PRO-ℓ GALOIS THEORY OF ZARISKI PRIME DIVISORS

by

Florian Pop

Abstract. — In this paper we show how to recover a special class of valuations (which generalize in a natural way the Zariski prime divisors) of function fields from the Galois theory of the functions fields in discussion. These valuations play a central role in the birational anabelian geometry and related questions.

Résumé (Théorie de Galois pro-ℓ des diviseurs premiers de Zariski)

Dans cet article nous montrons comment retrouver une classe spéciale de valuations de corps de fonctions (qui généralisent naturellement les diviseurs premiers de Zariski) à partir de la théorie de Galois des corps de fonctions en question. Ces valuations jouent un rôle central en géométrie anabélienne birationnelle et pour d’autres questions connexes.

1. Introduction

The aim of this paper is to give a first insight into the way the pro-ℓ Galois theory of function fields over algebraically closed base fields of characteristic ≠ ℓ encodes the Zariski prime divisors of the function fields in discussion. We consider the following context:

• ℓ is a fixed rational prime number.

• K|k are unction fields with k algebraically closed of characteristic ≠ ℓ.

• K(ℓ)|K is the maximal pro-ℓ Galois extension of K in some separable closure of K, and G_{K}(ℓ) denotes its Galois group.

2000 Mathematics Subject Classification. — Primary 12E, 12F, 12G, 12J; Secondary 12E30, 12F10, 12G99. Key words and phrases. — anabelian geometry, function fields, valuations, Zariski prime divisors, Hilbert decomposition theory, pro-ℓ Galois theory.

Supported by NSF grant DMS-0401056.

It is a Program initiated by Bogomolov [B] at the beginning of the 1990’s which has as ultimate goal to recover (the isomorphy type of) the field K from the Galois group G_{K}(ℓ). Actually, Bogomolov expects to recover the field K even from the Galois information encoded in PGal_{K}^{c}, which is the quotient of G_{K}(ℓ) by the second factor in its central series. Unfortunately, at the moment we have only a rough idea (maybe a hope) about how to recover the field K from G_{K}(ℓ), and \underline{not} a definitive answer to the problem. Nevertheless, this program is settled and has a positive answer, in the case k is an algebraic closure of a finite field, Pop [P4]; see also Bogomolov–Tschinkel [B–T2] for the case of function fields of smooth surfaces with trivial fundamental group and ℓ, char(k) ≠ 2.

It is important to remark that ideas of this type were first initiated by Neukirch, who asked whether the isomorphism type of a number field F is encoded in its absolute Galois group. The final result in this direction is the celebrated result by Neukirch, Iwasawa, Uchida (with previous partial results by Ikeda, Komatsu, etc.) which roughly speaking asserts that the isomorphy types of global fields are functorially encoded in their absolute Galois groups. Nevertheless, it turns out that the result above concerning global fields is just a first piece in a very broad picture, namely that of Grothendieck’s anabelian geometry, see Grothendieck [G1], [G2]. Grothendieck predicts in particular, that the finitely generated infinite fields are functorially encoded in their absolute Galois groups. This was finally proved by the author Pop [P2], [P3]; see also Spiess [Sp].

The strategy to prove the above fact is to first develop a “Local theory”, which amounts of recovering local type information about a finitely generated field from its absolute Galois group. And then “globalize” the local information in order to finally get the field structure. The local type information consists of recovering the Zariski prime divisors of the finitely generated field. These are the discrete valuations which are defined by the Weil prime divisors of the several normal models of the finitely generated field in discussion.

In this manuscript, we will mimic the Local theory from the case of finitely generated infinite fields, and will develop a geometric pro-ℓ Local theory, whose final aim is to recover the so called quasi-divisorial valuations of a function field K|k form G_{K}(ℓ) —notations as at the beginning of the Introduction. We remark that this kind of results played a key role in Pop [P4], where only the case k = \overline{𝔽}_{p} was considered.

We mention here briefly the notions introduced later and the main results proved later in the paper —notations as above.

Let v be some valuation of K(ℓ), and for subfields Λ of K(ℓ) denote by vΛ and Λv the value group, respectively residue field, of the restriction of v to Λ. And let T_{v} ⊆ Z_{v} be the inertia, respectively decomposition, group of v in G_{K}(ℓ) = Gal (K(ℓ)|K).

First recall, see Section 3, A), that a Zariski prime divisor v of K(ℓ) is any valuation of K(ℓ) whose restriction v|_{K} to K “comes from geometry”, i.e., the valuation ring of v|_{K} equals the local ring O_{X,x_{v}} of the generic point x_{v} of some Weil prime divisor of some normal model X → k of K|k. Thus vK ≅ ℤ and Kv|k is a function field satisfying td(Kv |k) = td(K|k) − 1. Now it turns out that Z_{v} has a “nice” structure as follows:

T_{v} ≅ ℤ_{ℓ} and Z_{v} ≅ T_{v} × G_{Kv}(ℓ) ≅ ℤ_{ℓ} × G_{Kv}(ℓ).

We will call the decomposition groups Z_{v} of Zariski prime divisors v of K(ℓ)|k divisorial subgroups of G_{K}(ℓ) or of K.

Now in the case k is an algebraic closure of a finite field, it turns out that a maximal subgroup of G_{K}(ℓ) which is isomorphic to a divisorial subgroup is actually indeed a divisorial subgroup of G_{K}(ℓ), see [P4]; this follows nevertheless from Proposition 4.1 of this manuscript, as k has no no-trivial valuations in this case.

On the other hand, if k has positive Kronecker dimension (i.e., it is not algebraic over a finite field), then the situation becomes more intricate, as the non-trivial valuations of k play into the game. Let us say that a valuation v of K(ℓ) is a quasidivisorial valuation, if it is minimal among the valuations of K(ℓ) having the properties: td(Kv |kv) = td(K|k) − 1 and vK/vk ≅ ℤ, see Definition 3.4, and Fact 5.5, 3). Note that the Zariski prime divisors of K(ℓ) are quasi-divisorial valuations of K(ℓ).

On the Galois theoretic side we make definitions as follows: Let Z be a closed subgroup of G_{K}(ℓ).

i) We say that Z a divisorial like subgroup of G_{K}(ℓ) or of K, if Z is isomorphic to a divisorial subgroup of some function field L|l such that td(L|l) = td(K|k), and l algebraically closed of characteristic ≠ ℓ.

ii) We will say that Z is quasi-divisorial, if Z is divisorial like and maximal among the divisorial like subgroups of G_{K}(ℓ).

Finally, for t ∈ K a non-constant function, let K_{t} be the relative algebraic closure of k(t) in K. Thus K_{t}|k is a function field in one variable, and one has a canonical projection p_{t} : G_{K}(ℓ) → G_{K_{t}}(ℓ).

In these notations, the main results of the present manuscript can be summarized as follows, see Proposition 4.1, Key Lemma 4.2, and Proposition 4.6.

Theorem 1.1. — Let K|k be a function field with td(K|k) > 1, where k is algebraically closed of characteristic ≠ ℓ. Then one has:

(1) A closed subgroup Z ⊂ G_{K}(ℓ) is quasi-divisorial ⟺ Z is maximal among the subgroups Z′ of G_{K}(ℓ) which have the properties:

i) Z′ contains closed subgroups isomorphic to ℤ_{ℓ}^{d}, where d = td(K|k).

ii) Z′ has a non-trivial pro-cyclic normal subgroup T′ such that Z′/T′ has no non-trivial Abelian normal subgroups.

(2) The quasi-divisorial subgroups of G_{K}(ℓ) are exactly the decomposition groups of the quasi-divisorial valuations of K(ℓ).

(3) A quasi-divisorial subgroup Z of G_{K}(ℓ) is a divisorial subgroup of G_{K}(ℓ) ⟺ p_{t}(Z) is open in G_{K_{t}}(ℓ) for some non-constant t ∈ K.

Among other things, one uses in the proof some ideas by Ware and Arason– Elman–Jacob, see e.g. Engler–Nogueira [E–N] for ℓ = 2, Engler–Koenigsmann [E–K] in the case ℓ ≠ 2, and/or Efrat [Ef] in general. And naturally, one could use here Bogomolov [B], Bogomolov–Tschinkel [B–T1]. We would also like to remark that this kind of assertions —and even stronger but more technical ones— might be obtained by employing the local theory developed by Bogomolov [B], and Bogomolov– Tschinkel [B–T1].

Concerning applications: Proposition 4.1 plays an essential role in tackling Bogomolov’s Program in the case the base field k is an algebraic closure of a global field (and hopefully, in general); and Proposition 4.6 is used in a proof of the so called Ihara/Oda–Matsumoto Conjecture. (These facts will be published soon).

Acknowledgments

I would like to thank the referee for the careful reading of the manuscript and the several suggestions which finally lead to the present form of the manuscript.

2. Basic facts from valuation theory

A) On the decomposition group (See e.g. [En], [BOU], [Z–S].) Consider the following context: K̃|K is some Galois field extension, and v is a valuation on K̃. For every subfield Λ of K̃ denote by vΛ and Λv the valued group, respectively the residue field of Λ with respect to (the restriction of) v on Λ. We denote by p = char(K̃v) 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(K̃|K), and K^{V}, K^{T}, and K^{Z} the corresponding fixed fields in K̃.

Fact 2.1. — The following are well known facts from Hilbert decomposition, and/or ramification theory for general valuations:

1) K̃v |Kv is a normal field extension. We set G_{v} := Aut(K̃v |Kv). Further, V_{v} ⊂ T_{v} are normal subgroups of Z_{v}, and 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 K̃v |Kv, thus it is the maximal Galois sub-extension of K̃v |Kv. Further, K^{V} |K^{T} is totally tamely ramified.

2) Let μ_{K̃v} denote the group of roots of unity in K̃v. There exists a canonical pairing as follows: ΨK̃ : T_{v} × vK̃/vK → μ_{K̃v}, (g, vx) ↦ (gx/x)v, and the following hold: The left kernel of ΨK̃ is exactly V_{v}. The right kernel of ΨK̃ is trivial if p = 0, respectively equals the Sylow p-group of vK̃/vK if p > 0. In particular, T_{v}/V_{v} is Abelian, V_{v} is trivial if char(K) = 0, respectively equals the unique Sylow p-group of T_{v} if char(K) = p > 0. Further, ΨK̃ is compatible with the action of G_{v}.

3) Suppose that v′ ≤ v is a coarsening of v, i.e., O_{v} ⊆ O_{v′}. Then denoting v_{0} = v/v′ the valuation induced by v on K̃v′, and by Z_{v_{0}} its decomposition group in G_{v′} = Aut(K̃v′|Kv), one has: T_{v} ⊆ Z_{v} are the preimages of T_{v_{0}} ⊆ Z_{v_{0}} in Z_{v′} via the canonical projection Z_{v′} → G_{v′}. In particular, T_{v′} ⊆ T_{v} and Z_{v} ⊆ Z_{v′}.

Fact 2.2. — Let ℓ be a rational prime number. In the notations and the context from Fact 2.1 above, suppose that K contains the ℓ^{∞} roots of unity, and fix once for all an identification of the Tate ℓ-module of 𝔾_{m,K} with ℤ_{ℓ}(1), say

ı : 𝕋_{ℓ} → ℤ_{ℓ}(1).

And let the Galois extensions K̃|K considered at Fact 2.1 satisfy K^{ℓ,ab} ⊆ K̃ ⊆ K(ℓ), where K^{ℓ,ab} is the maximal Abelian extension of K inside K(ℓ). Finally, we consider valuations v on K̃ such that Kv has characteristic ≠ ℓ. Then by the discussion above we have: V_{v} = {1}, and further: vK̃ is the ℓ-divisible hull of vK; and the residue field extension K̃v |Kv is separable and also satisfies the properties above we asked for K̃|K to satisfy.

1) For n = ℓ^{e}, there exists a unique sub-extension K_{n}|K^{T} of K̃|K^{T} such that K_{n}|K^{Z} is a Galois sub-extension of K̃|K^{Z}, and vK_{n} = \frac{1}{n} vK^{T} = \frac{1}{n} vK. On the other hand, the multiplication by n induces a canonical isomorphism \frac{1}{n} vK / vK ≅ vK/n. Therefore, the pairing ΨK̃ gives rise to a non-degenerate pairing

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

hence to isomorphisms θ^{v,n} : vK/n → Hom(T_{v}, μ_{n}), θ_{v,n} : T_{v}/n → Hom(vK, μ_{n}). In particular, taking limits over all n = ℓ^{e}, one obtains a canonical isomorphism of G_{v}-modules

θ_{v} : T_{v} → Hom(vK, ℤ_{ℓ}(1)).

2) 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 K̃ 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}] ⊂ K̃, it follows that v is totally ramified in K^{0}|K, and vK^{0} is ℓ-divisible. Therefore, K^{0}v = Kv, and the inertia group of v in K̃|K^{0} is trivial. In particular, T_{v} has complements in Z_{v}, and T_{v} ≅ ℤ_{ℓ}^{B}(1) as G_{v}-modules.

3) Since by hypothesis μ_{ℓ^{∞}} ⊆ K, the action of G_{v} on ℤ_{ℓ}^{B}(1) ≅ T_{v} is trivial. In particular, setting δ_{v} := |B| = dim_{𝔽_{ℓ}}(vK/ℓ) we finally have:

Z_{v} ≅ T_{v} × G_{v} ≅ ℤ_{ℓ}^{δ_{v}} × G_{v}.

B) Two results of F. K. Schmidt

In this subsection we will recall the pro-ℓ form of two important results of F. K. Schmidt and generalizations of these like the ones in Pop [P1], The local theory, A. See also Endler–Engler [E–E].

Let ℓ be a fixed rational prime number. We consider fields K of characteristic ≠ ℓ containing the ℓ^{∞} roots of unity. For such a field K we denote by K(ℓ) a maximal pro-ℓ Galois extension of K. Thus the Galois group of K(ℓ)|K is the maximal pro-ℓ quotient of the absolute Galois group G_{K} of K. We will say that K is ℓ-closed, if K(ℓ) = K, or equivalently, if every element x ∈ K is an ℓ^{th} power in K.

We next define the pro-ℓ core of a valuation on K(ℓ). This is the pro-ℓ correspondent of the core of a valuation on the separable closure of K, as defined in [P1], The local theory, A. The construction is as follows: In the above context, let v be a valuation of K(ℓ) which is pro-ℓ Henselian on some subfield Λ of K(ℓ). Consider the set V′ of all the coarsenings v′ of v such that Λv′ is pro-ℓ closed, and set V = V′ ∪ {v}. Note that V′ might be empty. By general valuation theory, V has an infimum whose valuation ring is the union of all the valuation rings O_{v′} (v′ ∈ V). We denote this valuation by

v_{pro-ℓ,Λ} = inf V

and call it the pro-ℓ Λ-core of v. Finally, for a given valuation v on K(ℓ) we denote by v_{pro-ℓ,K} the pro-ℓ K^{Z}-core of v, where K^{Z} is the decomposition field of v in K(ℓ). With this definition of a “core”, the Proposition 1.2 and Proposition 1.3 from [P1] remain true in the following form:

Proposition 2.3. — Let v be a non-trivial valuation of K(ℓ), and suppose that K^{Z} is not pro-ℓ closed. Then the pro-ℓ K^{Z}-core v_{pro-ℓ,K} of v is non-trivial and lies in V. Consequently:

(1) Kv is pro-ℓ closed ⟺ K^{Z}v_{pro-ℓ,K} is pro-ℓ closed ⟺ Z_{v} = T_{v}.

(2) If v has rank one or Kv is not pro-ℓ closed, then v equals its pro-ℓ K^{Z}-core.

Proof. — The proof is word by word identical with the one from loc.cit.

The following result is the announced pro-ℓ form of the results of F. K. Schmidt.

Proposition 2.4. — Suppose that K is not pro-ℓ closed. Let w^{1}, w^{2} be two valuations on K(ℓ) such that they are pro-ℓ Henselian on some sub-extension Λ|K of K(ℓ)|K. Then their pro-ℓ Λ-cores are comparable.

Consequently, let K ⊆ L ⊆ Λ ⊆ K(ℓ) be sub-extensions of K(ℓ)|K, and let v be a valuation on K(ℓ) that is pro-ℓ Henselian on Λ and equals its pro-ℓ Λ-core. Then the following hold:

(1) If Λ|L is normal, then v is pro-ℓ Henselian on L and equals its pro-ℓ L-core. In particular, Z_{v} is self-normalizing in G_{K}(ℓ).

(2) If Λ|L is finite, then v is pro-ℓ Henselian on L and v = v_{pro-ℓ,L}. In particular, Z_{v} is not a proper open subgroup of a closed subgroup of G_{K}(ℓ).

Proof. — The proofs of these assertions are identical to the ones in [P1], loc.cit., thus we will omit them here. For the fact that Z_{v} is self-normalizing, let N be its normalizer in G_{K}(ℓ), and set Λ = K^{Z} and L = K(ℓ)^{N}. Then Λ|L is normal, so v is pro-ℓ Henselian on L by the first part of (1) above. Thus finally Λ = K^{Z} = L, i.e., N = Z_{v}.

3. Zariski prime divisors and quasi-divisorial valuations

In the sequel we consider function fields K|k over algebraically closed base fields k of characteristic ≠ ℓ.

A) Zariski prime divisors

Remark/Definition 3.1. — Recall that a valuation v of K is called a k-valuation, if v is trivial on k. For a k-valuation v of K the following conditions are equivalent:

i) v is discrete, and its residue field Kv is a function field in td(K|k) − 1 variables. ii) td(Kv |k) = td(K|k) − 1. A k-valuation v on K with the above equivalent properties is called a Zariski prime

divisor of K|k. See Appendix, A), where a geometric description of the Zariski prime divisors of K|k is given.

The aim of this subsection is to give a first insight in the pro-ℓ Galois theory of the Zariski prime divisors of K|k. By abuse of language, we say that a k-valuation v of K(ℓ) is a Zariski prime divisor, if v is the prolongation of some Zariski prime divisor of K|k to K(ℓ). Since K(ℓ)|K is algebraic, hence K(ℓ)v |Kv is algebraic too, it follows that a k-valuation of K(ℓ) is a Zariski prime divisor if and only if td(K(ℓ)v |k) = td(K(ℓ)|k) − 1.

With this convention, for a Zariski prime divisor v of K(ℓ)|k we denote the decomposition group of v in G_{K}(ℓ) by Z_{v}. By general decomposition theory we have: Two Zariski prime divisors of K(ℓ)|k have the same restriction to K if and only if they are conjugated under G_{K}(ℓ).

Definition 3.2. — In the notations from above, we will say that Z_{v} is a divisorial subgroup of G_{K}(ℓ) or of the function field K|k (at the Zariski prime divisor v, if this is relevant for the context).

More generally, a closed subgroup Z ⊂ G_{K}(ℓ) which is isomorphic to a divisorial subgroup Z_{w} of a function field L|l with td(L|l) = td(K|k) and l algebraically closed of characteristic ≠ ℓ is called a divisorial like subgroup of G_{K}(ℓ) or of K|k.

By the remark above, the divisorial subgroups as well as the divisorial like subgroups of G_{K}(ℓ) form full conjugacy classes of closed subgroups of G_{K}(ℓ). Some of the significant properties of the divisorial subgroups are summarized in the following:

Proposition 3.3. — Let v be a Zariski prime divisor of K(ℓ)|k. Let Z_{v} ⊂ G_{K}(ℓ) be the divisorial subgroup at v, T_{v} the inertia group of Z_{v}, and G_{v} = Z_{v}/T_{v} the Galois group of the corresponding Galois residue field extension K(ℓ)v |Kv. Then the following hold:

(1) K(ℓ)v = (Kv)(ℓ), hence G_{v} = G_{Kv}(ℓ). Further, T_{v} ≅ ℤ_{ℓ} and Z_{v} ≅ T_{v}×G_{Kv}(ℓ) as pro-finite groups.

In particular, a divisorial like subgroup of G_{K}(ℓ) is any closed subgroup Z which is isomorphic to ≅ ℤ_{ℓ} × G_{L_{1}}(ℓ) for some function field L_{1}|l with l algebraically closed of characteristic ≠ ℓ and td(L_{1}|l) = td(K|k) − 1.

(2) T_{v} is the unique maximal Abelian normal subgroup of Z_{v}.

(3) Z_{v} is self-normalizing in G_{K}(ℓ), and it is maximal among the subgroups of G_{K}(ℓ) which have a non-trivial pro-cyclic normal subgroup.

Proof. — Assertion (1) follows immediately from Fact 2.2, 2), 3).

Concerning (2), let T ⊆ Z_{v} be an Abelian normal closed subgroup. Then the image \overline{T} of T in G_{Kv}(ℓ) is an Abelian normal subgroup of G_{Kv}(ℓ). Thus the assertion follows from the following:

Claim. If L|k is a function field, then G_{L}(ℓ) has no non-trivial Abelian normal subgroups.

In order to prove the Claim, recall that if L ≠ k, then L is a Hilbertian field; see e.g. [F–J], Ch.16 for basic facts concerning Hilbertian fields. Let L_{1}|L be a proper Galois sub-extension of L(ℓ)|L. Then by Kummer theory, L(ℓ)|L_{1} is not finite. Choose any proper finite sub-extension L_{2}|L_{1} of L(ℓ)|L_{1}. Then by Weissauer’s Theorem, L_{2} is a Hilbertian field. Since every finite split embedding problem with Abelian kernel over L_{2} is properly solvable, it follows that L_{2} has “many” finite Galois ℓ-extensions which are not Abelian. Therefore, L(ℓ)|L_{1} cannot be an Abelian extension.

To (3): First, since a Zariski prime divisor has rank one, it follows by Proposition 2.3 that it equals its absolute pro-ℓ core. Thus by Proposition 2.4, (1), it follows that its decomposition group is self-normalizing. Concerning the maximality: Let Z′ be a subgroup of G_{K}(ℓ) having a non-trivial pro-cyclic normal subgroup T′ and satisfying Z_{v} ⊆ Z′. We show that Z′ = Z_{v}, and that T′ = T_{v} provided T′ is a maximal pro-cyclic normal subgroup.

Indeed, since Z_{v} ⊆ Z′, and T′ is normal in Z′, we have: T′ is normal in G := T_{v}T′; and one has an exact sequence 1 → T′ → G → \overline{T}_{v} → 1, where \overline{T}_{v} = T_{v}/(T_{v} ∩ T′) is a quotient of T_{v} ≅ ℤ_{ℓ}. Let Λ = K(ℓ)^{G} be the fixed field of G, thus Λ(ℓ) = K(ℓ). Since Λ contains the algebraically closed field k, it contains the ℓ^{∞} roots of unity. Thus by Kummer Theory, the following two finite ℓ-elementary Abelian groups G^{ab}/ℓ and Λ^{×}/ℓ are (Pontrjagin) dual to each other. On the other hand, by the definition of G —see the above exact sequence, we have: Either G^{ab}/ℓ ≅ ℤ/ℓ, or G^{ab}/ℓ ≅ (ℤ/ℓ)^{2}. Applying again Kummer Theory in its pro-ℓ setting, we get:

a) If G^{ab}/ℓ ≅ ℤ/ℓ, then Λ^{×}/ℓ is cyclic. Thus Λ(ℓ)|Λ is pro-cyclic. And in turn, G ≅ ℤ_{ℓ}.

b) If G^{ab}/ℓ ≅ (ℤ/ℓ)^{2}, then Λ^{×}/ℓ is generated by exactly two elements. Thus G = G_{Λ}(ℓ) has ℤ_{ℓ} × ℤ_{ℓ} as a quotient. Hence taking into account the exact sequence above 1 → T′ → G → \overline{T}_{v} → 1, it follows that G ≅ ℤ_{ℓ} × ℤ_{ℓ}.

From this we conclude that G is abelian in both cases a) and b) above; hence T_{v} is a normal subgroup of G = T_{v}T′. Since v is pro-ℓ Henselian on the fixed field K^{T} of T_{v} and equals its pro-ℓ K^{T} -core, it follows by the above cited result of F. K. Schmidt, Proposition 2.4, (1), that Λ is pro-ℓ Henselian with respect to v. Thus denoting by K^{T′} the fixed field of T′ in K(ℓ), we have Λ ⊆ K^{T′}. Hence v is pro-ℓ Henselian on K^{T′}. Finally, as T′ is a normal subgroup of Z′, it follows by loc.cit. again that v is pro-ℓ Henselian on the fixed field K^{Z′} of Z′. Since Z_{v} is the decomposition group of v in G_{K}(ℓ), it finally follows that Z′ ⊆ Z_{v}. Thus Z′ = Z_{v}, and T′ = T_{v} provided T′ is a maximal pro-cyclic normal subgroup of Z′.

B) Quasi-divisorial valuations The ultimate goal of the Galois theory of the Zariski prime divisors of K|k is to identify these divisors as corresponding to the conjugacy classes of particular divisorial like subgroups. Nevertheless, the only kind of extra information one might use in such a characterization should be of group theoretic nature, originating in the pro-ℓ Galois theory of function fields. Obviously, the best one can expect is that “morally” the converse of Proposition 3.3 above should also be true; this means that if Z ⊂ G_{K}(ℓ) is a divisorial like subgroup, then it should originate from a Zariski prime divisor of K(ℓ)|k, which should be unique. Unfortunately, this cannot be true, as indicated below.

• First, every open subgroup Z′ ⊆ Z_{v} of a divisorial subgroup is a divisorial like subgroup. Indeed, such an open subgroup is a divisorial subgroup for some properly chosen finite sub-extension K′|K of K(ℓ)|K. Thus in general, a divisorial like subgroup is not a divisorial subgroup.

An obvious way to remedy this failure for the converse of Proposition 3.3 is by restricting ourselves to considering maximal divisorial like subgroup Z of G_{K}(ℓ) only, and then ask whether every such a maximal subgroup Z is divisorial.

• Unfortunately, there is a more subtle source of divisorial like subgroups of G_{K}(ℓ) coming from so called defectless valuations v on K(ℓ) of relative rational rank 1, which generalize in a natural way the Zariski prime divisors of K|k.

In order to explain these phenomena, we introduce notations/notions as follows: Let v be an arbitrary valuation on K(ℓ). Since k is algebraically closed, v k is a totally ordered ℚ-vector space (which is trivial, if the restriction of v to k is trivial). We will denote by r_{v} the rational rank of the torsion free group vK/vk, and by abuse of language call it the rational rank of v. Next remark that the residue field kv is algebraically closed too, and Kv |kv is some field extension (not necessarily a function field!). We will denote td_{v} = td(Kv |kv) and call it the residual transcendence degree. By general valuation theory, see e.g. [BOU], Ch.6, §10, 3, one has the following:

r_{v} + td_{v} ≤ td(K|k).

We will say that v has no (transcendence) defect, or that v is defectless, if the above inequality is an equality, i.e., r_{v} + td_{v} = td(K|k). See Appendix, B), for basic facts concerning defectless valuations, in particular for a “recipe” which produces all the defectless valuations v of K(ℓ).

Remark/Definition 3.4. — For a valuation v of K(ℓ) the following conditions are equivalent:

i) The valuation v is minimal among the valuations w of K(ℓ) satisfying r_{w} = 1 and td_{w} = td(K|k) − 1.

ii) v has no relative defect and satisfies: First, r_{v} = 1, and second, r_{v′} = 0 for any proper coarsening v′ of v.

A valuation of K with the equivalent properties i), ii), above is called quasi-divisorial. In particular, by Appendix, Fact 5.5, 2), b), it follows that if v is quasi-divisorial,

then Kv|kv is a function field with td(Kv |kv) = td(K|k)−1, and second, vK/vk ≅ ℤ.

Further, every Zariski prime divisor of K|k is a quasi-divisorial valuation. And a quasi-divisorial valuation v of K(ℓ) is a Zariski prime divisor if and only if v is trivial on k.

The aim of this subsection is to give a first insight in the pro-ℓ Galois theory of the quasi-divisorial valuation of K|k.

Proposition 3.5. — Let v be a valuation on K(ℓ) having no relative defect such that r_{v} = 1 and char(Kv) ≠ ℓ. Let T_{v} ⊂ Z_{v} and G_{v} = Z_{v}/T_{v} be defined as usual. Then the following hold:

(1) K(ℓ)v = (Kv)(ℓ), hence G_{v} = G_{Kv}(ℓ). Further, T_{v} ≅ ℤ_{ℓ} and Z_{v} ≅ T_{v}×G_{Kv}(ℓ) as pro-finite groups. In particular, Z_{v} is a divisorial like subgroup of G_{K}(ℓ).

(2) T_{v} is the unique maximal Abelian normal subgroup of Z_{v}.

(3) Suppose that td(K|k) > 1. Then Z_{v} is maximal among the divisorial like subgroups of G_{K}(ℓ) if and only if v is a quasi-divisorial valuation on K(ℓ).

Proof. — For assertion (1), recall that Kv |kv is a function field which satisfies td(Kv |kv) = td(K|k) − 1. One concludes by applying Fact 2.2, 2), 3) and taking into account that vK/ℓ ≅ ℤ/ℓ.

The proof of assertion (2) is identical with the proof of the corresponding assertion in the case of divisorial subgroups of G_{K}(ℓ) in Proposition 3.3.

To (3): First, suppose that Z_{v} is maximal among the divisorial like subgroups of G_{K}(ℓ). Let v′ be the unique coarsening of v which is a quasi-divisorial valuation of K(ℓ), as described in Appendix, Fact 5.5, 3). We claim that v = v′. Indeed, let v_{0} = v/v′ be the valuation induced by v on the residue field K(ℓ)v′. Then by general decomposition theory for valuations, see e.g. Fact 2.1, 3), it follows via the canonical exact sequence

1 → T_{v′} → Z_{v′} → G_{Kv′}(ℓ) → 1,

that Z_{v} is exactly the preimage of Z_{v_{0}} in Z_{v′}. Since v′ has r_{v′} = 1 and is defectless, by assertion (1) of the Proposition, it follows that Z_{v′} is divisorial like. On the other hand, Z_{v} ⊆ Z_{v′} is maximal among divisorial like subgroups of G_{K}(ℓ), hence Z_{v} = Z_{v′}. But then Z_{v_{0}} = G_{Kv′}(ℓ), and so v_{0} is pro-ℓ Henselian on the function field Kv′|kv′. Since Kv′|kv′ has transcendence degree equal to td(K|k) − 1 > 0, the valuation v_{0} must be the trivial valuation. Thus finally v = v′, and v is quasi-divisorial.

For the converse, suppose that v is quasi-divisorial. We show that Z_{v} is maximal among the subgroups Z′ of G_{K}(ℓ) having a non-trivial pro-cyclic normal subgroup T′. The proof is more or less identical with the proof of assertion (3) of Proposition 3.3. Indeed, in the above notations (which are identical with the ones from loc.cit.) and reasoning like there, we have: The fixed field Λ of G := T_{v}T′ is pro-ℓ Henselian with respect to v. Next recall that K^{T} is the fixed field of T_{v} in K(ℓ). Hence Λ ⊂ K^{T}.

Claim. The pro-ℓ, K^{T} -core v_{pro-ℓ,K^{T}} equals v.

To simplify notations, let L = K^{T} be the fixed field of T_{v} in K(ℓ), and w = v_{pro-ℓ,L} = v_{pro-ℓ,K^{T}}. Then L(ℓ) = K(ℓ), and G_{L}(ℓ) = T_{v}. And further, G_{L}(ℓ) ⊂ Z_{w}, as w is pro-ℓ Henselian on L. By contradiction, suppose that w < v is a proper coarsening of v. Since v is quasi-divisorial, it follows that wK is divisible, thus wL is divisible too. Therefore, T_{w} = 1 by Fact 2.2; and the canonical projection below is an isomorphism

π_{w} : Z_{w} → G_{Kw}(ℓ).

On the other hand, since K^{T} v = Lv is pro-ℓ closed, and w = v_{pro-ℓ,L}, it follows by Proposition 2.3, (1), that Lw is pro-ℓ closed too. Thus π_{w}(G_{L}(ℓ)) = G_{Lw}(ℓ) = 1, hence G_{L}(ℓ) is trivial. Contradiction! The Claim is proved.

We return to the proof of assertion (3): Taking into account that w = v_{pro-ℓ,K^{T}} is pro-ℓ Henselian on Λ, and that Λ ⊆ K^{T} = L, by using the Claim above and the properties of pro-ℓ core, we get:

v = v_{pro-ℓ,K^{T}} ≤ v_{pro-ℓ,Λ} ≤ v.

Hence finally v_{pro-ℓ,Λ} = v. Therefore, as in loc.cit., it follows that v is pro-ℓ Henselian on the fixed field K(ℓ)^{Z′}. Equivalently, Z′ ⊆ Z_{v}.

4. Characterization of the (quasi-)divisorial subgroups

A) Arithmetical nature of quasi-divisorial subgroups Recall the notations from the Introduction: A closed subgroup Z ⊂ G_{K}(ℓ) is called quasi-divisorial, if it is divisorial like and maximal among the divisorial like subgroups of G_{K}(ℓ). The aim of this section is to show that the quasi-divisorial subgroups are of arithmetical/geometrical nature. More precisely, we will prove the following:

Proposition 4.1. — Let K|k be a function field with d = td(K|k) > 1. Then the following hold:

(1) Every divisorial like subgroup of G_{K}(ℓ) is contained in a unique quasi-divisorial subgroup of G_{K}(ℓ).

(2) Suppose that Z is a quasi-divisorial subgroup of G_{K}(ℓ). Then there exists a unique quasi-divisorial valuation v on K(ℓ) such that Z_{v} = Z. Moreover, char(Kv) ≠ ℓ.

Proof. — The main step in the proof is the following Key Lemma, which in some sense plays the same role as the q-Lemma in the Local theory from [P1]. Nevertheless, its proof requires other techniques, see below.

Key Lemma 4.2. — In the context of the Theorem, let Z ⊂ G_{K}(ℓ) be a subgroup with the following properties:

i) Z contains closed subgroups Z_{0} isomorphic to ℤ_{ℓ}^{d} with d = td(K|k).

ii) Z has non-trivial Abelian normal subgroups.

Then there exists a non-trivial valuation v of K(ℓ) which is pro-ℓ Henselian on the fixed field K^{Z} of Z in K(ℓ) and satisfies the following: v has no relative defect, and r_{v} > 0, and char(Kv) ≠ ℓ, and T_{v} ∩ Z ≅ ℤ_{ℓ}^{r_{v}}.

Moreover, if Z has a pro-cyclic normal subgroup T such that Z/T has no nontrivial Abelian normal subgroups, then there exists a unique quasi-divisorial valuation v such that Z ⊆ Z_{v} and T = T_{v} ∩ Z.

Proof. — The main ingredient in the proof of Key Lemma is the following result, see e.g. Engler–Nogueira [E–N] for ℓ = 2, and Engler–Koenigsmann [E–K] and/or Efrat [Ef] in general. And naturally, one could use here Bogomolov [B], Bogomolov– Tschinkel [B–T1].

Fact 4.3. — Let Λ be a field such that G_{Λ}(ℓ) ≅ ℤ_{ℓ} ×^{|} ℤ_{ℓ}. Then there exists a nontrivial valuation w on K(ℓ) which is ℓ-Henselian on Λ such that wΛ is not ℓ-divisible, and the residual characteristic char(Λw) ≠ ℓ.

As a consequence of Fact 4.3 we have the following:

Fact 4.4. — Let K|k be as above, and d = td(K|k). Then by induction on d it follows: Every Abelian subgroup G ⊂ G_{K}(ℓ) is of the form G ≅ ℤ_{ℓ}^{s} for some s = s_{G} ≤ d.

Using these facts and the techniques developed in Pop [P1], Local Theory, one can easily prove the above Key Lemma as follows:

Let K^{Z} be the fixed field of Z in K(ℓ). Let T′ ≅ ℤ_{ℓ}^{r} be a maximal non-trivial Abelian normal subgroup of Z. We remark that Z contains a subgroup G ⊆ Z as the one in the Fact 4.3 above such that G ∩ T′ is non-trivial. Indeed, if r > 1, then T′ contains a subgroup G ≅ ℤ_{ℓ} × ℤ_{ℓ}. If on the other side r = 1, then we choose any g ∈ Z not in T′. Then denoting by G the closed subgroup generated by T′ and g, we get a subgroup G ⊆ Z with the desired properties.

Next let Λ be the fixed field of G in K(ℓ). Then by Fact 4.3 above, there is a valuation w on K(ℓ) which is pro-ℓ Henselian on Λ and satisfies: wΛ is not ℓ-divisible, and Λw has characteristic ≠ ℓ.

In particular, w is also pro-ℓ Henselian on the fixed field Λ^{T′∩G} of T′ ∩ G in K(ℓ). Let v be the pro-ℓ Λ^{T′∩G}-core of w. We claim that v has the properties we asked for in the Key Lemma.

Claim 1. v is pro-ℓ Henselian on K^{Z}.

Indeed, taking into account that T′ ∩ G is a normal subgroup of T′, it follows by Proposition 2.4, (1), that v is pro-ℓ Henselian on the fixed field K^{T′} of T′ in K(ℓ). On the other hand, since K^{T′} ⊆ Λ^{T′∩G}, and v equals its pro-ℓ Λ^{T′∩G}-core, it follows that v equals its pro-ℓ K^{T′}-core. Thus reasoning as above, since T′ is a normal subgroup in Z, it follows that v is pro-ℓ Henselian on the fixed field K^{Z} of Z in K(ℓ). And moreover, v equals its pro-ℓ K^{Z}-core.

Claim 2. v has no relative defect.

Indeed, let Λ_{0} be the fixed field of a subgroup Z_{0} ≅ ℤ_{ℓ}^{d} of Z ⊂ G_{K}(ℓ). Taking into account that v is pro-ℓ Henselian on Λ_{0}, applying Fact 2.2, 3), we have:

ℤ_{ℓ}^{d} ≅ Z_{0} = G_{Λ_{0}}(ℓ) ≅ ℤ_{ℓ}^{δ_{0}} × G_{Λ_{0}v}(ℓ),

where δ_{0} equals the dimension of the 𝔽_{ℓ} vector space vΛ_{0}/ℓ. And since Z_{0} is Abelian, G_{Λ_{0}v}(ℓ) is Abelian too, say G_{Λ_{0}v}(ℓ) ≅ ℤ_{ℓ}^{s}. Hence d = δ_{0} + s. On the other hand, we have the inequalities as follows:

a) δ_{0} ≤ δ_{v} = r_{v}, deduced from the exact sequence 0 → vk → vK → ℤ^{r_{v}} → 0. Here vk is a divisible group, and δ_{v} is the dimension of the 𝔽_{ℓ} vector space vK/ℓ.

b) s ≤ td(Λ_{0}v |kv) = td(Kv |kv) = td_{v}, by Fact 4.4.

Therefore we have: d = δ_{0} + s ≤ r_{v} + td_{v} ≤ d, and hence the inequalities at a), b) above are equalities, and r_{v} + td_{v} = d. Thus v has no relative defect.

In order to show that T_{v} ∩ Z ≅ ℤ_{ℓ}^{r_{v}}, we use the conclusion of the discussion above: We have namely proved that via π_{v} : Z_{v} → G_{Kv}(ℓ) one has: Z_{0}∩ker(π_{v}) ≅ ℤ_{ℓ}^{δ_{0}} = ℤ_{ℓ}^{r_{v}}. Therefore, since ker(π_{v}) = T_{v}, we have:

ℤ^{r_{v}} ≅ Z_{0} ∩ T_{v} ⊆ Z ∩ T_{v} ⊆ T_{v} ≅ ℤ^{r_{v}}.

Thus finally we get T_{v} ∩ Z ≅ ℤ_{ℓ}^{r_{v}}.

Finally, we address the last assertion of the Key Lemma, and consider the case when Z has a pro-cyclic normal subgroup T such that Z/T has no non-trivial Abelian subgroups. Let v be the valuation just constructed above. Since T_{v} is the center of Z_{v}, it follows that Z ∩ T_{v} ≅ ℤ_{ℓ}^{r_{v}} is contained in the center of Z. Hence by the hypothesis on Z and T it follows that Z ∩ T_{v} is contained in T ≅ ℤ_{ℓ}, thus in particular r_{v} = 1. Furthermore, since ℤ_{ℓ} ≅ T_{v} ∩ Z ⊆ T ≅ ℤ_{ℓ}, it follows that T_{v} ∩ Z is an open subgroup of T. Thus

π_{v}(T) ≅ T/(T ∩ T_{v}) = T/(Z ∩ T_{v})

is a finite cyclic subgroup of G_{Kv}(ℓ). Since G_{Kv}(ℓ) is torsion free, it follows that π_{v}(T) = 1, thus T = T_{v} ∩ Z.

In order to conclude, replace v by its minimal coarsening v′ which is a quasidivisorial valuation, as in Appendix, Fact 5.5, 3). Then by Fact 2.1, 3), one has Z_{v} ⊆ Z_{v′}. And by Proposition 3.5, (3), v′ is the only quasi-divisorial valuation of K(ℓ) such that Z_{v} ⊆ Z_{v′}.

This concludes the proof of the Key Lemma.

We now come to the proof of the Proposition 4.1. Assertion (1) is an immediate consequence of the Key Lemma and the fact that (by Proposition 3.3) the divisorial like subgroups satisfy the conditions i), ii), of the Key Lemma. For Assertion (2), the uniqueness of v follows from Proposition 2.4 by taking into account that a quasidivisorial valuation equals its absolute pro-ℓ core (as its residue field is not pro-ℓ closed if d > 1).

B) Characterization of the divisorial subgroups We now show that using the information encoded in “sufficiently many” 1-dimensional projections one can characterize the divisorial subgroups among all the quasi-divisorial subgroups. First some preparations:

Fact 4.5. — Let K|k be a function field in d = td(K|k) > 1 variables, and L|k a function subfield of K|k with td(L|k) > 0. In particular, we can and will view L(ℓ) as a subfield of K(ℓ). Finally let v be a valuation of K(ℓ), and denote by K^{Z} and L^{Z} its decomposition groups in G_{K}(ℓ), respectively G_{L}(ℓ). Then the following hold:

1) If v is defectless on K(ℓ), then v is defectless on L(ℓ). Moreover, if v is a quasidivisorial valuation of K(ℓ), then v is either trivial or a quasi-divisorial valuation on L(ℓ).

2) One has L^{Z} ⊆ K^{Z}. Further, denoting by L′ the relative algebraic closure of L^{Z} in K^{Z}, it follows that L′|L^{Z} is finite.

Proof. — The first assertion follows by the additivity of r_{v} and td_{v} in towers of fields in the case of defectless valuations. For the second assertion see the proof of Corollary 1.18 from [P1].

Next we define 1-dimensional projections of G_{K}(ℓ) as follows: For every t ∈ K\k, let K_{t} be the relative algebraic closure of k(t) in K. Then K_{t}|k is a function field in one variable. Moreover, if t is “general”, then K_{t} = k(t) is a rational function field over k. Turning our attention to Galois theory, the inclusion ı_{t} : K_{t} → K gives rise to a surjective restriction homomorphism

p_{t} : G_{K}(ℓ) → G_{K_{t}}(ℓ).

We now are ready to announce the recipe for detecting the divisorial valuations of K|k using the projections p_{t}.

Proposition 4.6. — Let K|k be a function field as usual, and suppose that with td(K|k) > 1. Then for a given quasi-divisorial subgroup Z ⊂ G_{K}(ℓ), the following assertions are equivalent:

i) Z is a divisorial subgroup of G_{K}(ℓ).

ii) ∃ t ∈ K\k such that p_{t}(Z) ⊆ G_{K_{t}}(ℓ) is an open subgroup.

Proof. — First, suppose that Z = Z_{v} is the divisorial subgroup defined by some divisorial valuation v on K(ℓ). Then choosing t such that t is a v-unit and tv is transcendental over k in Kv, we have: v is trivial on L := K_{t}. And in particular, v is pro-ℓ Henselian on L, i.e., L^{Z} = L. Therefore, in the notations from the Fact 4.5 above, it follows by loc.cit. that the relative algebraic closure L′ of L^{Z} = L in K^{Z} is a finite extension of L = K_{t}. Thus p_{t}(Z) = G_{L′}(ℓ) ≤ G_{K_{t}}(ℓ) is an open subgroup.

Conversely, suppose that p_{t}(Z) is an open subgroup of G_{K_{t}}(ℓ) for some nonconstant t ∈ K. Equivalently, the relative algebraic closure L′ of the field L := K_{t} in K^{Z} is a finite extension of L. Now denoting by L^{Z} the decomposition field of v in L(ℓ), it follows by Fact 4.5 that L ⊆ L^{Z} ⊆ L′. Therefore, L^{Z}|L is finite, thus L^{Z}|k is a function field in one variable over k. And since v is a pro-ℓ Henselian valuation of this function field, it follows that v must be trivial on L^{Z}. In particular, v is trivial on k. Hence v is a Zariski prime divisor of K.

5. Appendix

A) Geometric interpretation of the Zariski prime divisors Let K|k be a function field. A model X → k of K|k is any integral k-variety X → k whose function field k(X)|k is identified with K|k. In particular, we then identify the structure sheaf O_{X} with a sheaf of k-subalgebras of K|k. In particular, the restriction morphisms of O_{X}, say ρ_{UV} : O_{X}(U) → O_{X}(V) for V ⊂ U are simply the inclusions. Therefore, we have the following:

Fact 5.1. — On the set of all the models X_{i} → k of K|k there exists a naturally defined domination relation as follows: X_{j} ≥ X_{i} if and only if there exists a surjective morphism φ_{ji} : X_{j} → X_{i} which at the structure sheaf level is defined by inclusions. Let P_{K} be the set of all projective normal models of K. Some basic results in algebraic geometry guarantee the following, see e.g., Zariski–Samuel [Z–S], Ch.VI, especially §17, Mumford [M], Ch.I, etc.:

1. Every model X_{i} → k is contained as an open subvariety in a complete model X̃_{i} → k of K|k (Nagata’s Theorem).

2. Every complete model X → k is dominated by some X_{i} ∈ P_{K} (Chow’s Lemma).

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

4. Given any function f ≠ 0 in K, there exist projective models X_{i} → k such that for each x_{i} ∈ X_{i} one has: Either f of 1/f is defined at x_{i}.

5. Let v be a k-valuation of K|k with valuation ring O_{v}. Then for every model X → k of K|k, there exists at most one point x ∈ X such that O_{v} dominates the local ring O_{X,x} (Valuation criterion).

If such a point exists, we will say that x is the center of v on X.

6. If in the above context, X → k is proper, then every v has a center on X (valuation criterion).

Remark/Definition 5.2. — We denote by ℜ_{K|k} the space of all the k-valuations of K|k, and call it the Riemann space of K|k. There exists a canonical identification of ℜ_{K|k} with lim_{⟵}_{i} X_{i} as follows:

Let v be a k-valuation of K|k with valuation ring O_{v}. For every projective —thus proper— model X_{i} → k of K|k, let x_{i} ∈ X_{i} be the center of v on X_{i}. Clearly, if X_{j} ≥ X_{i}, say via a morphism ϕ_{ji} : X_{j} → X_{i}, then ϕ_{ji}(x_{j}) = x_{i}. Therefore (x_{i})_{i} ∈ ℜ_{K} = lim_{⟵}_{i} X_{i}. Further, for the local rings we have: O_{X_{i},x_{i}} ⊆ O_{X_{j},x_{j}}. Thus R_{v} := ∪_{X_{i}}O_{X_{i},x_{i}} is a k-subalgebra of K|k, which finally turns to be exactly the valuation ring O_{v}.

Conversely, given (x_{i})_{i} ∈ lim_{⟵}_{i} X_{i}, one has: O_{X_{i},x_{i}} ⊆ O_{X_{j},x_{j}} if j ≥ i. Thus O := ∪_{X_{i}}O_{X_{i},x_{i}} is a k-subalgebra of K|k. Using Fact 3.1, 4) above, it follows that for every f ≠ 0 from K one has: Either f of 1/f lie in O_{X_{i},x_{i}} for i sufficiently large; thus f of 1/f lie in O. Hence O is a k-valuation ring of a k-valuation v of K|k.

Remark/Definition 5.3. — We remark that for a point v = (x_{i})_{i} in ℜ_{K|k} as above, the following conditions are equivalent, one uses e.g. [BOU], Ch.IV, §3, or [P1], The Local Theory:

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) v is discrete, and Kv|k is a function field in td(K|k) − 1 variables.

iii) td(Kv |k) = td(K|k) − 1.

A valuation v on K with the above equivalent properties is called a Zariski prime divisor of K|k. As a corollary of the observations above we have:

The space of all Zariski prime divisors of K is the union of the spaces of Weil prime divisors of all normal models of K|k (if we identify every Weil prime divisor with the discrete valuation on K it defines).

B) Defectless valuations We consider function fields K|k as at the beginning of the Subsection 3, B). In the notations from there, for a valuation v of K we denote by r_{v} and td_{v} the rational rank, respectively the residual transcendence degree of v. We recall the following basic facts concerning defectless valuations.

Fact 5.4. — Let v be the composition v = v_{0} v′ of a valuation v′ on K(ℓ) and a valuation v_{0} on K_{0}(ℓ), where K_{0} := Kv′ is a field extension of the algebraically closed field k_{0} := kv′.

1) One has a canonical diagram with exact rows as follows:

0 → v_{0}k_{0} → vk → v′k → 0
↓ ↓ ↓
0 → v_{0}K_{0} → vK → v′K → 0

The vertical maps are inclusions, therefore one has an exact sequence of torsion free groups

(∗)  0 → v_{0}K_{0}/v_{0}k_{0} → vK/vk → v′K/v′k → 0 .

From this we deduce the following:

a) r_{(·)} is additive in the following sense: r_{v} = r_{v′} + r_{v_{0}}.

b) There exists a unique minimal coarsening say v′ of v such that r_{v′} = r_{v}. And this coarsening is defined by some convex divisible subgroup Δ′ of vK.

2) Further, one has td_{v} = td_{v_{0}}, thus td_{v′} ≥ td_{v} +r_{v_{0}}.

3) Hence v is defectless ⟺ both v′ and v_{0} are defectless.

Proof. — The only less obvious fact might be assertion b). In order to prove it, suppose first that v′ is a coarsening of v such that r_{v′} = r_{v}. Then from the exact sequence (∗) above we deduce that r_{v_{0}} = 0. Thus v_{0}K_{0}/v_{0}k_{0} is a torsion free group of rational rank equal to 0. Hence v_{0}K_{0} = v_{0}k_{0}. Since k_{0} = kv′ is algebraically closed, v_{0}k_{0} is divisible; hence the convex subgroup Δ_{v′} := v_{0}K_{0} of vK defining v′ is divisible. One gets Δ′ by taking the union of all the convex divisible subgroups of the form Δ_{v′} for v′ having r_{v′} = r_{v}.

Note that Δ′ = vK if and only if r_{v} = 0.

Fact 5.5. — Let v be a defectless valuation on K(ℓ).

1) The fundamental equality holds for every finite extension L|K of K. This means, that if v_{1}, …, v_{r} are the finitely many prolongations of v_{0} := v|_{K} to L, then

[L : K] = ∑_{i}e(v_{i}|v_{0}) f(v_{i}|v_{0}).

This is a non-trivial fact, see e.g., Kuhlmann [Ku]. It is a generalization of results by several people starting with Deuring, Grauert–Remmert’s Stability Theorem from the rigid algebraic geometry, etc.

2) In particular, using the description of the defectless valuations given by Fact 5.6 below, one gets the following:

a) vK / vk ≅ ℤ^{r_{v}}.

b) Kv |kv is a function field in td_{v} variables.

3) The unique minimal coarsening v′ of v such that r_{v} = r_{v′} —defined in Fact 5.4, 1), b), is defectless and is defined by the unique maximal divisible convex subgroup Δ′ of vK.

In particular, for every defectless valuation v of K(ℓ) with r_{v} = 1, there exists a unique coarsening v′ of v which is a quasi-divisorial valuation of K|k.

Fact 5.6. — A recipe which produces all possible valuations without relative defect on K|k with given invariants r = r_{v} and td = td_{v} is as follows:

First let v be a defectless valuation on K(ℓ), say with invariants r = r_{v} and td = td_{v}. Since vK(ℓ)/vK is a torsion group, it follows that vK/vk has rational rank equal to r. We do the following:

- Choose a system (t_{1}, …, t_{r}) of elements of K^{×} such that setting γ_{i} = v t_{i}, the resulting system (γ_{1}, …, γ_{r}) of elements of vK is linearly independent in vK/vk.

- Further choose a system of v-units T_{0} = (t_{r+1}, …, t_{d}) in K such that (t_{r+1}v, …, t_{d}v) is a transcendence basis of Kv |kv.

Using e.g. [BOU], Ch.6, §10, 3, it follows that T = (t_{1}, …, t_{d}) is a system of elements of K which is algebraically independent over k. Since v is defectless by hypothesis, it follows that d = td(K|k), i.e., T is a transcendence basis of K|k. Thus denoting by w_{T_{0}} and w_{T} the restrictions of v to the rational function fields k(T_{0}) ⊂ k(T ) respectively, the following hold:

i) w_{T_{0}} is the so called generalized Gauß valuation defined by v_{k} := v|_{k} and T_{0} on k(T_{0}), i.e., for every polynomial p(t) in the system of variables t_{0} = (t_{r+1}, …, t_{d}), say p(t_{0}) = ∑_{i} a_{i}t_{0}^{i} ∈ k[T_{0}] one has:

w_{T_{0}}(p(t_{0})) = min_{i} v_{k}(a_{i}) = v(p(t_{0})).

ii) w_{T} is the unique prolongation of w_{T_{0}} to k(T ) such that for all t_{i} (1 ≤ i ≤ r) one has:

w_{T} (t_{i}) = γ_{i} = v(t_{i}).

And finally, v|_{K} is one of the finitely many prolongations of v_{T} to the finite extension K|k(T ).

In particular, Γ := vk(T ) = w_{T} k(T ) = vk + γ_{1}ℤ + ⋯ + γ_{r}ℤ is a totally ordered group such that Γ/vk is a free Abelian group of rank r. And the residue field k(T )v_{T} is the rational function field kv(t_{r+1}v, …, t_{d}v).

The observation above can be “reversed” in order to produce defectless valuations on K(ℓ) as follows. Consider:

- Valuations v_{k} on k together with totally ordered groups of the form Γ = v_{k}k + γ_{1}ℤ + ⋯ + γ_{r}ℤ with Γ/v_{k}k a free Abelian group of rank r.

- Transcendence bases T = (t_{1}, …, t_{d}) of K|k.

We set T_{0} = (t_{r+1}, …, t_{d}). On the rational function fields k(T_{0}) we consider the generalized Gauß valuation w_{T_{0}} defined by v_{k} and T_{0}; and we denote by w_{T} the unique prolongation of w_{T_{0}} to k(T ) satisfying w_{T} (t_{i}) = γ_{i} if i ≤ r.

Then finally we have:

For every w_{T} obtained as indicated above, all its prolongations v to K(ℓ) are defectless valuations on K(ℓ).

Conversely, every valuation without relative defect on K|k is obtained by the above recipe.

In particular, a valuation v without relative defect on K(ℓ) is a Zariski prime divisor ⟺ v is trivial on k and has r_{v} = 1

Remark 5.7. — It would be very desirable to have a geometric description of the space of all the quasi-divisorial valuations of K(ℓ), thus generalizing the construction of all the Zariski prime divisors of K|k given in the subsection A) above. Unfortunately, at the moment we are not able to do this in a satisfactory way.

One could do this along the same lines as in subsection A) above for a special class of quasi-divisorial valuations, which are the so called constant reductions à la Deuring, see Roquette [R], followed by Zariski prime divisors of the residue function fields of such constant reductions. This situation arises in an arithmetical way as follows, see loc.cit.:

Let R be the valuation ring of a valuation v_{k} of the base field k. Let X_{0} = Proj R[X_{0}, …, X_{d}] be the d-dimensional projective space over R, where d = td(K|k). Let T be a transcendence basis of K|k, and identify k(T ) with the function field of X_{0} via t_{i} = X_{i}/X_{0}. Then the local ring of the generic point η of the special fiber of X_{0} is exactly the valuation ring of the Gauß T , v_{k}-valuation v_{T} on k(T ). Finally, if X → X_{0} is the normalization of X_{0} in the function field extension k(T ) ↪ K, then denoting by η_{i} the generic points of the special fiber of X, it follows that the corresponding local rings O_{X ,η_{i}} are exactly the valuation rings of the prolongations v_{i} of v_{T} to K. Obviously, if v_{i} is a constant reduction of K|k, then r_{v_{i}} = 0 and td_{v_{i}} = td(K|k). By abuse of language, we will say that a valuation v_{0} of K(ℓ) is a constant reduction, if td(Kv_{0}| kv_{0}) = td(K|k). Thus the constant reductions of K(ℓ) are exactly the prolongations to K(ℓ) of the “usual” constant reductions of K|k defined above.

Finally, if v_{0} is a given constant reduction of K(ℓ), and v_{1} is a Zariski prime divisor of the residue field K(ℓ)v_{0} = (Kv_{0})(ℓ) of v_{0}, then the composition of the two valuations v = v_{1}v_{0} is a quasi-divisorial valuation on K(ℓ) which we call a c.r.-divisorial valuation.

References

[B] F. A. Bogomolov, On two conjectures in birational algebraic geometry, in Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings, ed A. Fujiki et all, Springer Verlag Tokyo 1991. ↑

[B–T1] F. A. and Tschinkel Bogomolov Y., Commuting elements in Galois groups of function fields

(2002). ↑ [B–T2] , Reconstruction of function fields, 2003. ↑

[BOU] N. Bourbaki, Algèbre commutative, Hermann Paris 1964. ↑

[D] P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over Q, Math. Sci. Res. Inst. Publ. 16, 79–297, Springer 1989. ↑

[Ef] I. Efrat, Construction of valuations from K-theory, Mathematical Research Letters 6 ((1999)), 335–344. ↑

[En] O. Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. ↑ [E–E] O. and Engler Endler A. J., Fields with Henselian Valuation Rings, Math. Z. 152 (1977),

191–193. ↑ [E–K] A. J. and Koenigsmann Engler J., Abelian subgroups of pro-p Galois groups, Trans. AMS 350

(1998), 2473–2485. ↑ [E–N] A. J. and Nogueira Engler J. B., Maximal Abelian normal subgroups of Galois pro-2-groups,

J. Algebra 166 (994), 481ndash505. ↑ [F–J] M. and Jarden Fried M., Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete.

3. Folge A Series of Modern Surveys in Mathematics, Vol. 11, Springer Verlag 2004. ↑

[GGA] Geometric Galois Actions I, LMS LNS Vol 242, eds L. Schneps – P. Lochak, Cambridge Univ. Press 1998. ↑

[G1] A. Grothendieck, Letter to Faltings, June 1983 , See [GGA]. ↑

[G2] , Esquisse d’un programme, 1984 , See [GGA]. ↑

[Ik] M. Ikeda, Completeness of the absolute Galois group of the rational number field, J. reine angew. Math. 291 (1977), 1–22. ↑

[Ko] H. Koch, Die Galoissche Theorie der p-Erweiterungen, Math. Monogr. 10, Berlin 1970. ↑ [K] J. Koenigsmann, From p-rigid elements to valuations (with a Galois characterization of p-adic

fields), J. reine angew. Math. 465 (1995), 165–182. ↑

[Km] K. Komatsu, A remark to a Neukirch’s conjecture, Proc. Japan Acad. 50 (1974), 253–255. ↑ [Ku] F.-V. Kuhlmann, Book on Valuation Theory, See: http://math.usask.ca/~fvk/Fvkbook.htm

(in preparation). ↑ [K–P–R] F.-V. Kuhlmann Pank, Immediate and purely wild extensions of valued fields, Manuscripta

math. 55 (1986), 39–67. ↑

[M] D. Mumford, The ref book of varieties and schemes, LNM ???, Springer Verlag 1966. ↑

[N1] J. Neukirch, Über eine algebraische Kennzeichnung der Henselkörper, J. reine angew. Math. 231 (1968), 75–81. ↑

[N2] , Kennzeichnung der p-adischen und endlichen algebraischen Zahlkörper, Inventiones math. 6 (1969), 269–314. ↑

[N3] , Kennzeichnung der endlich-algebraischen Zahlkörper durch die Galoisgruppe der maximal auflösbaren Erweiterungen, J. reine angew. Math. 238 (1969), 135–147. ↑

[Pa] A. N. Parshin, Finiteness Theorems and Hyperbolic Manifolds, in: The Grothendieck Festschrift III, ed P. Cartier et all, PM Series Vol 88, Birkhäuser Boston Basel Berlin 1990. ↑

[P1] F. Pop, On Grothendieck’s conjecture of birational anabelian geometry, Ann. of Math. 138 (1994), 145–182. ↑

[P2] , On Grothendieck’s conjecture of birational anabelian geometry II , Heidelberg– Mannheim Preprint series Arithmetik II, N^{0} 16, Heidelberg 1995. ↑

[P3] , Alterations and birational anabelian geometry, 519–532, in: Resolution of singularities, Birkhäuser PM Series, Vol. 181; eds Herwig Hauser et al, Birkhäuser Verlag, Basel 2000. ↑

[P4] , Pro-ℓ birational anabelian geometry over alg. closed fields I , See:

    http://arxiv.org/pdf/math.AG/0307076. ↑

[R] P. Roquette, Zur Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten, J. reine angew. Math. 200 (1958), 1–44. ↑

[S] J.-P. Serre, Cohomologie Galoisienne, LNM 5, Springer 1965. ↑

[Sp] M. Spiess, An arithmetic proof of Pop’s Theorem concerning Galois groups of function fields over number fields, J. reine angew. Math. 478 (1996), 107–126. ↑

[U1] K. Uchida, Isomorphisms of Galois groups of algebraic function fields, Ann. of Math. 106 (1977), 589–598. ↑

[U2] , Isomorphisms of Galois groups of solvably closed Galois extensions, Tôhoku Math. J. 31 (1979), 359–362. ↑

[U3] , Homomorphisms of Galois groups of solvably closed Galois extensions, J. Math. Soc. Japan 33 (1981), ???–???. ↑

[W] R. Ware, Valuation Rings and rigid Elements in Fields, Can. J. Math. 33 (1981), 1338–1355. ↑

[Z–S] O. and Samuel Zariski P., Commutative Algebra, Vol II, Springer-Verlag, New York, 1975. ↑

Florian Pop, Department of Mathematics, University of Pennsylvania, DRL, 209 S 33rd Street, Philadelphia, PA 19104. USA • E-mail : pop@math.upenn.edu

    Url : http://math.penn.edu/~pop