts concerning the Local theory from [P1], but in a pro-ℓ setting. Our aim is to give to give a Galois theoretic recipe for finding information about the space of Zariski prime divisors in a functorial way.

Let 𝔎|𝔨 be a function field over some base field 𝔨, and td(𝔎|𝔨) = d > 0. We consider the family of all models X_{i} → 𝔨 of 𝔎|𝔨 such that the structure sheaf of X_{i} is a sheaf of sub-rings of 𝔎 with inclusions as structure morphisms. On the family of all the X_{i}’s there exists a naturally defined domination relation as follows: X_{j} ≥ X_{i} if there exists a surjective 𝔨-morphism φ_{ji} : X_{j} → X_{i} which at the structure sheaf level is defined by inclusions. Let 𝔓𝔯𝔬𝔧_{𝔎} be the subfamily of all projective, normal models of the function field 𝔎. The following is well known, see e.g. Zariski–Samuel [Z–S], Ch.VI, especially §17:

• Every complete model is dominated by some X_{i} ∈ 𝔓𝔯𝔬𝔧_{𝔎} (Chow Lemma).

• The set 𝔓𝔯𝔬𝔧_{𝔎} is increasingly filtered with respect to ≥, hence it is a surjective projective system.

• Denote ℜ_{𝔎} = lim_{⟵}_{i} X_{i} as topological spaces. We will call ℜ_{𝔎} the Riemann space of 𝔎. Then the points of ℜ_{𝔎} are in bijection with the space of all valuation 𝔨-rings of 𝔎. For v = (x_{i})_{i} in ℜ_{𝔎} one has: x_{i} is the center of v on X_{i} in the usual sense.

Fact/Definition 1.1. Using e.g. [BOU], Ch.IV, §3, one shows that for a point v = (x_{i})_{i} in ℜ_{𝔎} the following conditions are equivalent:

i) For i sufficiently large, x_{i} has co-dimension 1, or equivalently, x_{i} is the generic point of a prime Weil divisor of X_{i}. Hence v is the discrete 𝔨-valuation of 𝔎 with valuation ring O_{X_{i},x_{i}}.

ii) td(𝔎v | 𝔨) = td(𝔎|𝔨) − 1.

We will say that a point v = (x_{i})_{i} in ℜ_{𝔎} satisfying the above equivalent conditions is a Zariski prime divisor of 𝔎|𝔨. We denote the space of all Zariski prime divisors of 𝔎|𝔨 by 𝔇_{𝔎} ⊂ ℜ_{𝔎}. One has: The space 𝔇_{𝔎} ⊂ ℜ_{𝔎} of all Zariski prime divisors of 𝔎 is the union of the spaces P_{i} of prime Weil divisors of all the models X_{i} of 𝔎|𝔨 in 𝔓𝔯𝔬𝔧_{𝔎} (if we identify every prime Weil divisor with the discrete valuation on 𝔎 it defines).

More generally, let 𝔎′|𝔎 be an arbitrary algebraic extension, and v′ a valuation on 𝔎′. Then we will say that v′ is a Zariski prime divisor (of 𝔎 or of 𝔎′), if its restriction v to 𝔎 is a Zariski prime divisor of 𝔎|𝔨.

A) On the decomposition group

Consider the following context: 𝔎′|𝔎 is some Galois field extension, and v is a valuation on 𝔎′. We set p = char(𝔎′v) the residue characteristic. Further let Z_{v}, T_{v}, and V_{v} be respectively the decomposition group, the inertia group, and the ramification group of v in Gal(𝔎′|𝔎). We denote by 𝔎^{V}, 𝔎^{T}, and 𝔎^{Z} the corresponding fixed fields.

Fact 1.2. The following are well known facts from Hilbert decomposition, and/or ramification theory for general valuations:

1) 𝔎′v | 𝔎v is a normal field extension, and V_{v} ⊂ T_{v} are normal subgroups of Z_{v}. Moreover, one has a canonical exact sequence

1 → T_{v} → Z_{v} → G_{v} → 1 .

One has v(𝔎^{T}) = v(𝔎^{Z}) = v𝔎, and 𝔎v = 𝔎^{Z}v. Further, 𝔎^{T}v | 𝔎v is the separable part of the normal extension 𝔎′v | 𝔎v, thus it is the maximal Galois sub-extension of 𝔎′v | 𝔎v.

2) There exists a pairing Ψ_{𝔎}′ : T_{v} × v𝔎′ → μ_{𝔎′v}, (g, vx) ↦ (gx/x)v, and the following hold: The left kernel of Ψ_{𝔎}′ is exactly V_{v}, and the right kernel of Ψ_{𝔎}′ is the p-divisible hull of v𝔎 in v𝔎′. In particular, T_{v}/V_{v} is Abel Ian, and V_{v} is the unique Sylow p-group of T_{v}. Further, Ψ is compatible with the action of Aut(𝔎′v | 𝔎v).

Fact 1.3. In the above context, let 𝔎|𝔨 be as in the Introduction, and 𝔎′|𝔎 satisfying the conditions i), ii), iii), from there. Suppose that v is trivial on 𝔨. Then V_{v} = {1}, v𝔎′ is the ℓ-divisible hull of v𝔎, and the residue field extension 𝔎′v | 𝔎v also satisfies the properties i), ii), iii). In particular, we will denote G′_{𝔎v} = Gal (𝔎′v | 𝔎v)

Further, for every n = ℓ^{e}, there exists a unique sub-extension 𝔎_{n}|𝔎^{T} of N|𝔎^{T} such that v𝔎_{n} = \frac{1}{n} v𝔎^{T} (thus = \frac{1}{n} v𝔎). Clearly, 𝔎_{n}|𝔎^{Z} is a Galois subextension of 𝔎′|𝔎^{Z}. On the other hand, the multiplication by n gives rise to a canonical isomorphism \frac{1}{n} v𝔎 / v𝔎 ≅ v𝔎/n. Therefore, the pairing Ψ_{𝔎_{n}} gives rise to a non-degenerate pairing

Ψ_{n} : T_{v}/n × v𝔎/n → μ_{n} ⟶^{ı} ℤ/n(1),

hence there result canonical isomorphisms θ^{v,n} : v𝔎/n → Hom(T_{v}, μ_{n}), and θ_{v,n} : T_{v}/n → Hom(v𝔎, μ_{n}). In particular, taking limits over all n = ℓ^{e}, one obtains canonical isomorphisms

1^{ℓ}) θ^{v} : v̂𝔎 → Hom(T_{v}, ℤ_{ℓ}(1)) and θ_{v} : T_{v} → Hom(v𝔎, ℤ_{ℓ}(1)).

Definition/Remark 1.4. In the above context, an element g ∈ T_{v} is called a v-inertia element. A v-inertia element is called arithmetical inertia element, w.r.t. identification ı : 𝕋_{ℓ} → ℤ_{ℓ}(1), if it satisfies the following equivalent conditions:

(i) θ_{v}(g)(v𝔎) ⊆ ℤ(1) ⊂ ℤ_{ℓ}(1). (ii) θ^{v}(v𝔎)(g) ⊆ ℤ(1) ⊂ ℤ_{ℓ}(1). The arithmetical v-inertia elements build a subgroup 𝔗_{v} ⊂ T_{v} which is

dense in every closed subgroup of T_{v}. Moreover, if v𝔎 is a finitely generated group, then 𝔗_{v} is canonically isomorphic to the ℤ-dual group of v𝔎. This is so for instance if the valuation v is a discrete valuation on 𝔎.

2^{ℓ}) Recalling the notations and remarks from Introduction, one gets a commutative diagram of the form:

𝔎^{×}  ⟶^{v}  v𝔎
↓^{ȷ_{𝔎}}  ↓_{θ^{v}}
H^{1}(Gℓ_{𝔎} , ℤ_{ℓ}(1)) ⟶^{ȷ_{v}} H^{1}(T_{v}, ℤ_{ℓ}(1))

Next let B = (vx_{i})_{i} be an 𝔽_{ℓ}-basis of the vector space v𝔎 / ℓ. For every x_{i}, choose a system of roots (α_{i,n})_{n} in N such that α_{i,n}^{ℓ} = α_{i,n−1} (all n > 0), where α_{i,0} = x_{i}. Then setting 𝔎^{0} = 𝔎[(α_{i,n})_{i,n}] ⊂ N, it follows that v is totally ramified in 𝔎^{0}|𝔎, and v𝔎^{0} is ℓ-divisible. Therefore, 𝔎^{0}v = 𝔎v, and we finally obtain:

3^{ℓ}) In the above context, T_{v} has complements in Z_{v}, thus Z_{v} ≅ T_{v} ×^{|} G_{v}. Equivalently, we have Z_{v} ≅ T_{v} ×^{|} G′_{𝔎v} ≅ ℤ_{ℓ}^{B}(1) ×^{|} G′_{𝔎v} as profinite groups. In particular, vcd(Z_{v}) = vcd(G′_{𝔎v}) + |B|, where vcd denotes the virtual cohomological dimension.

4^{ℓ}) Let U_{v} ⊂ 𝔎^{×} denote the v-units, and U_{v}^{1} = 1 + 𝔪_{v} ⊂ U_{v} the principal v-units. Then U_{v}^{1} becomes ℓ-divisible in 𝔎^{Z}, and 𝔎v^{×} = U_{v}/U_{v}^{1}. We have a commutative diagram of the form:

U_{v}  ⟶  𝔎v
↓^{ȷ_{𝔎}}  ↓^{ȷ_{𝔎 v}}
H^{1}(Gℓ_{𝔎} , ℤ_{ℓ}(1)) ⟶^{ȷ_{v}} H^{1}(G_{𝔎v}^{ℓ}, ℤ_{ℓ}(1))

B) Recovering 𝔇_{𝔎} (compare [P1])

Definition/Remark 1.5. Let 𝔎|𝔨 be a function field, and suppose that td(𝔎|𝔨) = d > 0.

1) The decomposition group Z_{v} ⊂ Gℓ_{𝔎} of some divisorial valuation v ∈ 𝔇_{𝔎}_{ℓ} is called a divisorial subgroup.

2) A subgroup Z ⊂ Gℓ_{𝔎} which is isomorphic to a divisorial subgroup (of some function field) and has vcd(Z) = vcd(𝔎) is called a divisorial like subgroup.

Proposition 1.6. Let K be a finitely generated field of positive characteristic with d = td(K) > 0. Let Z_{v} ⊂ G_{K}^{ℓ} be a divisorial subgroup, say defined by a divisorial valuation v on K^{ℓ}. Let T_{v} be the inertia group of v. Then the following hold:

(1) Z_{v} is self-normalising in G_{K}^{ℓ}. Further, if Z_{v′} ≠ Z_{v} is another divisorial subgroup, then one has: Z_{v′} ∩ Z_{v} = 1.

(2) T_{v} ≅ ℤ_{ℓ}(1) as a G_{Kv}^{ℓ}-module. Further T_{v} is the unique maximal pro-ℓ abelian normal subgroup of Z_{v}. And Z_{v} ≅ T_{v} ×^{|} G_{Kv}^{ℓ} ≅ ℤ_{ℓ}(1) ×^{|} G_{Kv}^{ℓ}.

Proof. To (1): Both assertions follow using a result of F. K. Schmidt, see e.g., Pop [P1], Proposition 1.3; see also the proof of Proposition 1.14 from loc.cit.. Concerning (2), the only non-obvious part is the fact that in the case td(K) > 1, the Galois group G_{Kv}^{ℓ} has no non-trivial abelian normal subgroups. This follows from the Hilbertianity of Kv.

A first content of the local theory is that “morally” the converse of the above Proposition is also true, i.e., if Z ⊂ G_{K}^{ℓ} is a divisorial like subgroup, then it comes from a Zariski prime divisor.

Proposition 1.7. Let K be a finitely generated field of positive characteristic. Then one has:

(1) For every divisorial like subgroup Z of G_{K}^{ℓ} there exists a unique divisorial valuation v of K^{ℓ} such that Z ⊂ Z_{v} and char(Kv) ≠ ℓ.

(2) Moreover, if T ⊂ Z is the unique maximal pro-cyclic normal subgroup of Z, then T = Z ∩ T_{v}, where T_{v} is the inertia subgroup of Z_{v}.

Therefore, the space 𝔇_{K^{ℓ}} of all Zariski prime divisors of K^{ℓ} is in bijection with the divisorial subgroups of G_{K}^{ℓ}. This bijection is given by v ↦ Z_{v}.

Proof. The main step in the proof is the following ℓ-Lemma below, which replaces the q-Lemma from [P1], Local theory. After having the ℓ-Lemma, the remaining steps in the proof are similar to (but easier than) the ones from the proof of Theorem 1.16 from [P1], Local theory. We will skip the remaining details.

The ℓ-Lemma (revisited). In the context of Proposition 1.7, let Z ⊂ G_{K}^{ℓ} be a solvable closed subgroup with vcd(Z) = td(K) + 1. Then there exists a valuation v of K^{ℓ} satisfying: Z ⊂ Z_{v} is an open subgroup and Kv a finite field. In particular, Z ≅ ℤ_{ℓ}^{d}(1) ×^{|} G_{v}, with G_{v} some open subgroup of G_{k}.

There are several ways to prove the ℓ-Lemma above: First, one could develop the corresponding model theoretic machinery, and proceed as in [P1], Local theory. Second, one can apply the results from Ware [W], and Koenigsmann [Ko]; see also Efrat [Ef], Bogomolov–Tschinkel [B–T], Engeler– Koenigsmann [E–K].

2. The constant field and the ℓ-Frobenius

Let K be a finitely generated field of positive characteristic with td(K) > 0. Let k be the field of constants of K. We remark that for all prime numbers ℓ′ ≠ ℓ we have cd_{ℓ′}(G_{K}^{ℓ}) = 1, and that further, cd_{ℓ}(G_{K}^{ℓ}) = td(K) + 1. Thus in particular, both ℓ and td(K) are encoded in G_{K}^{ℓ}. Therefore, if v ∈ 𝔇_{K^{ℓ}} is a Zariski prime divisor, and Z_{v} is its decomposition group, then td(Kv) is encoded in Z_{v} via cd_{ℓ} of G_{Kv}^{ℓ} = Z_{v}/T_{v}.

A) Detecting the canonical projection pr : G_{K}^{ℓ} → G_{k}

Let T_{K}^{ℓ} ⊂ G_{K}^{ℓ} be the closed subgroup generated by all the inertia subgroups T_{v} (all v ∈ 𝔇_{K^{ℓ}}. Then T_{K}^{ℓ} is a normal subgroup, and π_{1,K}^{ℓ} = G_{K}^{ℓ}/T_{K}^{ℓ} is the Galois group of the maximal sub-extension of K^{ℓ}|K in which no Zariski prime divisor of K is ramified. Thus is X is any normal model of K, and π_{1}^{ℓ}(X) is its cyclotomic by pro-ℓ étale fundamental group, then π_{1,K}^{ℓ} is a quotient of it. (N.B., if X is a smooth complete model, then the two groups coincide.) Looking at the abelianization π_{1,K}^{ℓ,ab}, and using the result of Katz–Lang [K–L], it follows that π_{1,K}^{ℓ,ab} fits into an exact sequence of the form:

1 → (finite ℓ-group) → π_{1,K}^{ℓ,ab} → G_{k} → 1 ,

where k is the field of constants of k. Since G_{k} ≅ ℤ̂, it finally follows that in the above exact sequence, “(finite ℓ-group)” is exactly the torsion subgroup of π_{1,K}^{ℓ,ab}. Thus we have:

Fact 2.1. The canonical projection pr : G_{K}^{ℓ} → G_{k} is encoded in G_{K}^{ℓ} as follows: First consider G_{K}^{ℓ} → G_{K}^{ℓ}/T_{K}^{ℓ} =: π_{1,K}^{ℓ}. Then G_{k} is the quotient of π_{1,K}^{ℓ,ab} by its torsion subgroup (which turns out to be a finite ℓ-group).

We conclude this subsection by the following result for later use:

Fact 2.2. In the above context, ker(pr) = G_{K\overline{k}}^{ℓ} is encoded group theoretically in G_{K}^{ℓ}. In particular, we also “know” the geometric variant of π_{1,K}^{ℓ}, which by definition is the following (here remark that T_{K}^{ℓ} = T_{K\overline{k}}^{ℓ}):

\overline{π}_{1,K}^{ℓ} := G_{K\overline{k}}^{ℓ} / T_{K}^{ℓ}

1) We remark that in the case td(K) = 1, the structure of \overline{π}_{1,K}^{ℓ} is well known by Grothendieck’s specialisation Theorem: It is the pro-ℓ group on 2g_{X} generators and one relation (product of commutators), where g_{X} is the geometric genus of the complete normal model of K.

2) In general, \overline{π}_{1,K}^{ℓ} is a quotient of the pro-ℓ geometric fundamental group of any normal model X of K\overline{k}. Further, if K\overline{k} has a smooth projective model \overline{X}, then \overline{π}_{1,K}^{ℓ} is the pro-ℓ quotient of the geometric fundamental group of \overline{X}.

B) Detecting the constant field k of K

We first consider the case cd G_{K}^{ℓ} = 2, or equivalently, td(K) = 1. Then K is a global function field, and 𝔇_{K} is in bijection with the closed points of the complete normal model X of K. Let v ∈ 𝔇_{K^{ℓ}} be a Zariski prime divisor, say corresponding to a closed point x ∈ X. Let d_{v} = [κ(x) : k] be the degree of x. Then d_{v} is encoded in Z_{v} ⊂ G_{K}^{ℓ} in an obvious way: If namely G_{v} = pr(Z_{v}) is the image of Z_{v} via the canonical projection, then d_{v} = (G_{k} : G_{v}). In particular, the number N_{d} of points of degree d of X is encoded in G_{K}^{ℓ} as follows: This is exactly the number of conjugacy classes of divisorial subgroups Z_{v} of G_{K}^{ℓ} such that d_{v} = d.

On the other hand, by the Riemann Hypothesis for curves, it follows that N_{d} = q^{d} + O(q^{d/2}), where q = |k| is the cardinality of the constant field k of K, and O(·) is a constant of absolute value bounded by 2g_{X} the genus of X. We concude detecting q = |k| by the following obvious fact:

Lemma 2.3. The cardinality q = |k| of the constant field k of K is the unique natural number q such that for constants O(·) as above for “sufficiently many” exponents d one has

N_{d} = q^{d} + O(q^{d/2}).

Now in the case cd_{ℓ}G_{K}^{ℓ} > 2, we proceed by induction. Taking any Zariski prime divisor v of K^{ℓ}, let Kv be its residue field, and let k_{v} be the constant field of Kv. Then G_{Kv}^{ℓ} ≅ Z_{v}/T_{v} is “known”, and cd_{ℓ}G_{Kv}^{ℓ} < cd_{ℓ}G_{K}^{ℓ}. Thus by induction on cd_{ℓ}, the cardinlity q_{v} = |k_{v}| of the constant field k_{v} of Kv is encoded in G_{Kv}^{ℓ}. Finally, using the Weil bounds, we deduce the following:

Fact 2.4. The cardinality q = |k| of the field of constants of K is exactly the g.c.d. of the set of all the local cardinalities q_{v} = |k_{v}| (all v ∈ 𝔇_{K^{ℓ}}).

C) Detecting the ℓ-Frobenius of k

In order to detect the k-Frobenius of k, we recall the by Propostion 1.6 we have: Z_{v} ≅ T_{v} ×^{|} G_{Kv}^{ℓ} (all divisorial Z_{v}). Further, G_{Kv}^{ℓ} acts on T_{v} via its quotient G_{v} := pr(G_{Kv}^{ℓ}), which is the open subgroup of G_{k} of index d_{v}. (Here d_{v} is defined by |k_{v}| = |k|^{d_{v}}.) Moreover, the action of some σ_{v} ∈ G_{v} on T_{v} is given by the conjugation with any preimage σ′_{v} ∈ Z_{v} of σ_{v}. In other words, there exists a unique ℓ-adic unit ϵ_{σ_{v}} ∈ ℤ_{ℓ}^{×} such that σ_{v}(τ) = ϵ_{σ_{v}}τ for all τ ∈ T_{v}. Finally, the mapping

χ_{v} : G_{v} → ℤ_{ℓ}^{×}, σ_{v} ↦ ϵ_{σ_{v}}

is exactly the ℓ-cyclotomic character of G_{v}. We now have the following description of the the ℓ-Frobenius of the constant field k_{v}: Let G_{1,v} = ker(χ_{v}). Then the fixed field k_{0,v} of G_{1,v} in k^{ℓ} is exactly the ℓ^{∞}-cyclotomic extension of k_{v}. Thus setting G_{0,v} = Gal(k_{0,v}|k_{v}), we have the following:

Lemma 2.5. The ℓ-Frobenius of k_{v} is the unique element Frob_{0,v} of G_{0,v} which acts on T_{v} as the multiplication by q^{d_{v}}. Equivalently, we have χ_{v}(Frob_{0,v}) = q^{d_{v}} viewed as an ℓ-adic unit.

Finally, from this we easily deduce the ℓ-cyclotomic character and the ℓ-Frobenius of k as follows. First, as at the Fact 2.4 above, it follows that the subgroup generated by all the kernels G_{1,v} (all v) is exactly the kernel G_{1} ⊂ G_{k} of the ℓ-cyclotomic character χ : G_{k} → ℤ_{ℓ}^{×}. Thus we have:

Fact 2.6. Let G_{0} = G_{k}/G_{1}. Then G_{0} is canonically the Galois group of the ℓ^{∞} cyclotomic extension of k. Then the ℓ-Frobenius of k is the unique element Frob_{0} ∈ G_{0} such that for all v ∈ 𝔇_{K^{ℓ}} we have: Frob _{0}^{d_{v}} is the image of Frob_{0,v} via the canonical projection G_{0,v} → G_{0}.

In particular, Frob_{0} also defines the ℓ-cyclotomic character of k in a canonical way.

3. Geometric families of Zariski prime divisors

We will work in the following context: 𝔎|𝔨 is a function field with 𝔨, say a perfect field (maybe finite), and td(𝔎|𝔨) > 1. Let X → 𝔨 be a normal model of 𝔎|𝔨. For such a model X → 𝔨 of 𝔎|𝔨, we identify each Weil prime divisor with its generic point x_{1}, as well as with its local ring O_{X,x_{1}} (which is a discrete valuation ring of 𝔎|𝔨), thus with the Zariski prime divisor v_{x_{1}} of 𝔎|𝔨 defined by it.

Fact/Definition 3.1. In the above context:

1) We denote by 𝔇_{X} ⊂ 𝔇_{𝔎| 𝔨} the set of all Zariski prime divisors of 𝔎|𝔨 which are defined by Weil prime divisors of X.

2) We will say that a subset 𝔇 ⊂ 𝔇_{𝔎| 𝔨} is a geometric set of Zariski prime divisors, if it satisfies the following equivalent conditions:

(i) There exists a normal projective model X → 𝔨 of 𝔎|𝔨 such that 𝔇 and 𝔇_{X} are almost equal.^{2}

(ii) There exists a quasi-projective normal model X′ → 𝔨 of 𝔎|𝔨 such that 𝔇 = 𝔇_{X′}.

Our aim in this section is to recall a criterion for describing the geometric sets 𝔇 of divisorial valuations of 𝔎|𝔨. The source is Pop, [P2] and/or [P3], see loc.cit. for more details.

A) Alterations and lines on varieties

A line on a 𝔨-variety is by definition an integral 𝔨-subvariety 𝔩 ⊆ X, which is a curve of geometric genus equal to 0. We denote by X^{line} the union of all the lines on X.

Definition 3.2. We will say that a variety X → 𝔨 is very unruly if the set X^{line} is not dense in X. In particular, a curve X is very unruly if and only if its geometric genus g_{X} is positive.

Remarks 3.3. The following facts are more or less obvious:

1) Being very unruly is a birational notion.

2) If X is very unruly, and f : Y → X is a dominant generically finite morphism of 𝔨-varieties, then Y is very unruly. (This is a consequence of the Lüroth Theorem.)

3) Let ℓ ≠ char(𝔨) be a prime number, say ℓ ≠ char(𝔨). Let X → 𝔨 be a geometrically integral variety of dimension d = dim(X). Then X has ”many” finite ℓ-elementary covers Z → X of degree ≤ ℓ^{d} such that Z is geometrically integral, and Z^{line} is empty. In particular, Z is very unruly.

Indeed, since the question is of birational nature, we can suppose that X is a finite cover of the affine d-dimensional space 𝔸^{d}, where d = dim(X). To preceed, for every 1 ≤ j ≤ d, let X^{(j)} → 𝔸^{1} be a cyclic cover of degree ℓ, such that X^{(j)} → 𝔨 is geometrically integral, and has geometric genus ≥ 1. We set Y = ×_{j}X^{(j)}, and consider the resulting finite cover Y → 𝔸^{d}. Then we have:

2 We say that two sets are almost equal, if their symmetric difference is finite.

a) Y is geometrically integral, and Y^{line} is empty.

b) Aut_{𝔸^{d}}(Y) ≅ (ℤ/ℓ)^{d}, thus ℓ-elementary of degree ℓ^{d}. Finally we let Z′ = X ×_{𝔸^{d}} Y be the fibre product of X and Y over 𝔸^{d},

and let p′ : Z′ → X and q′ : Z′ → Y be the structural projections. Since q′ is a finite morphism, and there are no lines on Y, it follows (by Lüroth Theorem) that there are no lines on Z′. To conclude, we choose a connected component Z ⊂ Z′ such that the resulting p : Z ↪ Z′ ⟶^{p′} X is dominant. Then there are no lines on Z, and we have Aut_{X}(Z) ⊂ Aut_{𝔸^{d}}(Y), thus a finite ℓ-elementary group of order ≤ ℓ^{d}.

4) Let X → 𝔨 be a normal very unruly 𝔨-variety. Then almost all prime Weil divisors X_{1} of X are very unruly (when viewed as 𝔨-varieties).

The birational interpretation of the facts above is as follows. Let 𝔎|𝔨 be a regular function field with td(𝔎|𝔨 = d > 0. We will say that 𝔎|𝔨 is very unruly, if 𝔎|𝔨 has models X → 𝔨 which are very unruly.

By the Remark 3) above, every function field 𝔎|𝔨 has “many” finite ℓelementary extensions N|𝔎 such that N|𝔨 is very unruly.

Suppose that d > 1. We call a Zariski prime divisor of 𝔎|𝔨 very unruly, if 𝔎v | 𝔨 is very unruly. A Zariski prime divisor v of 𝔎|κ is very unruly if and only if there exists a normal model X → κ of 𝔎|𝔨, and a very unruly prime Weil divisor X_{1} of X such that v = v_{X_{1}}.

One has the following birational variant/sharpening of the remark 4) above. This is a very essential step in the Global theory. Its proof relies on de Jong’s theory of alterations.

Theorem 3.4. Let 𝔎|𝔨 be a regular function field with td(𝔎|𝔨) > 1, and X → 𝔨 a normal model for 𝔎|𝔨. For every Zariski prime divisor v of 𝔎|𝔨, let x_{v} the centre of v on X, and X_{v} the Zariski closure of x_{v} in X. Then X_{v} is a prime Weil divisor of X for almost all very unruly Zariski prime divisors v of K|κ.

In other words, the set of all very unruly Zariski prime divisors of 𝔎|𝔨 is almost equal to the set of all very unruly Weil prime divisors of the model X → 𝔨.

For a proof, see loc.cit.. As an application we have:

Corollary 1. Let 𝔎|𝔨 be a very unruly function field with td(𝔎|𝔨) > 1, and X → 𝔨 a normal model for 𝔎|𝔨. Then the set of Zariski prime divisors 𝔇_{X} defined by X is almost equal to the set of the very unruly Zariski prime divisors of 𝔎|𝔨.

B) Geometric sets of prime divisors

As in the previous subsection, let 𝔎|𝔨 be a regular function field. Let 𝔇 be a set of Zariski prime divisors of 𝔎|𝔨. For every finite extension 𝔎_{i}|𝔎, let 𝔇_{i} be the prolongation of 𝔇 to 𝔎_{i}. Thus in particular, 𝔇_{i} is a set of Zariski prime divisors of 𝔎_{i}|𝔨.

We have now the following first step in giving a recipe for detecting the geometric sets of Zariski prime divisors:

Proposition 3.5. In the context above, set d = td(𝔎|𝔨). A set 𝔇 of Zariski prime divisors of 𝔎|𝔨 is geometric if and only if the following conditions are satisfied:

(i) There exists a finite ℓ-elementary extension 𝔎_{0}|𝔎 of degree ≤ ℓ^{d} such that 𝔇_{0} is almost equal to the set of all very unruly prime divisors of 𝔎_{0}.

(ii) If 𝔎_{2}|𝔎 is any ℓ-elementary extension of degree ≤ ℓ^{d}, and 𝔎_{1} = 𝔎_{2}𝔎_{0} is the compositum, then 𝔇_{1} is almost equal to the set of all very unruly prime divisors of 𝔎_{1}.

Proof. Let X → 𝔨 be a quasi-projective normal model of 𝔎|𝔨. Let 𝔎_{2}|𝔎 be a finite field extension such that 𝔎_{2}|𝔨 is very unruly, and let Y → X be the normalisation of X in 𝔎_{2}|𝔎. Thus Y → 𝔨 is a normal model of 𝔎_{2}|𝔨. Further, the set of Zariski prime divisors 𝔇_{Y} of 𝔎_{2}|𝔨 is exactly the prolongation of 𝔇_{X} to 𝔎_{2}. On the other hand, by the Corollary above, it follows that 𝔇_{Y} is almost equal to the set of all very unruly Zariski prime divisors of 𝔎_{2}|𝔨. Therefore we have: If 𝔎_{2}|𝔨 is very unruly, then the prolongation of 𝔇_{X} to 𝔎_{2} is almost equal to the set of all very unruly Zariski prime divisors of 𝔎_{2}|𝔨.

Now coming back to the proof of the Proposition, we proceed as follows: Suppose first that 𝔇 is geometric. Thus 𝔇 and 𝔇_{X} are almost equal, and the same is true for their prolongations to any finite field extension 𝔎_{i}|𝔎 of 𝔎. Next consider a finite irreducible cover Z → X as in Remark 3.3, 3), and let 𝔎_{0}|𝔎 be the corresponding ℓ-elemetary extension of degree ≤ ℓ^{d}. Since Z contains no lines, 𝔎_{0} is very unruly; and the same is true for all finite extensions 𝔎_{i} of 𝔎_{0}, in particular, for the ones considered in the Theorem. Counting everything togehter we get: The prolongations of 𝔇, and that of 𝔇_{X}, to 𝔎_{i} are both almost equal to the set of all very unruly Zariski prime divisors of 𝔎_{i}|𝔨.

Conversally, given a set 𝔇 with the properties i) and ii), from the Theorem. To the given finite extension 𝔎_{0}|𝔎, we choose 𝔎_{2}|𝔎 to be the finite ℓ-elementary extension of degree ≤ ℓ^{d} defined by some irreducible finite cover Z → X as in the Remark 3.3, 3). Then 𝔎_{2}|𝔨 is very unruly, and the same is true for the finite extension 𝔎_{1} = 𝔎_{0}𝔎_{2} of 𝔎_{2}. As above, it follows that the set of all very unruly Zariski prime divisors of 𝔎_{1} is geometric. Therefore, if Y → X is the normalisation of X in the finite field extension 𝔎_{1}|𝔎, it follows that 𝔇_{Y} is almost equal to the set of all very unruly Zariski prime divisors of 𝔎_{1}|𝔨. On the other hand, by the hypotheses on i) and ii), it follows that 𝔇_{1} is almost equal to the set of all very unruly Zariski prime divisors of 𝔎_{1}|𝔨. Thus finally, 𝔇_{1} is almost equal to 𝔇_{Y}. But then the same is true restrictions of 𝔇_{1} and 𝔇_{Y} to 𝔎. These sets are on the other hand exactly 𝔇 and 𝔇_{X}.

Criterion 3.6.

Finally, we reamrk that the above Proposition gives an inductive way to reduce the description of geometric set of Zariski prime divisors of 𝔇 to a criterion for its Zariski prime divisors v to be very unruly. This can be done inductively on d = td(𝔎|𝔨) as follows:

1) Case d = 1:

P_{geom}(1): A set of Zariski prime divisors 𝔇 of 𝔎|𝔨 is geometric if and only if it is almost equal to the set of all Zariski prime divisors 𝔇_{𝔎| 𝔨}.

P_{v.u.}(1): 𝔎|𝔨 is very unruly if and only if the geometric genus of the unique complete normal model of 𝔎|𝔨 is > 0.

2) Case d > 1: Then by induction on d, we already have criteria P_{geom}(k) and P_{v.u.}(k) which assure that sets of Zariski prime divisors are geometric, respectively that function fields are very unruly (all 1 ≤ k < d). We now make the induction step d = td(𝔎|𝔨) as follows:

P_{geom}(d): A set of Zariski prime divisors 𝔇 of 𝔎|𝔨 is geometric if and only if the criterion given by Lemma 3.5 is satisfied. Here we remark that the assertion from loc.cit. “ v is a very unruly Zariski prime divisor of 𝔎|𝔨 ” is actually equivalent to “ 𝔎v | 𝔨 satisfies P_{v.u.}(d − 1) ”.

P_{v.u.}(d): 𝔎|𝔨 is very unruly if and only if for some set 𝔇 ⊂ 𝔇_{𝔎| 𝔨} satisfying P_{geom}(d) one has: P_{v.u.}(d − 1) holds for almost all 𝔎v |𝔨.

C) The case of a finite base field

Here we show that in the case the base field k is finite, and K|k is a (regular) function field with d = td(K|k) > 0, the above Criterion 3.6 for checking whether a set of Zariski prime divisors can be interpreted in G_{K}^{ℓ}.

Indeed, by Proposition 1.7, the Zariski prime divisors of K^{ℓ} are in bijection with the divisorial subgroups of G_{K}^{ℓ} via v ↦ Z_{v}. Thus a bijection 𝔇^{ℓ} ↦ Z_{𝔇^{ℓ}} from sets of Zariski prime divisors 𝔇^{ℓ} ⊂ 𝔇_{K^{ℓ}} to sets of divisorial subgroups Z of G_{K}^{ℓ}. Moreover, a set 𝔇^{ℓ} is the prolongation to K^{ℓ} of a set of Zariski prime divisors 𝔇 ⊂ 𝔇_{K} if and only if Z_{𝔇^{ℓ}} is invarinat under conjugation in G_{K}^{ℓ}. If this is the case, then every conjugacy class represents an element of 𝔇.

Let Z be a set of divisorial subgroups of G_{K}^{ℓ}, which is closed under conjugation in G_{K}^{ℓ}. Thus the corresponding set of Zariski prime divisors 𝔇_{Z}^{ℓ} is the prolongation to K^{ℓ} of a set of Zariski prime divisors 𝔇_{Z} ⊂ 𝔇_{K}. By induction on d = td(K) = cd (G_{K}^{ℓ}) − 1, we have the following Galois translation of Criterion 3.6:

1) Case d = 1:

Gal P_{geom}(1) is equivalent to: Z contains almost all conjugacy classes of divisorial subgroups of G_{K}^{ℓ}.

Gal P_{v.u.}(1) is equivalent to: \overline{π}_{1,K}^{ℓ} := G_{K\overline{k}}^{ℓ} / T_{K}^{ℓ} as defined at Fact 2.2 is non-trivial.

2) Case d > 1:

Then by induction on d, we already have the Galois translation Gal P_{geom}(δ) and Gal P_{v.u.}(δ) of the criteria P_{geom}(δ) and P_{v.u.}(δ) (all 1 ≤ δ < d). We now make the induction step d = td(K|k) as follows:

Gal P_{geom}(d): Z defines a geometric set of Zariski prime divisors 𝔇_{Z} of K|k if and only if (the Galois translation of) the criterion given by Lemma 3.5 is satisfied. Here we remark that the assertion from loc.cit. “ v is a very unruly Zariski prime divisor of K|k ” is actually equivalent to “ G_{Kv}^{ℓ} satisfies Gal P_{v.u.}(d − 1) ”.

Gal P_{v.u.}(d): K|k is very unruly if and only if for some set Z as above satisfying Gal P_{geom}(d) one has: For almost all conjugacy classes of Z_{v}, the condition Gal P_{v.u.}(d − 1) holds for G_{Kv}^{ℓ}.

Part II

4. Abstract Γ by pro-ℓ Galois theory

In this section we develop a “Γ by pro-ℓ Galois theory”, in other words a theory of “Γ by pro-ℓ Galois groups”, which in some sense has a flavor similar to one of the abstract class field theory. The final aim of this theory is to provide a machinery for recovering fields from their pro-ℓ Galois theory in an axiomatic way.

A) The axioms

The context is the following: Let ℓ be a fixed prime number, and Z_{ℓ} a quotient of ℤ_{ℓ}. Let further Γ_{0} be a profinite group endowed with an “Z_{ℓ}- cyclotomic character” χ_{Γ_{0}} : Γ_{0} → Z_{ℓ}^{×}.

Definition. An abstract (Γ by pro-ℓ) level δ ≥ 0 pre-Galois formation G = (G, pr, (Z_{v})_{v}) is defined by induction on δ to be a profinite group G endowed with extra structure as follows:

Axiom I) The level δ = 0:

A level δ = 0 pre-Galois formation is a pro-finite group G, which is an extension of a closed subgroup Γ ⊆ Γ_{0} by a pro-ℓ-group:

1 → \overline{G} ⟶ G ⟶^{pr} Γ → 1

We will say that G has level δ > 0, if the following Axioms II, III, are inductively satisfied.

Axiom II) Decomposition structure:

G is endowed with a set of of closed subgroups (Z_{v})_{v} closed under conjugation. We call Z_{v} the decomposition group at v, and suppose that:

i) Every decomposition group Z_{v} is endowed with a non-trivial pro-cyclic normal subgroup T_{v} ⊆ ker(pr), such that the set {T_{v}}_{v} is closed under conjugation.

ii) T_{v} ≅ Z_{ℓ} as abstract pro-ℓ groups (all v), and T_{v} ∩ T_{w} = {1} if v ≠ w, and the trivial group is the only accumulation point of the set {T_{v}}_{v}.

We call T_{v} the inertia subgroup at v, and let Γ_{v} = pr(G_{v}) be the image of G_{v} in Γ. We set G_{v} := Z_{v}/T_{v}, and call G_{v} the residue group at v. We remark that since T_{v} ⊆ ker(pr), the restriction of pr to Z_{v} factorizes through G_{v}. Thus G_{v} is an extension of Γ_{v} by a pro-ℓ group of the form

(∗)  1 → \overline{G}_{v} ⟶ G_{v} ⟶^{pr_{v}} Γ_{v} → 1

We also remark that G, as well as all the G_{v} act on Z_{ℓ} via their “cyclotomic characters” χ := χ_{Γ_{0}} ∘ pr, respectively χ_{v} := χ ∘ pr_{v}.

iii) The canonical action of G_{v} on T_{v} resulting from the exact sequence 1 → T_{v} → Z_{v} → G_{v} → 1 is via the cyclotomic character χ_{v} of G_{v}.

Axiom III) Induction:

Every G_{v} caries itself the structure of an abstract pre-Galois formation of level (δ − 1) via the exact sequence (∗) above.

Convention. Let G = (G, pr, (Z_{v})_{v}) be a level δ ≥ 0 pre-Galois formation. In order to have a uniform notation, we enlarge the index set v (which in the case δ = 0 is empty) by a new symbol, which we denote v_{∗}, by setting Z_{v}_{∗} = G and T_{v}_{∗} = {1}. We will say that v_{∗} is the “trivial valuation”. Thus G is the decomposition group of the trivial valuation, and its “inertia group” is the trivial group. In particular, the residue Galois group at v_{∗} is G_{v}_{∗} = G.

Definition/Remark 4.1. Let G = (G, pr, (Z_{v})_{v}) be a pre-Galois formation of some level δ ≥ 0. Consider any α such that 0 ≤ α ≤ δ.

1) By induction on δ it is easy to see that one can view G = (G, pr, (Z_{v})_{v}) canonically as a pre-Galois formation of level α.

2) We define inductively the system G^{(α)} = (G_{α,i})_{i} of the α-residual abstract pre-Galois formations of G as follows: First, if α = 0, then this system consists of G only. In general, if 0 < α ≤ δ, we recall that every residue group G_{v} carries the structure of an abstract pre-Galois formation of level (δ−1); thus of level (α − 1) by remark 1) above. Let G_{v} be this abstract pre-Galois formation. Then by induction, the system of the (α−1)-residual abstract pre-Galois formations G_{v}^{(α−1)} of each G_{v} are defined. We then set G^{(α)} = (G_{v}^{(α−1)})_{v}.

We remark that the “correct” notation for the system of the α-residual abstract pre-Galois formations (G_{α,i})_{i} is to index it by multi-indices of length α of the form v = (v_{i_{1}} … v_{i_{α}}), where v_{i_{1}} is a valuation of G, v_{i_{2}} is a valuation of G_{v}_{1}, etc..

3) For every multi-index v as above, let G_{v} be the profinite group on which the v-residual abstract pre-Galois formation G_{v} is based. Then we will call G_{v} a v-residual group of G. One can further elaborate here as follows: Given a multi-index v = (v_{i_{1}} … v_{i_{α}}), we can define inductively the following:

a) The decomposition group Z_{v} of G, as being —inductively on α— the pre-image of Z_{(v_{i_{2}}...v_{iα})} ⊆ G_{v_{i_{1}}} in Z_{v}_{i_{1}} via Z_{v_{i_{1}}} → G_{v_{i_{1}}} .

b) The inertia group T_{v} of Z_{v}, as being the kernel of Z_{v} → G_{v}. From the definition it follows that T_{v} ≅ Z_{ℓ}^{α}.

c) A canonical projection pr_{v} : G_{v} → Γ_{v} ⊂ Γ induced by pr.

4) We can/will associate to G its “geometric part” \overline{G} = (\overline{G}, pr, (\overline{Z}_{v})_{v}), which is defined as follows: \overline{G} := ker(pr), pr is trivial morphism, and \overline{Z}_{v} = Z_{v}∩ \overline{G}, etc.. Clearly, if G has level δ, then \overline{G} has level δ too. Further, one can define the geometric part \overline{G_{v}} of each v-residual pre-Galois formation G_{v}. Remark that the process of “going/taking v-residual” is compatible with “going/taking geometric parts”, i.e., \overline{G_{v}} = \overline{G}_{v}.

B) From abstract pre-Galois formations to abstract Galois ones

Definition/Remark 4.2. Let G = (G, pr, (Z_{v})_{v}) be an abstract pre-Galois formation of some level δ′ ≥ 0. Because of the action of G on Z_{ℓ} via its “cyclotomic character” χ, we can speak about the corresponding (abstract) G-twists Z_{ℓ}(r), and the resulting ℓ-adic cohomology. We do correspondingly the same for all closed subgroups of G, e.g., for Z_{v} and T_{v} (all v).

1) We denote L̂_{G} = H^{1}(G, Z(1)) and call it the ℓ-adic completion of the abstract pre-field formation defining G. Set further, κ̂_{G} := H^{1}(Γ, Z(1)) and L̂_{\overline{G}} = H^{1}(\overline{G}, Z(1)). Via the exact sequence 1 → \overline{G} ↪ G ⟶^{pr} Γ → 1, we get canonical embeddings

κ̂_{G} ↪ L̂_{G} ↪ (L̂_{\overline{G}})^{Γ} ↪ L̂_{\overline{G}}

We denote Ĥ_{G} = L̂_{G}/κ̂_{G}, and call it the geometric part of L̂_{G}.

Further, consider α satisfying 0 ≤ α ≤ δ′, and v some multi-index of length α. Then we do the same correspondingly for the δ-residual abstract pre-Galois formation G_{v}.

From now on suppose that δ′ > 0.

2) Let T ⊆ G be the closed subgroup generated by all the inertia groups T_{v} (all v). We denote π_{1,G} := G/T and call it the abstract fundamental group of G. Thus there exists a canonical exact sequence 1 → T → G → π_{1,G} → 1. Taking ℓ-adic cohomology, and taking into account that T acts trivially on Z_{ℓ}, it follows that we have an exact sequence of the form

0 → H^{1}(π_{1,G}, Z_{ℓ}(1)) ⟶^{can} H^{1}(G, Z_{ℓ}(1)) = L̂_{G} ⟶^{ȷ^{G}} (Hom(T, Z_{ℓ}))^{G}

We set Û_{G} = H^{1}(π_{1,G}, Z_{ℓ}(1)) and call it the unramified part of L̂_{G}. In particular, if no confusion is possible, we will identify Û_{G} with its image in L̂_{G}.

3) We now have a closer look at the structure of L̂_{G}. For an arbitrary v we have inclusions T_{v} ↪ Z_{v} ↪ G. Thus we can/will consider/denote restriction maps as follows:

ȷ^{v} : L̂_{G} = H^{1}(G, Z_{ℓ}(1))⟶^{res_{Zv}}H^{1}(Z_{v}, Z_{ℓ}(1))⟶^{res_{v}}Hom(T_{v}, Z_{ℓ})

Let T_{v} and T_{v}′ be conjugated, say by σ. It is well known that σ acts trivially on L̂_{G} = H^{1}(G, Z_{ℓ}(1)), and that H^{1}(σ) ∘ ȷ^{v} = ȷ^{v′}. Therefore we have: Let 𝔳 be a system of representatives for the G-conjugacy classes of the inertia (equivalently: decomposition) groups T_{v}. Then by the Axiom II), ii), the family (ȷ^{𝔳})_{𝔳} gives rise canonically to a continuous homomorphism ˆ⊕_{𝔳} ȷ^{𝔳} of ℓ-adically complete ℤ_{ℓ}-modules

ˆ⊕_{𝔳} ȷ^{𝔳} : L̂_{G} = H^{1}(G, Z_{ℓ}(1)) → ˆ⊕_{𝔳} Hom(T_{𝔳}, Z_{ℓ}) .

It obvious that ˆ⊕_{𝔳} ȷ^{𝔳} factors trough Hom(T, Z_{ℓ})^{G}. And one checks without difficulty that the resulting canonical homomorphism is an embedding (of ℓadically complete Z_{ℓ}-modules):

Hom(T, Z_{ℓ})^{G} ↪ ˆ⊕_{𝔳} Hom(T_{𝔳}, Z_{ℓ}) .

We will identify Hom(T, Z_{ℓ})^{G} with its image inside ˆ⊕_{𝔳} Hom(T_{𝔳}, Z_{ℓ}). Thus ȷ^{G} = ˆ⊕_{𝔳} ȷ^{𝔳} on L̂_{G}.

We will denote D̂iv_{G} := ˆ⊕_{𝔳} Hom(T_{𝔳}, Z_{ℓ}) and call it the abstract divisor group of G. We further say that the image of L̂_{G} in D̂iv_{G} is the divisorial quotient (or the divisorial part) of L̂_{G}.

We further set ℭ̂𝔩_{G} = coker(ȷ^{G}), and call it the abstract divisor class group of G. Therefore, we finally have a canonical exact sequence

0 → Û_{G} ↪ L̂_{G} ⟶^{ȷ^{G}} D̂iv_{G} ⟶^{can} ℭ̂𝔩_{G} → 0 .

4) Recall that \overline{G} = ker( pr), thus T ⊂ \overline{G}. Let T → T^{ab}, and \overline{G} → \overline{G}^{ab} be the abelianisations of T and \overline{G}. Thus there is a canonical homomorphism T^{ab} → \overline{G}^{ab}, which need not be injective. Let \overline{v} denote a representative for the \overline{G}-conjugacy classes of T_{v} (all v).

We say that G is complete curve like if the following holds:

i) The canonical map T^{ab} → \overline{G}^{ab} is injective, and the abelianisation homomorphism T → T^{ab} is injective on each T_{v}, i.e., T_{v} ≅ T ab ; \overline{v}.

ii) There exist generators τ ab ; \overline{v} of T ab ; \overline{v} such that ∑\overline{v} τ ab ; \overline{v} = 0, and this is the only pro-relation satisfied by the system of elements 𝔗 = (τ ab ; \overline{v})_{\overline{v}}.

Now consider 0 ≤ δ < δ′. We say that G is δ-residually complete curve like if all the δ-residual pre-Galois formations G_{v} are residually complete curve like. In particular, “0-residually complete curve like” is the same as “residually complete curve like”.

5) For every v consider the exact sequence 1 → T_{v} → Z_{v} → G_{v} → 1 given by Axiom II, iii). Recall that T_{v} ≅ Z_{ℓ}(1) as a Z_{v}-module, and as a G_{v}- module. Using the above exact sequence, we get an inflation homomorphism inf_{v} : H^{1}(G_{v}, Z_{ℓ}(1)) → H^{1}(Z_{v}, Z_{ℓ}(1)).

We set Û_{v} = ker(ȷ^{v}) and call it the abstract v-units in L̂_{G}. Thus the unramified part of L̂_{G} is exactly Û_{G} = ∩_{v}ker(ȷ^{v}). Since T_{v} = ker(Z_{v} → G_{v}), it follows that res_{Z_{v}}(Û_{v}) is the image of the inflation map infl_{v}. Therefore there exists a canonical continuous homomorphism, which we call the v-reduction homomorphism:

ȷ_{v} : Û_{v} → H^{1}(G_{v}, Z_{ℓ}(1)) =: L̂_{G_{v}}

where L̂_{G_{v}} is the ℓ-adic completion of the abstract v-residual field, i.e., the one attached to the v-residual pre-Galois formation G_{v}.

Denote L_{G,fin} = { x ∈ L̂_{G} | ȷ^{v}(x) = 0 for almost all v }. Remark/recall that κ̂_{G} ⊆ Û_{G} ⊆ L_{G,fin}. A closed submodule Δ ⊂ L̂_{G} is said to have finite co-rank, if Δ ⊂ L_{G,fin}, and Δ/ Û_{G} is a finite Z_{ℓ}-module (or equivalently, Δ is contained in ker(ȷ^{v}) for almost all v). Clearly, the sum of two finite co-rank submodules of L̂_{G} is again of finite co-rank. Thus the set of such submodules is inductive. And one has:

L_{G,fin} = ∪_{Δ} Δ (all finite co-rank Δ)

6) Next let v be arbitrary, and G_{v} the corresponding residual pre-Galois formation. To G_{v} we have the corresponding exact sequence as defined for G at point 3) above:

0 → Û_{G_{v}} ↪ L̂_{G_{v}} ⟶^{ȷ^{Gv}} D̂iv_{G_{v}}

We will say that G is ample, if for every finite co-rank submodule Δ ⊆ L̂_{G} there exists v such that:

i) Δ ⊆ Û_{v} and ȷ_{v}(Δ) ⊆ L_{G_{v},fin}.

ii) ker( Δ ⟶^{ȷ_{v}} L̂_{G_{v}} ⟶^{ȷ^{Gv}} D̂iv_{G_{v}} ) ⊆ Û_{G}.

Now consider 0 ≤ δ ≤ δ′. We say that G is ample up to level δ, if either δ = 0, or by induction on δ the following hold: First, if δ = 1, then G is ample in the sense defined above. Second, if δ > 1, then G_{v} is ample up to level (δ−1) for all v.

C) From abstract Galois formations to abstract field formations

Convention/Definition 4.3.

1) In order avoid too technical formulations, we will suppose from now —if not explicitly otherwise stated— that Z_{ℓ} = ℤ_{ℓ}. In particular, ℤ ⊂ ℤ_{(ℓ)} are subgroups/subrings of Z_{ℓ}.

2) Let M be an arbitrary Z_{ℓ}-module. We will say subsets M_{1}, M_{2} of M are ℓ-adically equivalent, if there exists an ℓ-adic unit ϵ ∈ Z_{ℓ} such that M_{2} = ϵ·M_{1} inside M. Correspondingly, given systems S_{1} = (x_{i})_{i} and S_{2} = (y_{i})_{i} of elements of M, we will say that S_{1} and S_{2} are ℓ-adically equivalent, if there exists an ℓ-adic unit ϵ ∈ Z_{ℓ} such that x_{i} = ϵ y_{i} (all i).

3) Let M be an arbitrary ℓ-adically complete module. We will say that a ℤ_{(ℓ)}-submodule M_{(ℓ)} ⊆ M of M is a ℤ_{(ℓ)}-lattice in M, (for short, a lattice) if M_{(ℓ)} is a free ℤ_{(ℓ)}-module, and it is ℓ-adically dense in M, and satisfies the following equivalent conditions:

a) M/ℓ ≅ M_{(ℓ)}/ℓ

b) M_{(ℓ)} has a ℤ_{(ℓ)}-basis 𝔅 which is ℓ-adically independent in M.

c) The condition b) above is satisfied for every ℤ_{(ℓ)}-basis of M_{(ℓ)}.

More general, let N ⊆ M_{(ℓ)} ⊆ M be ℤ_{(ℓ)}-submodules of M such that M/N is again ℓ-adically complete. We will say that M_{(ℓ)} is an N-lattice in M, if M_{(ℓ)}/N is a lattice in M/N.

4) Finally, in the context from 3) above, a true lattice in M is a free abelian subgroup M of M such that M_{(ℓ)} := M ⊗ ℤ_{(ℓ)} is a lattice in M in the sense of 3) above. And we will say that ℤ-submodule M ⊆ M containing N is a true N-lattice in M, if M/N is a true lattice in M/N.

Construction 4.4. Let G = (G, pr, (Z_{v})_{v}) be a pre-Galois formation which is both ample up to level δ and δ-residually complete curve like. Recall the last exact sequence from point 3) of subsection B):

0 → Û_{G} ↪ L̂_{G} ⟶^{ȷ^{G}} D̂iv_{G} ⟶^{can} ℭ̂𝔩_{G} → 0 .

The aim of this subsection is to describe the ℓ-adic equivalence class of a Û_{G}-lattice L_{(ℓ)} in L̂_{G}, which —in the case it exists— will called an abstract field formation defining G.

The case δ = 0, i.e., G complete curve like.

In the notations from Definition/Remark 4.2, 4) above, let 𝔗 = (τ ab ; \overline{v})_{\overline{v}} be the system of generators of the groups T ab ; \overline{v} as there. Let as call such a system a distinguished system of inertia generators. We remark that any two distinguished system of inertia generators are strictly ℓ-adically equivalent. Indeed, if τ ′ ; \overline{v} ∈ T ab ; \overline{v} is another generator of T ab ; \overline{v}, then ϵ\overline{v} τ ′ ; \overline{v} = τ\overline{v} for some ℓ-adic units ϵ\overline{v} ∈ Z_{ℓ}. Thus if 𝔗′ = (τ ′ ; \overline{v})_{\overline{v}} does also satisfy condition ii) from Definition/Remark 4.2, 4), then we have also ∑\overline{v} ϵ−1 ; \overline{v} τ ab ; \overline{v} = 0. By the uniqueness of the relation ∑\overline{v} τ ab ; \overline{v} = 0, it follows that ϵ\overline{v} = ϵ for some fixed ℓ-adic unit ϵ ∈ Z_{ℓ}, as claimed.

Now let 𝔗 = (τ ab ; \overline{v})_{\overline{v}} be a distinguished system generators of T. Further let F_{𝔗} be the Abelian pro-ℓ free group on the system 𝔗. Then one has a canonical exact sequence of pro-ℓ groups

0 → Z_{ℓ} τ → F_{𝔗} → T^{ab} → 0 ,

where τ = ∑\overline{v} τ ab ; \overline{v} in F_{𝔗} is the pro-ℓ sum of the generators τ ab ; \overline{v} (all \overline{v}). Taking ℓ-adically continuous Homs we get an exact sequence

0 → Hom(T^{ab}, Z_{ℓ}(1)) → Hom(F_{𝔗}, Z_{ℓ}(1)) → Hom(Z_{ℓ} τ, Z_{ℓ}(1)) → 0

and remark that last homomorphism is simply the summation:

φ ↦ (τ ↦ ∑\overline{v} φ(τ ab ; \overline{v})

Thus Hom(T^{ab}, Z_{ℓ}(1)) consists of all the homomorphisms φ ∈ Hom(F, Z_{ℓ}) with trivial “trace”. Next consider the system 𝔅 = (φ\overline{v})_{\overline{v}} of all the functionals φ\overline{v} ∈ Hom(F_{𝔗}, Z_{ℓ}) defined by φ\overline{v}(τ_{\overline{v}′}^{ab}) = 1 if \overline{v} = \overline{v}′, and 0, otherwise (all \overline{v}, \overline{v}′). We denote by D_{𝔗,(ℓ)} the ℤ_{(ℓ)}-submodule of Hom(F, Z_{ℓ}) generated by 𝔅, and set

D_{𝔗,ℓ}^{0} := \{ ∑\overline{v} a\overline{v} φ\overline{v} ∈ D_{𝔗,(ℓ)} | ∑\overline{v} a\overline{v} = 0 \} = D_{𝔗,(ℓ)} ∩ Hom(T^{ab}, Z_{ℓ})

We have: 𝔅 is an ℓ-adic basis of Hom(F, Z_{ℓ}), i.e., D_{𝔗,(ℓ)} is ℓ-adically dense in Hom(F, Z_{ℓ}), and there are no non-trivial ℓ-adic relations between the elements of 𝔅. We will say that 𝔅 = (φ\overline{v})\overline{v} is the “dual basis” to 𝔗. In particular, D_{𝔗,(ℓ)} is a lattice in Hom(T^{ab}, Z_{ℓ}), and D_{𝔗,(ℓ)}^{0} is a lattice in Hom(T^{ab}, Z_{ℓ}). Precisely, if φ_{\overline{v}_{0}} is fixed, then the system e\overline{v} = φ\overline{v} − φ_{\overline{v}_{0}} (all \overline{v} ≠ \overline{v}_{0}) is an ℓ-adically independent ℤ_{(ℓ)}-basis of D_{𝔗,(ℓ)}^{0}.

We now explain the dependence of D_{𝔗,(ℓ)} on the distinguished system of generators 𝔗 = (τ ab ; \overline{v})_{\overline{v}} of of T. Let namely 𝔗′ = (τ ′ ; \overline{v})_{\overline{v}} be another such system. Let ϵ ∈ Z_{ℓ} be the unique ℓ-adic unit such that ϵ · 𝔗′ = 𝔗. If 𝔅 = (φ′ ; \overline{v})_{\overline{v}} is the dual basis to 𝔗′, then φ′ ; \overline{v} = ϵ φ\overline{v} inside Hom(T^{ab}, Z_{ℓ}). Thus 𝔅′ = ϵ · 𝔅. In other words, 𝔅 and 𝔅′ are strictly ℓ-adically equivalent. Hence we have: ϵ·D_{𝔗,(ℓ)} = D_{𝔗′,(ℓ)} and ϵ·D_{𝔗,(ℓ)}^{0} = D_{𝔗′,(ℓ)}^{0}. Moreover, ϵ is unique modulo rational ℓ-adic units.

1) Therefore, all the subgroups of Hom(T^{ab}, Z_{ℓ}) the form D_{𝔗,(ℓ)}, respectively of the form D_{𝔗,(ℓ)}^{0}, are ℓ-adically rationally strictly equivalent (all distinguished 𝔗). Hence the ℓ-adic equivalence classes of D_{𝔗,(ℓ)} and D_{𝔗,(ℓ)}^{0} do not depend on 𝔗, but only on G.

2) We now consider G-invariants: First we remark that G acts on both T^{ab} and the distinguished systems of inertia generators 𝔗 by conjugation, and on Hom(T^{ab}, Z_{ℓ}(1)) functorially. Thus this action is takes place actually via the quotient pr : G → Γ. Thus we have:

a) If 𝔗 = ( τ ab ; \overline{v})\overline{v} is a distinguished system of inertia generators, then 𝔗^{γ} is also such a system. Taking into account Axiom II), iii), it follows that 𝔗^{γ} = (τ ab ; \overline{v})^{γ} = χ(γ) τ_{\overline{v}_{γ}}^{ab}, where χ is the cyclotomic character of Γ, and \overline{v}_{γ} is the corresponding Γ-conjugate of \overline{v}. Thus up to permutation of the coordinates we have 𝔗^{γ} = χ(γ) · 𝔗. Since for every v the image pr(Z_{v}) = Γ_{v} ⊆ Γ is open, it follows that the orbit of each T ab ; \overline{v} under Γ is finite, say of cardinality d\overline{v}.

b) From this one easily deduces the action of γ on the generators φ\overline{v}. It is given by (φ\overline{v})^{γ} = φ_{\overline{v}}^{γ}. Hence from this it follows that the Γ-invariant part of D_{𝔗,(ℓ)} and D_{𝔗,(ℓ)}^{0} can be described as follows:

• (D_{𝔗,(ℓ)})^{G} = \{ ∑\overline{v} a\overline{v} φ\overline{v} ∈ D_{𝔗,(ℓ)} | a\overline{v} = a_{\overline{v}}^{γ} for all \overline{v} and all γ ∈ Γ \}.
• (D_{𝔗,(ℓ)}^{0})^{G} = \{ φ ∈ (D_{𝔗,(ℓ)})^{G} | ∑\overline{v} a\overline{v} = 0 \}.

For every 𝔗, we denote Div_{𝔗,(ℓ)} = (D_{𝔗,(ℓ)})^{G} = D_{𝔗,(ℓ)} ∩ D̂iv_{G,(ℓ)}. We have the following:

Fact 4.5. In the above context, let L_{𝔗,(ℓ)} be the pre-image of Div_{𝔗,(ℓ)} in L̂_{G}. Further consider all the finite co-rank submodules Δ of L̂_{G} containing Û_{G}. Then the following assertions are equivalent:

(i) L_{𝔗,(ℓ)} is a Û_{G}-lattice in L̂_{G}.

(ii) Δ ∩ L_{𝔗,(ℓ)} is a Û_{G}-lattice in Δ (all Δ as above).

Proof. Clear.

The case: δ > 0.

For every v we denote by G_{v} the corresponding v-residual abstract pre-Galois formation, etc.. In particular, G_{v} is both (δ − 1)-residually complete curve like, and ample up to level (δ − 1). Now let Δ be a finite co-rank Z_{ℓ}- submodule of L̂_{G} such that Δ ∩ Û_{G} = 1. For every v such that Δ ⊂ Û_{v}, we set Δ_{v} = ȷ_{v}(Δ). Since G is ample up to level δ, there exists some v such that:

(∗) Δ ⊂ Û_{v}, and Δ_{v} has finite co-rank in L̂_{G_{v}}, and Δ_{v} ∩ Û_{G_{v}} = 1. In particular, Δ and Δ_{v} have the same Z_{ℓ}-rank.

Fact 4.6. For every v and the corresponding G_{v}, let a Û_{G_{v}}-lattice L_{v,(ℓ)} in L̂_{G_{v}} be given. Then up to ℓ-adic equivalence, there exits at most one Û_{G}- lattice L_{(ℓ)} in L̂_{G} such that for every finite co-rank Z_{ℓ}-module Δ ⊂ L̂_{G} with Δ ∩ Û_{G} = 1 and v as at (∗) above the following hold:

i) L_{Δ} := Δ ∩ L_{(ℓ)}, L_{Δ_{v}} := Δ_{v} ∩ L_{v,(ℓ)} are lattices in Δ, respectively Δ_{v}.

ii) The lattices ȷ_{v}(L_{Δ}) and L_{Δ_{v}} are ℓ-adically equivalent.

Proof. Let L_{(ℓ)}, L′_{(ℓ)} be Û_{G}-latices in L̂_{G} satisfying i), ii), above. For Δ as in Fact above, set L′_{Δ} = Δ ∩ L′_{(ℓ)}. Then by hypothesis ii) it follows that both latices ȷ_{v}(L_{Δ}) and ȷ_{v}(L′_{Δ}) are ℓ-adically equivalent to the lattice L_{Δ_{v}} inside Δ_{v}. Thus they are equivalent. After replacing L′_{(ℓ)} by some multiple, we can suppose that L_{Δ} = L′_{Δ}. As Δ was arbitrary, it finally follows that L_{(ℓ)} = L′_{(ℓ)}.

Definition/Remark 4.7. Let G be an abstract pre-Galois formation which is both δ-residual complete curve like and ample up to level δ. An abstract field formation L_{(ℓ)} defining G —if it exists— is defined inductively on δ as follows:

First, if δ = 0, suppose that G satisfies the equivalent conditions (i), (ii), from Fact 4.5. Then define L_{(ℓ)} to be one of the L_{𝔗,(ℓ)}.

Second, if δ > 0, suppose that for all v, corresponding v-residual abstract field formations L_{v,(ℓ)} defining G_{v} exist. We define L_{(ℓ)} to be any Û_{G}-lattice in L̂_{G} which satisfies the conditions i), ii) from Fact 4.6.

We will say that G as above is a level δ abstract Galois formation, if there exist abstract field formations L_{(ℓ)} defining G as explained above.

We remark the following:

Let G be a level δ abstract Galois formation, and L_{(ℓ)} an abstract field formation defining G. Then ȷ^{v}(L_{(ℓ)}) is a lattice in Hom(T_{v}, Z_{ℓ}(1)) ≅ Z_{ℓ} (all v). Recalling that 𝔳 are representatives for the conjugacy classes of the valuations v, we will set Div_{L_{(ℓ)}} = ⊕_{𝔳} ℤ_{(ℓ)}𝔳, and ℭ𝔩_{L_{(ℓ)}} = Div_{L_{(ℓ)}}/ȷ^{G}(L_{(ℓ)}), and call them the abstract divisor group of L_{(ℓ)}, respectively the abstract ideal class group of L_{(ℓ)}. Thus one has a commutative diagram of the form

(∗)

0 → Û_{G} ↪ L_{(ℓ)} ⟶^{ȷ^{G}} Div_{L_{(ℓ)}}  ⟶^{can} ℭ𝔩_{L_{(ℓ)}} → 0
↓  ↓  ↓  ↓^{ȷ_{ℭ𝔩}}
0 → Û_{G} ↪ L̂_{G}  ⟶^{ȷ^{G}}  D̂iv_{G}  ⟶^{can}  ℭ̂𝔩_{G} → 0

where the first three vertical morphisms are the canonical inclusions, whereas the last one is the ℓ-adic completion homomorphism.Thus in particular, ℭ𝔩_{L_{(ℓ)}} does not necessarily embed in ℭ̂𝔩_{G}.

D) Example: Concrete Galois formations / concrete field formations

Let 𝔨 be a perfect base field, e.g., a finite field or a number field. Let further 𝔎|𝔨 be a field extension such that the relative (separable) algebraic closure of 𝔨 in 𝔎 is finite over 𝔨. Consider any field extension 𝔎′|𝔎 satisfying the conditions i), ii), iii), from Introduction. Suppose that 𝔎′ contains a separable closure 𝔨^{sep} = \overline{𝔨} of 𝔨. We set as usually G = Gal(𝔎′|𝔎), and Γ_{0} = G_{𝔨}, and let pr : G → Γ := pr(G) ⊆ Γ_{0} be the canonical projection. We further set Z_{ℓ} = ℤ_{ℓ}, and endow Γ_{0} with its Z_{ℓ}-cyclotomic character χ_{Γ_{0}} : Γ_{0} → Z_{ℓ}^{×}. Then we get:

Fact 4.8. G = (G, pr, / O) is a level δ = 0 pre-Galois formation.

Nevertheless, in the case 𝔎|𝔨 is a function field with d = td(𝔎|𝔨) > 0, we can refine the pre-Galois formation above by making it into a pre-Galois formation of level δ for every 0 ≤ δ ≤ d as follows: First, let us endow every function field 𝔎|𝔨 as above with a geometric set 𝔇_{X} of Zariski prime divisors of 𝔎|𝔨. Here X → 𝔨 is a quasi-projective normal model of 𝔎|𝔨, and 𝔇_{X} is its set of Weil prime divisors. Without loss of generality, we will suppose that X geometrically integral, or equivalently, that the function field 𝔎|𝔨 is regular. Second, for every field extension 𝔎′|𝔎 as above, let 𝔇′_{X} be the prolongation of 𝔇_{X} to 𝔎′. For every v ∈ 𝔇′_{X} we have: The residue field 𝔎v is again a function field over 𝔨 with td(𝔎v | 𝔨) = (d − 1). Furthermore, using the facts presented in Section 2, it is an easy exercise to show that one has the following:

j) Let T_{v} ⊂ Z_{v} be the inertia, respectively the decomposition group of v in G. Then T_{v} ≅ ℤ_{ℓ} is pro-ℓ cyclic. Further, if v ≠ w, then T_{v} ∩ T_{w} = {1}.

jj) For every v, the residue field extension 𝔎′v | 𝔎v satisfies the conditions i),

ii), iii) from the Introduction. jjj) Let G_{v} = Gal(𝔎′v | 𝔎v) be the residual Galois group at v. Then denoting

pr_{v} : G_{v} → Γ the canonical projection, it follows that Γ_{v} := pr_{v}(G_{v}) is an open subgroup of Γ.

Thus the field extension 𝔎′|𝔎 defines in a canonical way “local” field extensions 𝔎′v|𝔎v which also satisfy the conditions j), jj), jjj), above. Thus if d > 1, each such function field 𝔎v|𝔨 comes equipped with a geometric set of Zariski prime divisors 𝔇_{X_{v}} (with X_{v} a quasi-projective normal model of 𝔎v | 𝔨).

Now consider any δ with 0 ≤ δ ≤ d. Then by induction on the transcendence degree over 𝔨, we can suppose that each G_{v} endowed with the projection pr_{v} : G_{v} → Γ and the set of decomposition groups defined by 𝔇′_{X_{v}} is an pre-Galois formation of level (δ − 1). Thus we have the following:

Proposition/Definition 4.9. Let G_{X} = (G, pr, (Z_{v})_{v}) be as constructed above. Then G_{X} is in a canonical way a level δ abstract pre-Galois formation, for every δ satisfying 0 ≤ δ ≤ td(𝔎|𝔨).

An abstract pre-Galois formation of the form G_{X} will be called a concrete pre-Galois formation of level δ.

Remarks 4.10. Let G_{X} = (G, pr, (Z_{v})_{v}) be a concrete pre-Galois formation as constructed/defined above. Then 𝔎̂ = H^{1}(G′_{𝔎} , Z_{ℓ}(1)) = L̂_{G_{X}}. In order to compute Û_{G_{X}}, D̂iv_{G_{X}}, and ℭ̂𝔩_{G_{X}} we do the following: First, let H(X) denote the group of principal divisors on X, and consider the canonical exact sequence 0 → H(X) ⟶ Div(X) ⟶^{pr} ℭ𝔩 (X) → 0 .

1) Passing to ℓ-adic completions, we get an exact sequence of ℓ-adic complete groups of the form: 0 → T_{ℓ,X} ⟶ Ĥ(X) ⟶ D̂iv(X) ⟶ ℭ̂𝔩 (X) → 0, where T_{ℓ,X} = lim_{⟵} _{ℓ^{n}}ℭ𝔩 (X) (with multiplication as homomorphisms), and the last three objects the corresponding ℓ-adic completions.

2) On the other hand one has Div(X) = ⊕_{𝔳∈𝔇_{X}} 𝔳𝔎. For every 𝔳, and the fixed prolongation v of 𝔳 to 𝔎′, consider the commutative diagram from subsection A), 2^{ℓ}). Recalling that 𝔎^{×} → H(X), x ↦ ⊕ 𝔳(x) is surjective, we get a commutative diagram of the following form:

𝔎̂  → D̂iv(X) = ˆ⊕_{v} v𝔎 → ℭ̂𝔩 (X) → 0
↓^{ȷ_{𝔎}}  ↓^{⊕ȷ_{v}}  ↓^{ȷ_{X}}
H^{1}(Gℓ_{𝔎} , ℤ_{ℓ}(1)) ⟶^{ȷ^{G_{X}}} ˆ⊕_{v} H^{1}(T_{v}, ℤ_{ℓ}(1)) ⟶^{can} P̂_{X} → 0

where the vertical maps are isomorphisms, and P̂_{𝔎} is simply the quotient of the middle group by the second. We remark/recall that ⊕ȷ_{v} is defined as follows, see Definition/Remark 1.4: For every v, let γ_{v} = 1 · v be the unique positive generator of v𝔎 = ℤv. Then there exists a unique generator of the arithmetical inertia τ_{v} ∈ 𝔗_{v} such that ȷ_{v}(γ_{v})(τ_{v}) = 1. Hence, the commutativity of the diagram above follows directly from the definition of the homomorphisms.

3) From this we deduce: D̂iv_{G_{X}} = D̂iv(X) and ℭ̂𝔩_{G_{X}} = ℭ̂𝔩(X).

4) Finally, we remark that one has a canonical exact sequence of the form 1 → Û(X) → Û_{G_{X}} → T_{ℓ,X} → 0, where U(X) is the group of global unit sections on X. Thus this gives a complete description of Û_{G_{X}}.

We next want to discuss the issue of a concrete pre-Galois formation being δ-residually complete curve like, respectively ample up to level δ. Let us say that a quasi-projective normal variety X → 𝔨 is almost complete if the group of global units is U(X) = 𝔨^{×}. Remark that projective normal varieties are almost complete in the sense above.

Proposition 4.11. Let 𝔨 be a perfect base field, and 𝔎|𝔨 a function field. Let G_{X} be a concrete Galois formation as introduced/defined in Proposition above. Consider some δ ≤ td(𝔎|𝔨), and let us view G_{X} as an abstract pre-Galois formation of level δ. We denote by G_{X_{v}} the residual pre-Galois formations of G_{X} with v multi-indices of length ≤ δ. In particular, X_{v} → 𝔨 are quasi-projective normal varieties. Then one has:

(1 ) G_{X} is δ-residually complete curve like if (and only if ) δ = td(𝔎|𝔨)−1, and all the δ-residual varieties X_{v} are complete normal curves.

(2 ) Suppose that either 𝔨 is a hilbertian field, or that δ < td(𝔎|𝔨). Further suppose that X_{v} is almost complete (all v as above). Then G_{X} is ample up to level δ.

(3 ) Let ȷ_{𝔎} : 𝔎^{×} → 𝔎̂ be the completion homomorphism, and inside 𝔎̂ set 𝔎_{(ℓ)} := ȷ_{𝔎} (𝔎^{×}) ⊗ ℤ_{(ℓ)}. Suppose that G_{X} is δ-residually curve like, and ample up to level δ, thus δ = td(𝔎|𝔨) − 1. Then L_{X,(ℓ)} := Û(X) · 𝔎_{(ℓ)} is an abstract field formation defining G_{X}, called the concrete field formation defining G_{X}.

Proof. To (1) and (3): Clear. It is nevertheless more/quite difficult to prove the “only if” part of (1), which we will not directly use, thus omit the proof here...

To (2): In the notations from subsection 4.2, let Δ ⊂ 𝔎̂ be a co-finite rank submodule such that Δ ∩ Û_{G_{X}} = 0. Then by definition, the set of all 𝔳 ∈ 𝔇_{X} such that Δ ⊂ Û_{v} has a finite complement in 𝔇_{X}. Thus there exists an open subset X′ ⊂ X such that for all 𝔳 ∈ 𝔇_{X′} one has: Δ ⊂ Û_{v}. In other words, after replacing X by X′, it is sufficient to prove that there exist infinitely many v ∈ 𝔇_{X} such that Δ = ker(ȷ^{G_{X}}) is mapped isomorphically into 𝔎̂v by the reduction map ȷ_{v} : Û_{v} → 𝔎̂v. This now immediately follows from the Lemma below.

We begin with the following little preparation. For the beginning, let 𝔎|𝔨 be any field extension, and ℓ ≠ char (𝔨) a fixed rational prime number. We will say that a finitely generated subgroup Δ ⊂ 𝔎^{×}, say of rational rank r, is 𝔨, ℓ-pure in 𝔎, if Δ (mod 𝔨^{×} · 𝔎^{×}^{ℓ}) has the same rank as Δ. Equivalently, setting 𝔏 = 𝔎[Δ^{\frac{1}{ℓ}}] one has: Δ is 𝔨, ℓ-pure in 𝔎 ⇔ [𝔏 : 𝔎] = ℓ^{r} and 𝔏 is linearly disjoint to the compositum 𝔎 𝔨′, where 𝔨′ is the Galois extension of 𝔨 obtained by adjoining to 𝔨 all the ℓ^{∞} roots from all the elements of 𝔨.

Now consider the context of the Proposition. One has an exact sequence of the form 1 → 𝔨^{×} → U(X) → H_{U(X)} → 0, where H_{U(X)} ⊂ 𝔎^{×}/𝔨^{×} is a finite free ℤ-module, say of rank r. Let Δ be a complement of 𝔨^{×} in U(X). Then Δ is a finite free ℤ-module of rank r. Setting 𝔏 = 𝔎[Δ^{\frac{1}{ℓ}}], one has: First, since 𝔎^{×}/𝔨^{×} is torsion free, [𝔏 : 𝔎] = ℓ^{r}. Second, since 𝔎|𝔨 is a regular field extension, 𝔏 is 𝔎-linearly disjoint to 𝔎 𝔨^{s}. In particular, Δ is 𝔨, ℓ-pure in 𝔎.

Lemma 4.12. Under the hypothesis of the Proposition above, there exist infinitely many v such that U(X) ⊂ U_{v} consists of non-principal v-units, and moreover, Δv is 𝔨, ℓ-pure in 𝔎v. In particular, Û(X) is mapped isomorphically into 𝔎̂v by the reduction map ȷ_{v}.

Proof. First remark that by induction on td(𝔎|𝔨), there exist “many” subfield extensions κ|𝔨 of 𝔎|𝔨 such that 𝔎|κ is a regular function field in one variable, and Δ ∩ κ = {1} (WHY?). Thus κ|𝔨 is a function field. We remark that under the hypothesis of point (2) of the Proposition, κ is hilbertian. Let κ̃ ⊂ κ′ ⊂ 𝔎′ denote the Galois extension of κ obtained by adjoining to κ all the ℓ^{∞} roots from all the elements of κ. Since κ is Hilbertian, it follows by Kuyk’s Theorem, see e.g. [We], Satz 9.8, that κ[μ_{ℓ^{∞}}] is hilbertian too. Therefore, by loc.cit., the same holds for κ̃, as it is an Abelian extension of κ[μ_{ℓ^{∞}}].

By the remarks above, 𝔏 = 𝔎[Δ^{\frac{1}{ℓ}}] is linearly disjoint to 𝔎κ̃ over 𝔎. To simplify notations, set 𝔏̃ = 𝔏κ̃ and 𝔎̃ = 𝔎κ̃. Then [𝔏̃ : 𝔎̃] = [𝔏 : 𝔎]. Since κ̃ is hilbertian, there exist infinitely many places ṽ of 𝔏̃|κ̃ which are inert in the function field extension 𝔏̃|𝔎̃.^{3} In particular, there exist such places ṽ that Δ consists of ṽ-units.

Therefore, for such places ṽ we have: 𝔏̃ṽ | 𝔎̃ṽ is a separable extension of the same degree as 𝔏̃|𝔎̃. On the other hand, 𝔏̃ṽ = 𝔎̃ṽ [ (Δṽ)^{\frac{1}{ℓ}}]. Therefore, 𝔏ṽ is linearly disjoint to 𝔎̃ṽ over 𝔎ṽ. On the other hand we have: First, 𝔏ṽ = 𝔎ṽ [ (Δṽ)^{\frac{1}{ℓ}}], and second, κ̃ ⊂ 𝔎̃ṽ. From this we deduce that Δṽ is κ, ℓ-pure in 𝔎ṽ. In particular, it follows that Δ does not contain non-trivial principal units, and second, that Δṽ is 𝔨, ℓ-pure in 𝔎ṽ. Finally, the ṽ-reduction homomorphism ȷṽ : Uṽ → 𝔎ṽ maps U(X) injectively into 𝔎ṽ.

In order to finish the proof of the Lemma, it is sufficient to remark that almost all places ṽ of 𝔎|κ are actually defined by Weil prime divisors 𝔳 of X. In order to do this, we remark that the field extension κ ↪ 𝔎 id defined by some rational dominant rational map ψ : X ---^{>} Y, where Y is some model Y → 𝔨 of the filed extension κ|𝔨. Since 𝔎|κ is a function field of one variable, it follows that the general fiber of ψ : X ---^{>} Y has dimension 1. Thus making a base change of ψ to κ (via the canonical morphism Spec κ ↪ Y), it follows that X_{κ} → κ is a geometrically integral normal model of 𝔎|κ. Thus X_{κ} is an open subset in its completion. Thus almost all places of 𝔎|κ are defined by closed points of X_{κ}, thus by Zariski prime divisors 𝔳 of X. (Precisely, by Zariski prime divisors corresponding to points x_{1} ∈ X of co-dimension 1, such that ψ(x_{1}) is not the generic point of Y.)

3 This is nothing but an equivalent way to say that κ̃ is a hilbertian field.

E) The case of a finite base field

Let K be a finitely generated field with d = td(K) > 0. Let k be the constant field of K. Further let G_{K}^{ℓ} = Gal(K^{ℓ}|K) be as usually defined. The aim of this section is to show that the system of all concrete pre-Galois formations G_{X} are actually group theoretically encoded in G_{K}^{ℓ}. Moreover, the extra information concerning such a pre-Galois formation, e.g., δ-residually complete curve like, and/or ample up to level δ, is also encoded in the Galois theory.

1) First, by Section 2, A) and C), the projection pr : G_{K}^{ℓ} → G_{k} and the cyclotomic character χ : G_{K}^{ℓ} → ℤ_{ℓ}^{×} are group theoretically encoded in G_{K}^{ℓ}.

2) Second, by Section 1, b), the set of all Zariski prime divisors 𝔇_{K^{ℓ}} of K^{ℓ} is in bijection with the divisorial subgroups Z_{v} of G_{K}^{ℓ}. And T_{v} ⊂ Z_{v} is the unique pro-ℓ cyclic normal subgroup of Z_{v}. Thus in particular, G_{Kv}^{ℓ} = Z_{v}/T_{v} is encoded in Z_{v}.

3) Moreover, by subsection C) of Section 3, the geometric sets of Zariski prime divisors are encoded in G_{K}^{ℓ}.

From 1)—3) above we deduce that the pre-Galois formations G_{X} based on K are encoded in G_{K}^{ℓ}.

4) Further, using Proposition 4.13, the fact that G_{X} is complete curve like is equivalent to the follows two facts: First, cd G_{K}^{ℓ} = 2, thus td(K) = 1. And second, (Z_{v})_{v} is the set of \underline{all} divisorial subgroups of G_{K}^{ℓ}.

In the same way, if d = td(K) > 1, then combining Proposition 4.13 with the discussion above, we see that the fact that some G_{X} is δ-residually complete curve like is encoded in the Galois information carried by G_{X}: One must have δ = td(K) − 1, and all the (δ − 1)-residual varieties X_{v} must be complete curves.

5) Finally, the questions whether some given G_{X} is ample up to level δ follows immediately using Proposition 4.13: Recall first the discussion preceding Proposition 4.13 above. By point 4) of loc.cit., Û_{G_{X}} ≅ Û(X). Thus X is almost complete if and only if Û_{G_{X}} equals k̂, or equivalently, it is finite. The same is true correspondingly for the residual varieties X_{v}. Thus applying Proposition 4.13, 2), we have: Suppose that all residual varieties X_{v} are almost complete (all multi-indices v of length ≤ δ). Then G_{X} is ample up to level δ if and only if δ < td(K).

6) Nevertheless, the questions about the existence of canonically defined abstract fields formations L_{(ℓ)} defining G_{X} is more subtle... Recalling the notation K_{(ℓ)} = ȷ_{K}(K^{×}) ⊗ ℤ_{(ℓ)} inside K̂, we show the following:

Theorem 4.13. In the above context, let G_{X} be δ-residually complete curve like, and let all residual varieties X_{v} be almost complete. Thus in particular, δ = td(K) − 1, and G_{X} is ample up to level δ. Then the “concrete” field formation L_{K,(ℓ)} := K_{(ℓ)} ⊗ ℤ_{(ℓ)} defining G_{X} is —up to ℓ-adic equivalence— the unique abstract field formation defining G_{X}.

Proof. Indeed, let us consider the exact sequence defined at 2) of loc.cit. above. Since T_{ℓ,X} = 0, we get

(∗)  0 → Ĥ(X) ⟶ D̂iv(X) ⟶ ℭ̂𝔩 (X) → 0 .

Case δ = 1, i.e., X is a complete curve, and G_{X} is complete curve like. Then the above exact sequence has a quite well understood shape, and everything is completely encoded in G_{K}^{ℓ}. Indeed, the concrete pre-Galois formation G_{X} is encoded in G_{K}^{ℓ}. Further, let us consider the “geometric part” \overline{G}_{X} of G_{X}. Then \overline{G} = G_{K\overline{k}}^{ℓ}. Choosing canonical generators of inertia τ ab ; \overline{v} (all \overline{v}) as indi-

cated at Definition/Remark 1.4, it is well known that the resulting 𝔗 = (τ ab ; \overline{v})_{\overline{v}} is a distinguished system of inertia generators. Consider the commutative diagram mentioned in Remark 4.10, which in our context looks like the following:

1 → k̂  →  K̂  →  D̂iv(X) → ℭ̂𝔩 (X) → 0
↓^{ȷ_{k}}  ↓^{ȷ_{K}}  ↓^{⊕ȷ_{v}}  ↓^{can}
1 → H^{1}(G_{k}, ℤ_{ℓ}(1)) → H^{1}(G_{K}^{ℓ}, ℤ_{ℓ}(1)) ⟶^{ȷ^{G_{X}}} ˆ⊕_{v} H^{1}(T_{v}, ℤ_{ℓ}(1)) → P̂_{X} → 0

It then immediately follows that the canonical lattice L_{𝔗,(ℓ)} as considered in Fact 4.5 is exactly the (pre-image under ȷ_{K} of the) pre-image of H(X)⊗ℤ_{(ℓ)} in K̂, where H(X) is the group of principal divisors on X. In other words,

L_{𝔗,(ℓ)} = K_{(ℓ)} := ȷ_{K}(K^{×}) ⊗ ℤ_{(ℓ)} inside K̂ = K̂_{G_{X}} .

General case td(K) > 0: By induction on td, suppose that we know that K_{v,(ℓ)} is —up to ℓ-adic equivalence— the unique abstract field formation defining G_{X_{v}} (all v). By Proposition 4.6, we have only to show that the k̂- lattice K_{(ℓ)} satisfies the conditions i), ii), from loc.cit.. Thus consider any finite co-rank submodule Δ of K̂, say containing k̂. Then Δ/k̂ is a finite free Z_{ℓ}-module, say of rank r. From the exact sequence (∗) above, it follows that Δ ∩ K_{(ℓ)} is a k̂-lattice in Δ. Further, by Proposition 4.13 it follows that there exists “many” v such that Δ ⊂ Û_{v}, and the reduction map ȷ_{v} : Û_{v} → K̂v is injective on Δ.

For any such v we have: Δ ∩ ȷ(K^{×}) is mapped by ȷ_{v} injectively into Kv^{×}. Further, ȷ_{v}(Δ) is the ℓ-adic closure of ȷ_{v}(Δ ∩ ȷ(K^{×})) in K̂v. Therefore, ȷ_{v}(Δ) has finite co-rank in K̂v, and further,

ȷ_{v}(Δ ∩ K_{(ℓ)}) ⊆ ȷ_{v}(Δ) ∩ K_{v,(ℓ)} .

Both sides of the above inclusion are finite ℤ_{(ℓ)}-modules, and they have equal ℓ-adic completions, namely equal to ȷ_{v}(Δ). Therefore the above inclusion is actually an equality. Thus the conditions i), ii), from Fact 4.6 are verified.

5. From abstract field formations to concrete ones

A) Morphisms of pre-Galois formations

Let Γ_{0,G} and Γ_{0,H} be given profinite groups endowed with “pro-ℓ cyclotomic characters” χ_{Γ_{0,G}} : Γ_{0,G} → ℤ_{ℓ}^{×} and χ_{Γ_{0,H}} : Γ_{0,H} → ℤ_{ℓ}^{×}. Let further Φ_{0} : Γ_{0,G} → Γ_{0,H} be an open homomorphism which is compatible with the cyclotomic characters.

Definition. Let G = (G, pr_{G}, (Z_{v})_{v}) and H = (H, pr_{H}, (Z_{w})_{w}) be given abstract pre-Galois formations of some levels δ_{G} and δ_{H}. Let δ be an integer satisfying 0 ≤ δ ≤ δ_{G}, δ_{H}.

1) A level δ = 0 morphism Φ : G → H is any open group Φ_{0}-morphism Φ : G → H, i.e., satisfying Φ_{0} ∘ pr_{G} = pr_{H} ∘ Φ.

2) A level δ > 0 morphism Φ : G → H is defined to be any open group Γ-homomorphism satisfying further conditions as follows:

i) For almost all w, there exits v such that Φ(T_{v}) ≠ {1} and Φ(T_{v}) ⊆ T_{w}, and if w ≠ w_{∗} is non-trivial, the set of conjugacy classes of v with this property is finite. Further, for almost all conjugacy classes of v with Φ(T_{v}) ≠ {1}, there exists w such that Φ(T_{v}) ⊆ T_{w}.

In the context above, if Φ(T_{v}) ≠ {1} and Φ(T_{v}) ⊆ T_{w}, we will say that w corresponds to v. In particular, if w ≠ w_{∗} corresponds to v, then v ≠ v_{∗} is non-trivial. And w_{∗} corresponds to all v such that Φ(T_{v}) = 1. And to almost all conjugacy classes of valuations v there do correspond (conjugacy classes of) valuations w. We remark that if w corresponds to v, then Φ induces a group homomorphism

Φ_{v} : G_{v} → H_{w}

which is compatible with the projection homomorphisms pr_{v} : G_{v} → Γ_{G} and pr_{w} : H_{w} → Γ_{H}. This means Φ_{0} ∘ pr_{v} = pr_{w} ∘ Φ_{v}.

ii) If w corresponds to v, then inductively on δ and on δ_{G}, the residual group homomorphism Φ_{v} : G_{v} → H_{w} gives rise to: A level (δ − 1) morphism of pre-Galois formations G_{v} → H_{w}, if w is non-trivial, respectively, a level δ morphism G_{v} → H, if w is trivial (all w, and v).

Remarks 5.1. In the above context, let Φ : G → H be a level δ morphism of pre-Galois formations.

1) Then Φ gives rise to a Kummer homomorphism

φ̂ : L̂_{H} = H^{1}(H, Z_{ℓ}(1)) → H^{1}(G, Z_{ℓ}(1)) = L̂_{G} .

which is compatible with the embeddings L̂_{H} ↩ κ̂_{H} ⟶^{can} κ̂_{G} ↪ L̂_{G}.

From now on suppose that δ > 0.

2) Let G_{v} and H_{w} be the δ-residual groups of G, respectively H, see Remark 4.1, especially 3). Then by induction on δ, it is clear how to define the fact that w corresponds to v. Using the elaboration from loc.cit., one has the following: w corresponds to v if and only if Φ(T_{v}) ⊆ T_{w}.

3) In particular, let w correspond to v, and suppose that w is a multiindex of length δ_{w}. Then Φ : G_{v} → H_{w} gives rise to a residual level (δ − δ_{w}) morphism of abstract pre-Galois formations Φ_{v} : G_{v} → H_{w}. Thus Φ_{v} also gives rise to a Kummer v-homomorphism

φ̂_{v} : L̂_{w} = H^{1}(H_{w}, Z_{ℓ}(1)) → H^{1}(G_{v}, Z_{ℓ}(1)) = L̂_{v}

4) For every finite co-rank module Δ ⊂ L̂_{H}, and let φ̂ (Δ) ⊂ L̂_{G} be its image by the Kummer homomorphism. Then by condition i) above, it follows that φ̂ (Δ) has finite co-rank. Thus in the notations from Definition/Remark 4.2, 5), it follows that φ̂ maps L_{H,fin} into L_{G,fin}.

Definition/Remark 5.2. Let Φ : G → H be a level δ morphism of pre-Galois formations. We will say that:

1) Φ is proper, if first, each w corresponds to some v, and vice-versa, to each v there is some w corresponding to it (which might be trivial). Second, if residually w corresponds to v, then the residual morphism Φ_{v} : G_{v} → G_{w} is —inductively— a proper one.

2) Φ defines H as a level δ quotient of G, or that H is a level δ quotient of G via Φ, if Φ is proper, and both Φ(G) = H and Φ(T_{G}) = T_{H}.

In particular, if H is a quotient of G, then we can identify Γ_{H} with the corresponding quotient of Γ_{G}.

We remark the following: Let Φ : G → H be a quotient of G. Then in the notations from Remark/Definition 4.2, 2), Φ gives rise to a commutative diagram as follows:

1 → T_{G} → G → π_{1,G} → 0
↓^{Φ}  ↓^{Φ}  ↓
1 → T_{H} → H → π_{1,H} → 0

where the vertical homomorphisms are all induced by Φ and in particular, surjective. Thus taking ℓ-adic cohomology, and using the remarks from loc.cit., we obtain a commutative diagram of the form:

(∗)  \begin{aligned}0 → Û_{H} & → & L̂_{H} & → & D̂iv_{H} & → & ℭ̂𝔩_{H} → 0 \\ ↓φ̂ & & ↓φ̂ & & ↓ & & ↓ \\ 0 → Û_{G} & → & L̂_{G} & → & D̂iv_{G} & → & ℭ̂𝔩_{G} → 0 \end{aligned}

Here the vertical homomorphisms are induced by the Kummer homomorphism. The first three ones are inclusions, and the last one is the canonically resulting one.

3) For later use, we also remark the following: Suppose that Φ(T_{G}) = T_{H}. Then Hom(T_{H}, Z_{ℓ}(1)) is pure in Hom(T_{G}, Z_{ℓ}(1)), i.e., the torsion of quotient of the latter group by the former one is trivial. Taking Γ-invariants, from this we finally deduce that Û_{G} · φ̂ (L̂_{H}) is pure in L̂_{G}. This the case e.g., if H is a quotient of G via Φ.

Defnition 5.3. Let G and H be level δ abstract Galois formations. Let further Φ : G → H be a level δ morphism of abstract pre-Galois formations, and φ̂ : L̂_{H} → L̂_{G} be its Kummer homomorphism. We say Φ is a level δ morphism of abstract Galois formations, if for a proper choice of abstract field formations L_{G,(ℓ)} and L_{H,(ℓ)} defining G, respectively H, we have φ̂ (L_{H,(ℓ)}) ⊆ L_{G,(ℓ)} · φ̂ (Û_{H}).

B) From abstract field formations to field formations

We will give a procedure by which —under certain hypothesis— we can identify in a canonical way “true Û_{G}-lattices” inside any abstract field formation defining a given Galois formation G. In the special case of finitely generated fields 𝔎 of positive characteristic, such true lattices will be ℓ-adically equivalent to the image ȷ_{𝔎} (𝔎^{×}) of the completion homomorphism ȷ_{𝔎} : 𝔎^{×} → 𝔎̂.

To begin with, let G be a level one Galois formation, thus in particular, a complete curve like pre-Galois formation. Recall the notations from Construction 4.4, Case δ = 0. In the notations from there, let 𝔗 be a distinguished system of inertia generators of T_{G}^{ab}, and 𝔅 the dual basis to 𝔗. Then we defined D_{𝔗} to be the ℤ_{(ℓ)}-submodule of D̂iv_{G} generated by 𝔅. And further, L_{𝔗,(ℓ)} to be the pre-image of D_{𝔗,(ℓ)} in L̂_{G} (and called it the abstract 𝔗-field defining G).

Now let Div_{𝔗} be the ℤ-submodule of D̂iv_{G} generated by the ℓ-adic basis 𝔅 dual to 𝔗. And further, set L_{𝔗} for its pre-image in L̂_{G}. Then L_{𝔗} is a true Û_{G}-lattice in L̂_{G}. And correspondingly, we can define the abstract divisor class group ℭ𝔩_{L_{𝔗}} of L_{𝔗}. Thus we have a canonical exact sequence of the from:

0 → Û_{G} ↪ L_{𝔗} ⟶^{ȷ^{G}} Div_{𝔗} ⟶^{can} ℭ𝔩_{L_{𝔗}} → 0 .

Definition 5.4. In the notations from above we define:

1) A level one Galois formation G_{i} is called projective line like, if the following hold: First, Û_{G_{i}} consists of the “constants” κ̂_{G_{i}} of L̂_{G_{i}} only. Second, there is a distinguished system of inertia generators 𝔗_{i} such that ℭ𝔩_{L_{𝔗i}} ≅ ℤ. We will call L_{𝔗_{i}} the 𝔗_{i}-field formation defining G_{i}.

2) Let G be a level δ > 0 Galois formation. A 1-dimensional projection of G is any level 1 morphism Φ_{i} : G → G_{i} into a level 1 Galois formation G_{i} satisfying Γ_{G} = Γ_{G_{i}}, thus also κ̂_{G} = κ̂_{G_{i}}, and further conditions as follows:

i) Suppose δ = 1, i.e., G is complete curve like. Then Φ is proper, and there exist distinguished systems of inertia generators 𝔗 = (τ ab ; \overline{v})_{\overline{v}} and 𝔗_{i} = (τ_{\overline{v}_{i}}^{ab})_{\overline{v}_{i}} of G, respectively G_{i}, such that if \overline{v}_{i} corresponds to \overline{v}, then τ_{\overline{v}_{i}}^{ab} is a integer multiple of Φ(τ_{\overline{v}_{i}}^{ab}) (all \overline{v}_{i}).

ii) Suppose that δ > 1. Then there exists v such that Φ_{i}(T_{v}) = {1}. And for all such v, the resulting residual morphism Φ_{i,v} : G_{v} → G_{i} is —inductively— a 1-dimensional projection.

3) A 1-dimensional projection Φ : G → G_{i} is called a rational projection if G_{i} is projective line like.

4) One defines correspondingly 1-dimensional quotients, respectively rational quotients.

Remarks 5.5. In notations as above we have:

1) Let G_{i} be a projective line like Galois formation, 𝔗_{i} one of its distinguished systems of inertia generators. Let κ̂_{i} be “constants” of G_{i}, and L_{i} = L_{𝔗_{i}} the 𝔗_{i}-field formation defining G_{i}. Let 𝔗′ = ϵ^{−1} · 𝔗_{i} with ϵ ∈ Z_{ℓ}^{×} be a further distinguished system of inertia generators of T_{i}. Then in corresponding notations we have a commutative diagram of the from

0 → Û_{G_{i}} → L_{i} → Div_{L_{i}} → ℭ𝔩_{L_{i}} → 0
↓≅  ↓  ↓≅  ↓
0 → Û_{G_{i}} → L′_{i} → Div_{L′_{i}} → ℭ𝔩_{L′_{i}} → 0

where the vertical maps are the multiplication by ϵ. Since ℭ𝔩_{L_{i}} ≅ ℤ, it follows that all the maps are actually isomorphisms. Thus the fact that a Galois formation G_{i} is projective line like does not depend on the specific distinguished system of inertia generators 𝔗_{i}.

a) In particular, every abstract field formation L_{i,(ℓ)} defining G_{i} contains a field formation L_{i} defining G_{i}. And all these field formations are ℓ-adic equivalent κ̂_{i}-latices in L̂_{G_{i}}.

b) Finally, let L_{i} = L_{𝔗_{i}} be a given a field formation defining G_{i}. An element x_{i} ∈ L_{i} is called a generating element if ȷ^{G_{i}}(x_{i}) = φ_{\overline{v}}′_{i} − φ_{\overline{v}_{i}} for some valuations \overline{v}′_{i} ≠ \overline{v}_{i} of \overline{G}. We denote (x_{i})_{∞} = \overline{v} and call it the pole of x_{i}. If x_{i} is a generating element as above, we will denote

P_{x_{i}} = { x′_{i} ∈ L_{i} | x′_{i} generating, and (x′_{i})_{∞} = (x_{i})_{∞} }

We further set P_{i} = ∪_{x_{i}}P_{x_{i}} the set of all generating elements from L_{i}.

2) Let G be a level δ ≥ 1 Galois formation, and L_{(ℓ)} an abstract field formation defining G. Let Φ_{i} : G → G_{i} be a rational projection of G, and φ̂_{i} : L̂_{i} → L̂_{G} the corresponding Kummer homomorphism. Then by induction on δ, one shows that there exists a unique (properly chosen) abstract field formation L_{i,(ℓ)} defining G_{i} such that φ̂_{i} (L_{i,(ℓ)}) ⊆ L_{(ℓ)}. Further one has:

a) Let L_{i,(ℓ)} be the unique abstract field formation defining G_{i} such that φ̂ (L_{i,(ℓ)}) ⊂ L_{(ℓ)}. Then φ̂ (L_{i}) ⊂ L_{(ℓ)} for any field formation L_{i} which is contained in L_{i,(ℓ)}.

b) Further, by Remark 5.2, 1), and 2), φ̂_{i} (L̂_{i}) ∩ Û_{G} = φ̂_{i} (κ̂_{i}) = κ̂_{G}, and Û_{G} · φ̂_{i} (L̂_{i}) is pure in L̂_{G}. Thus φ̂_{i} (L_{i,(ℓ)}) ∩ Û_{G} = κ̂_{G}, and further, Û_{G} · φ̂ (L_{i,(ℓ)}) is pure in L_{(ℓ)}.

Definition 5.6. In notations as above, let the level δ Galois formation G be endowed with a family of rational quotients Φ_{i} : G → G_{i}. Let φ̂_{i} : L̂_{i} → L̂_{G} be the corresponding Kummer homomorphism (all i), and let L_{i,(ℓ)} be the unique abstract field formation defining G_{i} such that φ̂_{i} (L_{i,(ℓ)}) ⊆ L_{(ℓ)}.

1) For every v, let I_{v} be the set of all i_{v} such that Φ_{i_{v}}(T_{v}) = {1}, if δ = 2; respectively such that the resulting rational projection Φ_{i_{v}} : G_{v} → G_{i_{v}} is a rational quotient, if δ > 2. We will say that (Φ_{i})_{i} is an ample family of rational quotients of G if the following hold:

i) Û_{G} together with the φ̂_{i} (L_{i,(ℓ)}) (all i) generate L_{(ℓ)}.

ii) If δ > 2, then —by induction on the level— Φ_{i_{v}} (all i_{v} ∈ I_{v}) is an ample family of rational quotients of G_{v}.

2) In the above context, suppose that (Φ_{i})_{i} is an ample family of rational quotients of G. Let a field formation L_{i} ⊂ L_{i,(ℓ)} defining G_{i} be given (all i). Let L be the ℤ-submodule of L_{(ℓ)} generated by the φ̂_{i} (L_{i}) (all i). We will say that G endowed with Φ_{i} (all i) is a Galois formation, respectively that L endowed with the L_{i} (all i) is a field formation defining G if one has:

i) L ∩ Û_{G} is trivial, and φ̂_{i} (L_{i}) is pure in L (all i).

ii) L · Û_{G} is a true Û_{G}-lattice in L_{(ℓ)}.

Definition/Remark 5.7. Let {G, (Φ)_{i}} and {H, (Φ_{j})_{j}} be Galois formations. Further let Φ : G → H be a level δ morphism of abstract Galois formations, and φ̂ : L̂_{H} → L̂_{G} the Kummer morphism. Then by definition, there exit abstract field formations L_{G,(ℓ)} ⊂ L̂_{G} and L_{H,(ℓ)} ⊂ L̂_{H} such that φ̂(L_{H,(ℓ)}) ⊂ L_{G,(ℓ)}.

Then Φ is called a morphism of Galois formations if for every j there exists an i and an isomorphism Φ_{ij} : G_{i} → G_{j} such that Φ_{j} ∘ Φ = Φ_{ij} ∘ Φ_{i}.

In the above notations, φ̂_{i}, and φ̂_{j} denote the Kummer morphisms defined by Φ_{i}, and Φ_{j}, respectively. Let L_{G} endowed with (L_{i})_{i}, and L_{H} endowed with (L_{j})_{j} be field formations defining G, respectively H. Let i and j be such that Φ_{j} ∘ Φ = Φ_{ij} ∘ Φ_{i}. Then one has:

1) Let L_{j,(ℓ)} be the unique abstract field formation defining H_{j} such that φ̂_{j}(L_{j,(ℓ)}) ⊆ L_{H,(ℓ)}. Then L_{j,(ℓ)} is also the unique abstract field formation defining H_{j} such that φ̂ ∘ φ̂_{j} (L_{j,(ℓ)}) ⊆ L_{G,(ℓ)}.

2) Moreover, after multiplication by a rational ℓ-adic unit ϵ ∈ ℤ_{(ℓ)}^{×}, we have φ̂ ∘ φ̂_{j}(L_{j}) = φ̂_{i} (L_{i}). Nevertheless, in order to have a “good” theory, we cannot assume that this equality holds simultaneously for all j and i as above. This is a fine point of the whole story, as we will see bellow...

C) Example: Concrete morphisms of concrete Galois formations

We consider notations as in subsection D) of the the previous Section. Let 𝔨 and 𝔩 be perfect base fields of characteristic ≠ ℓ. Let ı : 𝔏|𝔩 ↪ 𝔎|𝔨 be an embedding of function fields such that the relative algebraic closure of 𝔩 in 𝔨 is finite. Further let 𝔎′|𝔎 and 𝔏′|𝔏 be field extensions as at the beginning of subsection D) of Section 4), and consider concrete pre-Galois formations G_{X} and G_{Y} as introduced/defined in Proposition/Definition 4.13. Suppose that ı has a prolongation to an embedding of fields

ı′ : 𝔏′ ↪ 𝔎′

Thus in particular, \overline{𝔩}|𝔩 ↪ \overline{𝔨}|𝔨 via ı′. Therefore, ı gives rise to a canonical projection of Galois groups

Φ_{0} : G_{𝔨} → G_{𝔩} , Φ : G′_{𝔎} → G′_{𝔏}

which are compatible with the projections G′_{𝔎} → G_{𝔨} and G′_{𝔏} → G_{𝔩}, and with the cyclotomic characters. Moreover, by the hypothesis on ı, the homomorphism Φ is open. Thus Φ defines a level δ = 0 morphism of pre-Galois formations G_{X} → G_{Y}. Moreover, the Kummer homomorphism

φ̂ : 𝔏̂ = H^{1}(G′_{𝔏} , ℤ_{ℓ}(1)) → H^{1}(G′_{𝔎} , ℤ_{ℓ}(1)) = 𝔎̂

defined by Φ is nothing but the ℓ-adic completion of ı.

Consider concrete Galois formations G_{X} and G_{Y} based on 𝔎|𝔨, respectively 𝔏|𝔩, of levels td(𝔎|𝔨), respectively td(𝔏|𝔩). Let 0 ≤ δ ≤ td(𝔏|𝔨) be given. Thus G_{X} and G_{Y} in a canonical way level δ (concrete) pre-Galois formations. We claim that Φ gives rise canonically to a level δ morphism of abstract pre-Galois formations Φ : G_{X} → G_{Y}.

Indeed, the 𝔨|𝔩-embedding of function fields ı : 𝔏 → 𝔎 is defined by some rational dominant map f : X ---^{>} Y which maps 𝔩 into 𝔨. We have: f is defined at almost all points x_{1} of co-dimension 1 of X; its value f(x_{1}) at almost all such points is a point of co-dimension 1 in Y; and almost all points y_{1} of codimension 1 of Y are images of such points via f. Moreover, for any such y_{1} there exist only finitely many pre-images x_{1} of co-dimension 1. In birational terms this means the following: For every Zariski prime divisor 𝔳 of 𝔎|𝔨, let 𝔴 := 𝔳|_{𝔏} be its restriction to 𝔏. Thus 𝔴 is either trivial, or it is a Zariski prime divisor of 𝔏|𝔨. The geometric arguments explained above show the following: There are only finitely many 𝔳 ∈ 𝔇_{X} such that the corresponding 𝔴 is nontrivial \underline{and} does not lie in 𝔇_{Y}. Second, almost all 𝔴 ∈ 𝔇_{Y} are restrictions of some 𝔳 ∈ 𝔇_{X}.

Using general valuation theory, this translates into the following facts concerning G_{X} and G_{Y}: Let v be a prolongation of 𝔳 ∈ 𝔇_{X} to 𝔎′. Define T_{v} ⊂ Z_{v} correspondingly. Then one has: Let w = v|_{𝔏}′ be the restriction of v to 𝔏′. (N.B., w might be trivial!) Then Φ(Z_{v}) ⊂ Z_{w}, Φ(T_{v}) ⊂ T_{w} are open subgroups. In particular, the residual homomorphism

Φ_{v} : G_{v} = G′_{𝔎v} → G′_{𝔏w} = G_{w}

has an open image. Further, this homomorphism is defined by the canonical embedding of function field extensions over 𝔨|𝔩 coming from valuation theory:

𝔏′w | 𝔏w ↪ 𝔎′v | 𝔎v .

Hence, proceeding by induction we obtain the following:

Proposition/Definition 5.8.

(1 ) In the above notations, ı : 𝔏′ ↪ 𝔎′ gives rise in a canonical way to a level δ morphism Φ : G_{X} → G_{Y} of pre-Galois formations, for all δ such that 0 ≤ δ ≤ td(𝔏|𝔨). The Kummer homomorphism φ̂ : 𝔏̂ → 𝔎̂ is the completion of the embedding ı : 𝔏 ↪ 𝔎, thus it maps 𝔏_{(ℓ)} isomorphically into 𝔎_{(ℓ)}.

If concrete pre-Galois formations G_{X} and G_{Y} as above are given, a morphism of pre-Galois formations Φ : G_{X} → G_{Y} is called a concrete morphism, if the Kummer homomorphism

φ̂ : 𝔏̂ = H^{1}(G′_{𝔏} , ℤ_{ℓ}(1)) → H^{1}(G′_{𝔎} , ℤ_{ℓ}(1)) = 𝔎̂

defined by Φ is —up to multiplication by an ℓ-adic unit— the ℓ-adic completion of some embedding of function fields ı : 𝔏 | 𝔩 ↪ 𝔎 | 𝔨.

(2 ) For a given concrete morphism Φ : G_{X} → G_{Y} as above, and its Kummer homomorphism φ̂, the embedding of function fields ı : 𝔏 | 𝔩 ↪ 𝔎 | 𝔨 defining φ̂ is unique.

Proof. The first assertion (1) follows immediately from the discussion preceding the Proposition. Let us prove (2). Let ı′, ı′′ be both embeddings as above defining φ̂. For every y ∈ 𝔏^{×}, set y′ = ı′(y), and y′ = ı′(y). Since both ı′, ı′′ define φ̂, it follows that ȷ_{𝔎} (y′) = ȷ_{𝔎} (y′′). Since ker(ȷ_{𝔎}) is contained in 𝔨^{×}, it follows that there exists a_{y} ∈ 𝔨^{×} such that y′′ = a_{y}y′. Now consider some y ∈ 𝔏 which is transcendental over 𝔩, thus its image y′ is transcendental over 𝔨. Then we have:

a_{y}y′ + 1 = y′′ + 1 = ı′′(y + 1) = a_{y+1}ı′(y + 1) = a_{y+1}(y′ + 1)

or equivalently, (a_{y} − a_{y+1})y′ = a_{y+1} − 1. By the hypothesis on y we deduce: a_{y} = 1 = a_{y+1}. Therefore, y′ = y′′, and so ı(y′) = ı′′(y).

Finally, since the set of all transcendental elements of 𝔏|𝔩 generates 𝔏, it follows that ı′ = ı′′.

We now turn our attention to rational projections of a concrete Galois formation G_{X} as above. Let 𝔵 ∈ 𝔎 be an arbitrary non-constant function, and let 𝔎_{𝔵} be the relative algebraic closure of 𝔨(𝔵) in 𝔎. Then 𝔎_{𝔵}|𝔨 is a function field in one variable. We endow 𝔎_{𝔵}|𝔨 with its unique projective normal model X_{𝔵} → 𝔨, and consider the pre-Galois formation G_{X_{𝔵}} defined by the algebraic extension 𝔎′_{𝔵}|𝔎_{𝔵}, where 𝔎′_{𝔵} is the relative algebraic closure of 𝔎_{𝔵} in 𝔎′.

By the facts explained above, it follows that the canonical inclusion of field extensions ı_{𝔵} : 𝔎′_{𝔵}|𝔎_{𝔵} ↪ 𝔎′|𝔎 gives rise to a morphism Φ_{𝔵} : G_{X} → G_{X_{𝔵}}. Clearly, Φ_{𝔵} : G′_{𝔎} → G′_{𝔎_{𝔵}} is surjective (by construction), but though, Φ viewed as morphism of concrete Galois formations G_{X} → G_{X_{𝔵}} need not necessarily define G_{X_{𝔵}} as a quotient of G_{X}. Nevertheless, since X is normal, the rational map f_{𝔵} : X ---^{>} X_{𝔵} defining ı is defined at all points of co-dimension 1. This means that for every 𝔳 ∈ 𝔇_{X}, if 𝔳 is not trivial on 𝔎_{𝔵}, then f_{𝔵}(𝔳) is a closed point 𝔴 ∈ 𝔇_{X_{𝔵}}. In particular, Φ defines G_{X_{𝔵}} as a quotient of G_{X} if and only if f_{i} is surjective. This is the case in particular, if X is itself complete (thus projective).

Convention: In order to simplify the discussion, let us assume from now on that 𝔨 is algebraically closed, and that X → 𝔨 as well as all the residual varieties X_{v} → 𝔨 defining the residual pre-Galois formations G_{v} of X are projective (and normal).

In particular, since 𝔨 is algebraically closed, 𝔨^{×} is ℓ-divisible. Thus the completion homomorphism ȷ_{𝔎} : 𝔎^{×} → 𝔎̂ has ker(ȷ_{𝔎}) = 𝔨^{×}, and ȷ_{𝔎} (𝔎^{×}) is isomorphic to the true lattice of all principal divisors H(X) inside the free Abelian group Div(X).

Second, with ı_{𝔵} : 𝔎_{𝔵} ↪ 𝔎 as defined above, Φ_{𝔵} : G_{X} → G_{𝔵} defines G_{𝔵} as a 1-dimensional projection of G (all 𝔵).

Remark 5.9. We say that a function 𝔵 ∈ 𝔎 is general, if Φ_{𝔵} : G_{X} → G_{𝔵} is a rational quotient of G_{X}. Then following hold:

1) First, a “general” function 𝔵 —in the sense of Bertini— is also general in the sense above. This follows by the fact that for such functions 𝔵, the corresponding 𝔎_{𝔵} is the rational function field in the variable 𝔵, and moreover, the resulting Φ_{𝔵} : G → G_{𝔵} is a rational quotient of G. Indeed, this nothing but the following well known fact: Consider the rational map f_{𝔵} : X ---^{>} X_{𝔵}, and further let f_{𝔵,\overline{η}} : X\overline{η} → \overline{𝔎}_{𝔵} be the geometric generic fiber of f_{𝔵}. Then under the canonical projection X\overline{η} → X the geometric fundamental group of X is a quotient of the fundamental group of X\overline{η}.

We denote L_{𝔵} = ȷ_{𝔵}(𝔎_{𝔵}) the image of 𝔎_{𝔵} in 𝔎̂_{𝔵} via the completion morphism ȷ_{𝔵}. We remark that since 𝔨 is algebraically closed, the set of 𝔵-generating elements is P_{𝔵} = { 𝔵 − a | a ∈ 𝔨 } (all general 𝔵).

Now suppose that td(𝔎|𝔨) > 1. Then one has:

2) The Kummer homomorphism φ̂_{𝔵} : 𝔎̂_{𝔵} → 𝔎̂ maps each L_{𝔵} isomorphically into L := ȷ_{𝔎} (𝔎^{×}). Further, it is well known that Û_{G_{X}} together with the L_{𝔵} (all general 𝔵) generate L_{X} = Û_{G_{X}} · L. Then an easy induction argument shows that Φ_{𝔵} (all general 𝔵) is an ample family of rational quotients of G_{X}.

3) Moreover, a “Bertini type argument” shows that L_{X} endowed with the L_{𝔵} (all general 𝔵) is actually a field formation defining G_{X}.

Therefore we have the following:

Proposition/Definition 5.10. In the above context —including the Convention—, we consider G_{X} as a Galois formation of level δ = td(𝔎|𝔨), and endow it with its canonical true Û_{G_{X}} -lattice L_{X} := Û_{G_{X}} ·L. For Φ_{𝔵} : G_{X} → G_{X_{𝔵}} (all general 𝔵), let L_{𝔵} ⊂ 𝔎̂_{𝔵} be the image of 𝔎×_{𝔵} under the completion functor. Then we have:

(1 ) G_{X} endowed with Φ_{𝔵} (all general 𝔵) is a Galois formation, which we call a concrete Galois formation.

(2 ) Further, L_{X} endowed with the family L_{𝔵} (all general 𝔵) is a field formation defining G_{X}.

A field formation of the type above will be called a level δ = td(𝔎|𝔨) concrete field formation.

We now can announce one of the main results concerning the Galois formations:

Theorem 5.11. Let {G, (Φ_{i})_{i}} be a level δ Galois formation, and in the usual notations, L endowed with (L_{i})_{i} is a field formation defining G. Let further {G_{X}, (Φ_{𝔵})_{𝔵}} be a concrete Galois formation, and L_{X} endowed with (L_{𝔵})_{𝔵} be a field formation defining G_{X}. Then one has:

(1 ) Up to conjugation by concrete automorphisms of {G_{X}, (Φ_{𝔵})_{𝔵}}, there exists at most one isomorphism of Galois formations Φ : {G, (Φ_{𝔵})_{𝔵}} → {G, (Φ_{i})_{i}}.

(2 ) Suppose that {G, (Φ_{i})_{i}} is a concrete pre-Galois formation as well, say of the form {G_{Y} , (Φ_{𝔶})_{𝔶}}, with Y → 𝔩 a projective normal model of some function field 𝔏|𝔩 with 𝔩 algebraically closed. Then every isomorphisms Φ as above is a concrete one. Moreover, the function field embedding ı : 𝔏|𝔩 ↪ 𝔎|𝔨 defining Φ maps 𝔩 isomorphically onto 𝔨, and 𝔎 | ı(𝔏) is purely inseparable.

Proof. Since (1) follows from (2), it suffices to prove assertion (2). Thus let Φ : {G, (Φ_{𝔵})_{𝔵}} → {G_{Y} , (Φ_{𝔶})_{𝔶}} be an isomorphism of (concrete) Galois formations.

For the beginning, let us view Φ as an isomorphism of abstract Galois formations. Then, first, the Kummer morphism φ̂ : L̂_{G_{Y}} → L̂_{G_{X}} is actually an isomorphism. Second, if L′_{X,(ℓ)} and L′_{Y,(ℓ)} are properly chosen abstract field formations defining G_{X}, respectively G_{Y}, we have: φ̂ maps L′_{Y,(ℓ)} isomorphically onto L′_{X,(ℓ)}. Since L′_{X,(ℓ)}, L′_{Y,(ℓ)} are ℓ-adically equivalent to L_{X,(ℓ)}, respectively L_{Y,(ℓ)}, we get: After multiplication of φ̂ by a properly chosen ℓ-adic unit, we can suppose that φ̂ maps L_{Y,(ℓ)} isomorphically onto L_{X,(ℓ)}. Second, it follows by Remark, 5.7, 2), that φ̂ maps Û_{G_{Y}} isomorphically onto Û_{G_{X}}.

In orde to simplify notations, let us identify 𝔎̂_{𝔵} and 𝔏̂_{𝔶} with their images via φ̂_{𝔵}, respectively φ̂_{𝔶}. In this notation, via the isomorphism Φ, one gets a bijection L_{𝔶} ↦ L_{𝔵}, defined by the fact that 𝔎̂_{𝔵} = φ̂(𝔏̂_{𝔶}).^{4}

• φ̂ maps 𝔏_{(ℓ)} isomorphically onto 𝔎_{(ℓ)}.

Indeed, we have L_{𝔵,(ℓ)} = 𝔎̂_{𝔵} ∩ 𝔎_{(ℓ)}, and correspondingly for L_{𝔶,(ℓ)}. Since Φ is an isomorphism, we have φ̂(L_{𝔶,(ℓ)}) = L_{𝔵,(ℓ)}, provided φ̂ (𝔏̂_{𝔶}) = 𝔎̂_{𝔵}. Since the set of all general 𝔵 multiplicatively generates 𝔎^{×}, it follows that the set of all the elements in all the L_{𝔵,(ℓ)} multiplicatively generates 𝔎_{(ℓ)}. Since the same holds correspondingly for 𝔏, we are done.

Next recall that L_{𝔎} := ȷ_{𝔎} (𝔎^{×}) and L_{𝔏} := ȷ_{𝔏} (𝔏^{×}) are true latices in 𝔎_{(ℓ)}, respectively 𝔏_{(ℓ)}. Since L_{𝔎} is a true lattice in the ℤ_{(ℓ)}-module 𝔎_{(ℓ)}, it follows that every x′ ∈ 𝔎_{(ℓ)} is of the form x′ = (m/n)x with m/n a positive rational number which is also an ℓ-adic unit, and x ∈ L_{𝔎}. For such an x′ and the corresponding x we define their ℓ-adic value as being the maximum over all non-negative integers k such that ℓ^{−k′}x′ ∈ 𝔎_{(ℓ)}, respectively ℓ^{−k}x ∈ L_{𝔎}. It is clear that x and x′ have the same ℓ-adic value. We will say that x (and x′) is primitive, if its ℓ-adic value is 0, or equivalently, if ℤ x is pure in L_{𝔎}.

4 N.B., this does not mean yet mean that we have a bijection 𝔶 ↦ 𝔵.

Now consider some y ∈ L_{𝔏}. Then by discussion above, and taking into account that φ̂ maps 𝔏_{(ℓ)} isomorphically onto 𝔎_{(ℓ)}, we have: There is a unique positive rational number which is also an ℓ-adic unit ϵ_{y} = m_{y}/n_{y}, and a unique element x ∈ L_{𝔎} such that φ̂ (y) = (m_{y}/n_{y})x. Moreover y is primitive iff x is primitive. Thus φ̂ gives rise to a canonical bijection

φ̃ : L_{𝔏} → L_{𝔎} , y ↦ x := ϵ_{y}^{−1}ϕ(y) .

mapping the primitive elements of L_{𝔏} bijectively onto those of L_{𝔎}, and respecting the ℓ-adic value of the elements. We say that φ̂ is y-normed, if ϵ_{y} = 1. And if y = ȷ_{𝔏} (𝔶), we will also say that φ̂ is 𝔶-normed.

Now suppose that y = ȷ_{𝔏} (𝔶) with 𝔶 a general element. Then for x ∈ L_{𝔎} with φ̂ (y) = (m_{y}/n_{y})x, choose some 𝔵 ∈ 𝔎 such that ȷ_{𝔎} (𝔵) = x. Then we see that φ̂ maps 𝔎_{𝔶,(ℓ)} into 𝔎_{𝔵,(ℓ)}. Thus 𝔵 is a general element of 𝔎, and we have φ̂ (L_{𝔶,(ℓ)}) = L_{𝔵,(ℓ)}. Therefore, φ̂ (L_{𝔶}) = ϵ_{𝔶} · L_{𝔵}, and φ̃(L_{𝔶}) = L_{𝔵}. Thus we have:

Fact 5.12. In the above context, the following assertions are equivalent:

(i) φ̂ is 𝔶-normed.

(ii) φ̂ is y′-normed for some y′ ∈ L_{𝔶}.

(iii) φ̂ (L_{𝔶}) = L_{𝔵}

Naturally, a priori, it might be possible that φ̂ is not normed with respect to any element y ∈ L_{𝔏}. To “artificially” remedy this, we fix some 𝔶, and replace φ̂ by its (1/ϵ_{𝔶})-multiple, which we again denote by φ̂. Precisely, without loss of generality, we will suppose from the beginning that:

• φ̂ is 𝔶-normed for some fixed 𝔶.

As we will see below, the above apparently “artificial” way to remedy the lack of normation of φ̂ is actually not at all artificial, but it has a deep arithmetic meaning.

First let us denote M_{𝔎} := φ̂ (L_{𝔏}) ∩ L_{𝔎}, M_{𝔏} := φ̂^{−1}(L_{𝔎}) ∩ L_{𝔏}. Thus M_{𝔎} = φ̂ (M_{𝔏}), and for u ∈ 𝔏^{×} and t ∈ 𝔎^{×} such that φ̂ ∘ ȷ_{𝔏} (u) = ȷ_{𝔎} (t) we have: ȷ_{𝔏} (u) ∈ M_{𝔏} if and only if ȷ_{𝔎} (t) ∈ M_{𝔎}. If u and t are as above, we will say that u is a pre-image of t via (the 𝔶-normed morphism) φ̂.

Lemma 1. Consider any t ∈ 𝔎 and u ∈ 𝔏 as above. Then we have the following: ȷ_{𝔎} (𝔨 t + 1) ⊆ M_{𝔎} and ȷ_{𝔏} (𝔨 u + 1) ∈ M_{𝔏}.

Proof of Lemma 1. By symmetry, it is sufficient to prove the assertion for t. In the notaions form Fact above we have: Since φ̂ is 𝔶-normed, it follows by loc.cit. above that L_{𝔶} ⊂ M_{𝔏} and L_{𝔵} ⊂ M_{𝔎}. Then we have: If t and 𝔵 are algebraically 𝔨-dependent, then 𝔎_{t} ⊆ 𝔎_{𝔵}, as 𝔎_{𝔵} is relatively algebraically closed in 𝔎. As φ̂ is 𝔶-normed, the assertion of Lemma 1 follows from the Fact above. Therefore, we may/will assume that t and 𝔵 are algebraically independent over 𝔨. (Thus by symmetry, u and 𝔶 are also algebraically independent over 𝔨.) For almost all a ∈ 𝔨, and almost all b = b_{a} ∈ 𝔨^{×}, by the “birational Bertini”we have: both t′_{a,b} := t/(b 𝔵 + a) and t′′_{a,b} := t/(b 𝔵 + a + 1) are general elements of 𝔎. In particular, since both ȷ_{𝔎} (t) and ȷ_{𝔎} (b 𝔵+a) are elements of M_{𝔎}, it follows that both ȷ_{𝔎} (t′_{a,b}) and ȷ_{𝔎} (t′′_{a,b}) are in M_{𝔎}. Since they are general elements, it follows that L_{t′_{a,b}} and L_{t′′_{a,b}} are contained in M_{𝔎}.

In the same way, one can see that for a fixed c ∈ 𝔨^{×}, and almost all a, b ∈ 𝔨 as above, the function t_{a,b,c} = (ct + b𝔵 + a + 1)/(t + b𝔵 + a) is a general element of 𝔎 too. On the other hand, we have:

t_{a,b,c} = \frac{b 𝔵 + a + 1}{b 𝔵 + a} \frac{c t′′_{a,b} + 1}{t′_{a,b} + 1} .

From this we finally deduce: The images via ȷ_{𝔎} of both the denominators as well as of the numerators of the fractions above are elements of M_{𝔎}. Thus finally, ȷ_{𝔎} (t_{a,b,c}) ∈ M_{𝔎}. Therefore, reasoning as above we have: L_{t_{a,b,c}} is contained in M_{𝔎}. In particular, ȷ_{𝔎} (t_{a,b,c} − 1) ∈ M. On the other hand, t_{a,b,c} − 1 = [(c − 1)t + 1]/(t + b𝔵 + a). Since ȷ_{𝔎} (t + b𝔵 + a) ∈ M_{𝔎}, we deduce that ȷ_{𝔎} ((c − 1)t + 1) ∈ M_{𝔎} for all c ∈ 𝔨^{×}. Equivalently, since t ∈ M_{𝔎} already, we get: ȷ_{𝔎} (𝔨 t + 1) ⊆ M_{𝔎}, as claimed.

Lemma 2. The 𝔶-normed morphism φ̂ respects co-lineations, i.e., let u_{i} ∈ 𝔏^{×} and pre-images t_{i} ∈ 𝔎^{×} be given (i = 1, 2, 3). Then (u_{i})_{i} are linearly 𝔩-(in)dependent if and only if (t_{i})_{i} are linearly 𝔨-(in)dependent.

Proof of Lemma 2. See e.g. Bogomolov [Bo], second Remark, p.51.

Now let 𝔎_{0} = ȷ−1_{𝔎}(M_{𝔎}) ∪ {0} be the pre-image of M_{𝔎} in 𝔎 together with 0 added. We claim that 𝔎_{0} ⊂ 𝔎 is a subfield. Indeed, since M_{𝔎} is a subgroup of L_{𝔎}, its pre-image ȷ−1_{𝔎}(M_{𝔎}) in 𝔎^{×} is a subgroup too. We check that 𝔎_{0} is closed with respect to addition. Indeed, for t, t′ ∈ 𝔎_{0} we have: t′′ = t′/t ∈ 𝔎_{0}, and further, t + t′ = t(t′′ + 1). As by Lemma 1 we have t′′ + 1 ∈ 𝔎_{0}, we finally get t + t′ = t(t′′ + 1) ∈ 𝔎_{0}. Moreover, we remark that M_{𝔎} = P(𝔎_{0}) is by definition exactly the projectivization of (𝔎_{0}, +).

Consider/define correspondingly 𝔏_{0} = ȷ−1_{𝔏}(M_{𝔏}) ∪ {0} inside (𝔏, +). Then by the definition of 𝔎_{0} and 𝔏_{0}, the Kummer isomorphism φ̂ : 𝔏̂ → 𝔎̂ defines a bijective mapping

φ := φ̂|_{P(𝔏)} : P(𝔏) = M_{𝔏} → M_{𝔎} = P(𝔎) ,

which, by Lemma 2, respects co-lineations.Therefore, by the Fundamental theorem of projective geometries, see e.g. Artin, φ is the projectivisation φ = P(ϕ) of some linear ı_{0}-isomorphism ϕ : (𝔏_{0}, +) → (𝔎_{0}, +), i.e., ı_{0} : 𝔩 → 𝔨 is an isomorphism of fields, and ϕ is a isomorphism of Abelian groups, such that ϕ(au) = ı_{0}(a)ϕ(u) for all a ∈ 𝔩 and u ∈ 𝔏_{0}. Moreover, ϕ is unique up to composition by homotheties of the form l_{a} ∘ ϕ ∘ l_{b} (all a ∈ 𝔨, b ∈ 𝔩).

Further, as 𝔨^{×} = ker(ȷ_{𝔎}) and 𝔩^{×} = ker(ȷ_{𝔏}), it follows that ϕ(𝔩) = 𝔨. We set ϕ_{0} := (1/ϕ(1))ϕ, and claim that ϕ_{0} is a field isomorphism. Indeed, for a fixed y ∈ 𝔏_{0}, consider ϕ_{y} : 𝔏_{0} → 𝔎_{0} defined by ϕ_{y}(u) = ϕ_{0}(y u). Then ϕ_{y} is a linear ı_{0}-isomorphism. Set x = ϕ_{0}(y). Then considering projectivisations, and using the fact that φ = P(ϕ_{0}) is multiplicative, it follows that for all u ∈ 𝔏_{0} we have: P(ϕ_{y})(u) = P(ϕ_{0})(yu) = P(l_{x}) ∘ P(ϕ_{0})(u), where l_{x} is the multiplication by x on 𝔎_{0}. Therefore, there exist a ∈ 𝔨^{×} and b ∈ 𝔩^{×} such that l_{b} ∘ϕ_{y} ∘l_{b} = l_{x} ∘ϕ_{0}. In other words, a ϕ_{0}(y b u) = x ϕ_{0}(u) for all u ∈ 𝔏_{0}. Setting u = 1, and taking into account that ϕ_{0}(1) = 1, we have: a ı_{0}(b) x = x. Thus a ı_{0}(b) = 1, the effects of l_{a} and l_{b} cancel each other. Hence we have

ϕ_{0}(y u) = x ϕ_{0}(u) = ϕ_{0}(y) ϕ_{0}(u), (all u, y ∈ 𝔏_{0}).

Therefore, ϕ_{0} is a field isomorphism as claimed.

In order to conclude, we have to show that both 𝔏|𝔏_{0} and 𝔎|𝔎_{0} are purely inseparable field extensions. By symmetry, it is sufficient to prove this for 𝔎|𝔎_{0}. Let p be the characteristic exponent of 𝔨. For every general element 𝔵 ∈ 𝔎, consider the smallest integer m = m_{𝔵} such that 𝔵^{m} ∈ 𝔎_{0}. We claim that m is a power of p. Indeed, by Fact 5.12, for all primitive elements x′_{0} ∈ 𝔎_{𝔵} one has: m is the smallest positive integer such that x′_{0}^{m} ∈ 𝔎_{0}. This means that 𝔎_{0,𝔵} := 𝔎_{𝔵} ∩ 𝔎_{0} is a subfield of 𝔎_{𝔵} such that 𝔎_{0,𝔵} is contained in the m^{th} powers of 𝔎_{𝔵}. But then m must be a power of the characteristic exponent of 𝔨. Equivalently, 𝔎_{𝔵} is purely inseparable over 𝔎_{0,𝔵}. Finally, since the set of all general elements 𝔵 generates the multiplicative group 𝔎^{×}, it follows that for every x ∈ 𝔎 there exists some p-power p^{e} such that x^{p^{e}} ∈ 𝔎_{0}.

The Theorem 5.11 is proved.

6. Concluding the Proof of the Theorem (Introduction)

We now consider the context from the Theorem (Introduction). Let k be the constant field of K. Recall that the canonical projection pr_{K} : G_{K}^{ℓ} → G_{k} as well as the Frobenius Frob_{k} of k are group theoretically encoded in G_{K}^{ℓ}. Further, the same is true correspondingly about L. On the other hand, we see that the “recipe” for deducing pr and Frob are actually invariant under isomorphisms, thus under Φ. Therefore we finally have: K and L have the same field of constants, say k, and Φ is a G_{k} isomorphism, i.e., Φ ∘ pr_{L} = pr_{K}.

Further, by Theorem 4.13 it follows that the ℤ_{(ℓ)}-lattice K_{(ℓ)} is encoded group theoretically in G_{K}^{ℓ}. Correspondingly, the same is true concerning L. Again, by the way K_{(ℓ)} is encoded in G_{K}^{ℓ}, one immediately sees that the Kummer homomorphism

φ̂_{K} : L̂ → K̂

maps L_{(ℓ)} isomorphically onto K_{(ℓ)}. Moreover, let K′|K be a finite (Galois) extension of K inside K^{ℓ}, and via the isomorphism Φ the corresponding finite (Galois) extension L′|L. Then using the functoriality of Kummer theory, we have a commutative diagram of the form

L̂ ⟶^{φ̂_{K}}  K̂
↓^{can}  ↓^{can}
L̂′ ⟶^{φ̂_{K′}} K̂′

which induces a corresponding one with ( )_{(ℓ)} in stead of ˆ( ). Suppose K′|K is Galois, and Φ′ : Gal(K′|K) → Gal(L′|L) is the isomorphism induced by Φ. Then for all σ ∈ Gal(K′|K), and τ = Φ(σ) we have:

σ ∘ φ̂_{K′} = φ̂_{K′} ∘ τ i.e., Φ(σ) = φ̂_{K′}^{−1} ∘ σ ∘ φ̂_{K′} .

Next let us take limits over the constant extensions of K, i.e., ones of the form K′ = Kk′, with k′|k a finite extension of the constant field k of K, we deduce that a similar result as Theorem 4.13 holds also for 𝔎 = K\overline{k}, respectively 𝔏 = L\overline{k}. Precisely, as Φ is a G_{k} isomorphism, it maps Gℓ_{𝔎} ⊂ G_{K}^{ℓ} isomorphically onto Gℓ_{𝔏} ⊂ G_{L}^{ℓ}. Moreover, the resulting Kummer isomorphism φ̂_{𝔎} : 𝔏̂ → 𝔎̂ maps 𝔏_{(ℓ)} isomorphically onto 𝔎_{(ℓ)}. We should also remark here that φ̂_{𝔎} is a G_{k}-morphism, thus in particular, K̂ as well as K_{(ℓ)} are the G_{k}-invariants of 𝔎̂, respectively 𝔎_{(ℓ)}. The same correspondingly for L.

Proposition 6.1. In the above context, the 1-dimensional projections Φ_{x} : Gℓ_{𝔎} → Gℓ_{𝔎_{x}} of Gℓ_{𝔎} are group theoretically encoded in Gℓ_{𝔎}, in a way which is invariant under isomorphisms.

Moreover, if Φ_{𝔵} : Gℓ_{𝔎} → Gℓ_{𝔎_{𝔵}} is a rational projection defined by a general element 𝔵 of 𝔎, then the corresponding projection Φ_{𝔶} : Gℓ_{𝔏} → Gℓ_{𝔏_{𝔶}} is a rational projection, and 𝔶 is a general element of 𝔏.

Proof. The first assertion is Bogomolov’s Lemma 4.2 from [Bo]. On the other hand, it is clear that the criterion from loc.cit. is invariant under isomorphisms. To prove the last assertion, we remark that the fact Φ_{x} is rational, as well as the fact that Φ_{x} corresponds to a rational projection defined by a general element, are both assertions completely described by the inertia structure of Gℓ_{𝔎_{x}}. On the other hand, the inertia elements in Gℓ_{𝔎_{x}} are group theoretically encoded in the Galois theory by the Local theory.

Conclusion:

We finally apply Theorem 5.11, and deduce that there exist finite pure inseparable extensions 𝔎_{0}|𝔎 and 𝔏_{0}|𝔏 such that φ̂_{𝔎_{0}} : 𝔏̂_{0} → 𝔎̂_{0} is the ℓ-adic completion of a unique field isomorphism ı_{0} : 𝔎_{0} → 𝔏_{0}. Let K_{0} be the the purely inseparable closure of K in 𝔎_{0}, and define L_{0} correspondingly. Then considering G_{k}-invariants, it follows that ı_{0} maps L_{0} isomorphically onto N_{0}, and φ̂_{K_{0}} : L̂_{0} → K̂_{0} is exactly the ℓ-adic completion of ı_{K}. Now taking limits over all finite extensions K′|K_{0} inside K_{0}^{ℓ} = K^{ℓ}K_{0}, and the corresponding finite extensions L′|L_{0} inside L_{0}^{ℓ}, we finally get an isomorphism ϕ : L^{i}^{ℓ} → K^{i}^{ℓ} defining Φ, i.e., Φ(g) = ϕ^{−1} g ϕ for all g ∈ G_{K}^{ℓ}.

We still have to prove that any two automorphisms ϕ′ and ϕ′′ both defining Φ differ by a power of Frobenius. Indeed, setting ϕ := ϕ′′ ∘ ϕ′^{−1} we get: ϕ is an automorphism of L^{i}^{ℓ} which maps L^{i} onto itself, and induces the identity on G_{L^{i}}^{ℓ}. We claim that such an automorphism ϕ is a power of Frobenius. Indeed, let v be an arbitrary Zariski prime divisor of L^{i}^{ℓ}. Then w := v ∘ ϕ is also a Zariski prime divisor of L^{i}^{ℓ}. Moreover, by the usual formalism, we have Z_{w} = ϕ^{−1}Z_{v} ϕ inside G_{L^{i}}^{ℓ}. Since the conjugation by ϕ is the identity on G_{L^{i}}^{ℓ}, it follows that Z_{w} = Z_{v}. Thus by Proposition 1.6, (1), it follows that v and w are equivalent valuations on L^{i}^{ℓ}. In particular, v(x) > 0 if and only if w(x) > 0 (all x ∈ L^{i}^{ℓ}). Let char(k) = p. We claim that y := ϕ(x) is some p-power of x. First, if y and x are algebraically independent, then there exists a Zariski prime divisor v of L^{i}^{ℓ} such that v(x) = 1 and v(y) = 0. But v(y) = v ∘ ϕ(x) = w(x), contradiction! Therefore, y = ϕ(x) is always algebraic over \overline{𝔽}_{p}(x) inside L^{i}^{ℓ}. Now let f(Y) ∈ \overline{𝔽}_{p}(x)[Y] be the minimal polynomial of y over \overline{𝔽}_{p}(x) of y. We show that f(Y) is totally inseparable. Indeed, for a “general” constant a ∈ \overline{𝔽}_{p}, we replace x by x_{a} = x + a, thus y by y_{a} = y + ϕ(a). Then the minimal polynomial of y_{a} over \overline{𝔽}_{p}(x) is f_{a}(Y) := f(Y − a), and the following holds: The distinct roots y_{a} =: y_{1}, …, y_{r} of f_{a}(Y) have disjoint zero divisors in 𝔇1_{𝔎^{ℓ}}. On the other hand, all these zero divisors are zeros of x_{a}. Further, by the fact that all the zeros of x_{a} are as well zeros of y_{a} = ϕ(x_{a}), it follows that r = 1, i.e., f_{a}(Y) is purely inseparable. Thus f(Y) is purely inseparable too. Thus finally we have: ϕ(x) = x^{p^{e}}, where e = e_{x} depending on x. We next show that e = e_{x} does not depend on x. Indeed, since ϕ is a field automorphism, we have for all x, y ∈ L^{i}^{ℓ} the following:

x^{e_{x+y}} + y^{e_{x+y}} = (x + y)^{e_{x+y}} = ϕ(x + y) = ϕ(x) + ϕ(y) = x^{e} + y^{e_{y}} .

Considering an arbitrary y ∈ L^{i}^{ℓ} which is algebraically independent over \overline{𝔽}_{p}(x), we therefore must have e_{x+y} = e_{y} = e. Now choosing a transcendence basis T of L over 𝔽_{p} (say, which contains x), we see that the restriction of ϕ to \overline{𝔽}_{p}(T ) is Frob^{e}. Therefore, ϕ = Frob^{e} on L^{i}^{ℓ}.

The Theorem (Introduction) is proved.

Mathematisches Institut, Universität Bonn

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