DIE BÖSE FARBE

A. BAUDISCH, M. HILS, A. MARTIN-PIZARRO AND F. O. WAGNER

abstract. We construct a bad field in characteristic zero.

1. Introduction

Morley rank is a model-theoretical generalization of Zariski dimension which can be extended to definable sets in any mathematical structure. A structure is called ω-stable if Morley rank is ordinal-valued. Morley degree is the analogous to the number of irreducible components of maximal Zariski dimension. An early conjecture due to B. Zil’ber stated that a structure of both Morley rank and degree 1 (called a strongly minimal set) arose from a classical geometry: the trivial (or degenerated) one, a vector space geometry or Zariski geometry over an algebraically closed field. This conjecture was refuted in 1988 by E. Hrushovski [9], who modified in a clever way Fraïssé’s original construction of the universal homogeneous model of a hereditary class of finite relational structures with the amalgamation property. This method was later divided by B. Poizat into two steps: first, the construction of a generic structure of rank ω and secondly, the collapse to a strongly minimal set. This procedure has been used in several applications: for example, E. Hrushovski fused two strongly minimal sets with the DMP in disjoint languages into a strongly minimal set [8] (cf. also [3]). In the aforementioned article, he also mentioned that this procedure could also be generalized to the case where both strongly minimal sets were expansions of a common vector space structure over a finite field. The fusion over a vector space was first proved by the second author and A. Hasson [7] in the 1-based case (moreover, they also studied the non-collapsed fusion of rank ω). The collapse to a strongly minimal fusion was attained finally by the first and third authors together with M. Ziegler [4]. Using similar arguments they also proved [5] the existence of a field of arbitrary prime characteristic of Morley rank 2 equipped with a definable additive subgroup of rank 1 after collapsing Poizat’s red fields [15] of rank ω · 2.

A bad field is a field of finite Morley rank equipped with a definable non-trivial proper divisible multiplicative subgroup. They appeared first in the study of simple groups of finite Morley rank, whose Borel subgroups (i.e. maximal solvable subgroups) are of the form K^{+} ⋊ T with 1 < T < K^{×}. According to the algebraicity conjecture due to Cherlin-Zil’ber (an algebraic variation of Zil’ber’s above conjecture) a simple infinite group of finite Morley rank is algebraic. The non-existence of bad fields would simplify the study of Borel subgroups. Due to a result by the fourth author [16, 17] bad fields are unlikely to exist in positive characteristic. After applying deep results due to J. Ax [1], B. Poizat [15] found a candidate of rank ω ·2 with a multiplicative subgroup of rank ω based on his construction [14] of a field of rank ω · 2 equipped with a definable set of rank ω. The collapse of the latter to rank 2 was explained by J. Baldwin and K. Holland in [2]. Poizat’s fields refuted a conjecture of C. Berline and D. Lascar, which stated that the rank of a field (in case of an infinite ordinal) was a monomial of the form ω^{α}. Together with M. Ziegler, the first and third author summarized and completed [6] the above results and exhibited a simpler axiomatization. Let us remark that a field of ordinal Morley rank is algebraically closed [11].

Date: May 24, 2006.

1991 Mathematics Subject Classification. Primary: 03C65; Secondary:03C50.

Key words and phrases. Model Theory, Fields of finite Morley Rank.

Third author was suppported by european research contract FP6-MOB-5-009541.

Le quatrième auteur est membre junior de l’Institut Universitaire de France.

Following Poizat’s notation we denote the predicate for the multiplicative subgroup by green. In this article we will collapse the green fields and therefore obtain a bad field in characteristic 0. This construction is extremely related to the red collapse [5]; however we will use results due to Ax-Poizat instead of locally finiteness of 𝔽_{p}-vector spaces.

2. Algebraic Lemmata

This section summarizes results coming from algebraic geometry which will be needed for our purposes. Let us fix some notation: ℂ denotes an algebraically closed field of characteristic 0. A variety V will always be a subvariety of some cartensian product (ℂ^{∗})^{n}. A torus is a connected algebraic subgroup of (ℂ^{∗})^{n}. It is described by finitely many equations of the form: x_{1}^{r_{1}} · . . . · x_{n}^{r_{n}} = 1. Linear dimension (as ℚ-vector spaces modulo torsion) equals algebraic dimension (as varieties) for tori which we will denote as l. dim_{ℚ}(T) or dim(T). Given a closed subvariety V in (ℂ^{∗})^{n}, its minimal torus is the smallest torus T such that V lies in some coset ā · T (with ā ∈ V). In this case, we define the codimension of an irreducible variety V as cd(V) := dim(T) − dim(V) = l. dim_{ℚ}(V) − dim(V), where l. dim_{ℚ}(V) := dim(T). A subvariety W ⊆ V is cd-maximal if cd(W′) > cd(W) for every subvariety W ⊊ W′ ⊆ V. Clearly, irreducible components of V and tori cosets maximally contained in V are examples of cd-maximal subvarieties.

Observe that any connected algebraic subgroup of a torus is again a torus.

We now state a result proved by B. Poizat [15, Corollaires 3.6 und 3.7]:

Theorem 2.1. Let V (x̄, z̄) be a uniformly definable family of varieties in (ℂ^{∗})^{n}. There exists a finite collection of tori {T_{0}, …, T_{r}}, such that for any torus T ⊆ (ℂ^{∗})^{n}, any member Vb̄ = V (x̄, b̄) of the family and any irreducible component W of Vb̄ ∩ ā · T (with ā in W) there is some i in {0, …, r} with W ⊆ ā · T_{i} and dim(T_{i}) − dim(V ∩ ā · T_{i}) = dim T − dim W.

Moreover, the minimal torus of every cd-maximal subvariety of Vb̄ belongs to the collection {T_{0}, …, T_{r}}.

We will assume throughout this article that the above tori are all distinct, and T_{0} = (ℂ^{∗})^{n} and T_{1} = {1}^{n}.

Note 2.2. Only the second claim of theorem 2.1 is needed for our purposes.

Corollary 2.3. Let V (x̄, z̄) be a uniformly definable family of varities. Then:

(1) If T is the minimal torus of Vb̄, there exists some θ(z̄) ∈ tp(b̄), such that T is the minimal torus of every Vb̄′ with ⊧ θ(b̄′). In particular, there is a definable neighbourhood of b̄ where l. dim_{ℚ} and cd remain constant.

(2) Suppose V (x̄, b̄) decomposes into m irreducible components W_{k} with d_{k} := dim(W_{k}), l_{k} := l. dim_{ℚ}(W_{k}) and c_{k} := cd(W_{k}). Then, there is some θ(z̄) ∈ tp(b̄) such that for all b̄′ ⊧ θ we have that Vb̄′ decomposes into exactly m irreducible components (W′_{k} : 1 ≤ k ≤ m) and (possibly after permutation) dim(W′_{k}) = d_{k}, l. dim_{ℚ}(W′_{k}) = l_{k} and cd(W′_{k}) = c_{k}.

Proof. The first claim follows from 2.1, since irreducible components are cd-maximal. So their minimal tori lie in {T_{0}, …, T_{r}}.

The second claim follows easily from a classical result of algebraic geometry and (1), since the decomposition into irreducible components is definable. □

3. δ-Arithmetic

Let now ℂ denote a large algebraically closed field of characteristic 0, i.e. a universal model of ACF_{0}. We consider the reduct L_{mult} := {·, 1, 0, =} ⊆ L_{Ring} = {+, ·, 1, 0, =} and T_{mult} := T h_{L_{mult}}(ℂ). The prime model of T_{mult} is obtained by adjoining 0 to the set of roots of unity μ(ℂ) ⊆ ℂ. The structure so obtained is L_{mult}- isomorphic to ℚ/ℤ (multiplicatively), after adding a new element. In particular, T_{mult} is modular non-trivial, and its geometry is projective over ℚ.

Given A ⊆ ℂ, denote by ⟨A⟩ the divisible hull of A^{∗} := A\{0} in ℂ^{∗} together with 0, equivalently, the algebraic closure of A in ℂ with respect to L_{mult}. This yields the prime model of T_{mult} over A. For a tuple ā ∈ ℂ^{∗} we have that MR(ā) in T_{mult} agrees with its linear dimension over ℚ (modulo torsion). We will write l. dim_{ℚ}(ā), and let dim(ā/B) denote Morley rank in ACF_{0}, equivalently, the dimension of its locus over B. The latter agrees with the transcendence degree of ā over B.

Consider now ⟨·⟩-closed subsets B of ℂ as an L_{Ring}-structure. Generally these structures need not be subfields of ℂ, however we will (after a possible Morleyzation L_{Morley} of ACF_{0} consisting of L_{mult} and relational symbols) formally work with these structures. Therefore, we will consider addition on ⟨·⟩-closed sets and refer to their fraction fields, their algebraic closure, etc. However this shall not confuse the unfortunate reader. Let D be the class of such structures as above described. Embeddings in D are elementary embeddings as L_{Ring}-structures, i.e. L_{Morley}- embeddings. A structure A ∈ D is finitely generated over B ⊆ A if A = ⟨āB⟩ for some finite tuple ā ∈ A. Finitely generated elements of D over B correspond to divisible hulls of subgroups of ℂ^{∗} of finite rank over ⟨B⟩ (adjoining 0). In case that ⟨AB⟩ is finitely generated over ⟨B⟩, we define

δ(A/B) := 2 dim(A/B) − l. dim_{ℚ}(A/B).

Clearly, δ(A/B) equals δ(⟨AB⟩/⟨B⟩). Since T_{mult} is modular, the following holds:

Lemma 3.1.

(1) δ(āb̄/C) = δ(b̄/C) + δ(ā/b̄C).

(2) Given C ⊆ B ∈ D, we have that δ(ā/B) ≤ δ(ā/B ∩ ⟨Cā⟩). (Submodularity)

Lemma 3.2. Given ā and B, let W be the locus of ā over acl(B). Then:

δ(ā/ acl(B)) = dim(W) − cd(W)

Moreover,

δ(ā/B) = dim(W) − cd(W) − l. dim_{ℚ}(⟨āB⟩ ∩ acl(B)/B).

Proof. For the first claim it suffices to observe that the smallest torus coset (over an algebraically closed set B) containing ā is B-definable in L_{mult}. Hence, its dimension equals l. dim_{ℚ}(ā/B). Modularity of T_{mult} gives the second statement. □

Definition 3.3.

• Let M ⊆ N ∈ D with l. dim_{ℚ}(N/M) = n ≥ 2. The extension N/M is minimal prealgebraic (of length n) if δ(N/M) = 0 and δ(N′/M) > 0 for every N′ = ⟨N′⟩ ∈ D with M ⊊ N′ ⊊ N.

• Let B ⊆ ℂ. A strong type p(x̄) ∈ S^{n}(B) (in ACF_{0}) is minimal prealgebraic, if the extension ⟨Bā⟩/⟨B⟩ is minimal prealgebraic of length n for some ā ⊧ p (in particular, ā is multiplicatively independent over B).

• A formula φ(x̄) of Morley degree 1 is minimal prealgebraic if its generic type is minimal prealgebraic.

Note 3.4.

• The condition δ(N′/M) > 0 is equivalent to δ(N/N′) < 0 since 0 = δ(N/M) = δ(N/N′) + δ(N′/M).

• If N/M is minimal prealgebraic and n̄ is a multiplicative basis of N over M, then stp(n̄/M) is minimal prealgebraic.

• Minimal prealgebraicity is invariant under parallelism class and multiplicative translation for strong types. In particular, the notion of minimal prealgebraic for stationary formulae in Definition 3.3 is well-defined.

4. Codes

We first aim to encode minimal prealgebraic extensions. Note that every strong type is the generic type of some variety. Hence, we can define the following:

Definition 4.1. A variety V = V (x̄, b̄) is a code variety if it is minimal prealgebraic. Equivalently, if for all B = ⟨B⟩ ∋ b̄ the extension B ⊆ ⟨Bā⟩ is minimal prealgebraic for some B-generic ā ∈ V.

Note that a multiplicative translation of a code variety is again a code variety. Let M ∈ D and N = ⟨Mā⟩ where tp(ā/M) is minimal prealgebraic of length

n. Consider N′ = ⟨N′⟩ with M ⊊ N′ ⊊ N. Modularity of T_{mult} yields some L_{mult}-basis ā′ ∈ ⟨ā⟩ for N′ over M of length m. Modulo torsion we have that a′_{j} = ∏ a_{i}^{λ_{ij}} for some λ_{ij} ∈ ℚ. After substitution with suitable powers or roots, we may assume that λ_{ij} ∈ ℤ and (λ_{ij})_{i<n} are coprime for j < m. Then, the equations (∏_{i<n} x_{i}^{λ_{ij}} = 1 : j < m) determine a torus T of dimension d = n − m. Since T is ∅-definable, it follows that ā′ is the canonical basis [āT] of the coset āT.

Similarly, given a torus T in (ℂ^{∗})^{n} of dim(T) = d, the element ā′ := [āT] generates a substructure N′ := ⟨Mā′⟩ ⊆ N with l. dim_{ℚ}(N′/M) = m = n − d.

Lemma 4.2. Let V (x̄, b̄) ⊆ ℂ^{n} be a variety, T ⊆ ℂ^{n} some torus and ā in V generic over B ∋ b̄. Then, an element ā_{1} ∈ W := V ∩ āT is generic in W over B[āT] if and only if it is generic in Vb̄.

Proof. Note that ā′ := [āT] is definable multiplicatively over any ā_{2} ∈ āT. In particular, it is definable over ā_{1} ∈ W. Hence:

dim(ā_{1}/Bā′) = dim(ā_{1}ā′/B) − dim(ā′/B) = dim(ā_{1}/B) − dim(ā′/B)
≤ dim(ā/B) − dim(ā′/B) = dim(ā/Bā′),

which proves the claim. □

The following result was already stated in [15]; however we exhibit a proof for completeness since similar ideas will appear later on.

Lemma 4.3. Let V (x̄, z̄) be a uniformly definable family of varieties. If Vb̄ = V (x̄, b̄) is a code variety then there is a formula θ(z̄) ∈ tp(b̄) such that Vb̄_{1} is a code variety for every b̄_{1} ⊧ θ.

Proof. Let {T_{0}, …, T_{r}} be the collection of tori associated to V (x̄, z̄) as in 2.1. Take some B containing b̄. Set

n = l. dim_{ℚ}(Vb̄) = 2k
k = dim(Vb̄).

Clearly ⟨Bḡ⟩ ∩ acl(B) = ⟨B⟩ for B–generic ḡ ∈ Vb̄.

Let θ(z̄) express:

(1) dim(Vz̄) = k and l. dim_{ℚ}(Vz̄) = n (in particular Vz̄ ≠ ∅),

(2) given generic ḡ in Vz̄, i = 2, …, r and W some irreducible component of V ∩ ḡT_{i} of maximal dimension, then cd(W) > dim(W) if V ∩ ḡT_{i} is infinite.

The existence of such a formula θ follows from Corollary 2.3.

Now we show that ⊧ θ(b̄). Let ḡ in Vb̄ be generic and take T ≠ T_{0}, T_{1} some torus with V ∩ ḡT infinite. Choose some irreducible component W ⊆ V ∩ ḡT of maximal dimension. By Lemma 4.2 we have that ḡ is generic in W over acl((b̄, [ḡT]). Hence, ḡ ∉ acl(b̄, [ḡT]) and [ḡT] ∉ acl(b̄). By Lemma 3.2 and miminal prealgebraicity of the extension ⟨ḡb̄⟩/⟨b̄⟩

dim(W) − cd(W) = δ(⟨ḡb̄⟩/ acl(b̄, [ḡT])) = δ(⟨ḡb̄⟩/ acl(b̄, [ḡT]) ∩ ⟨ḡb̄⟩)) < 0.

Now let b̄_{1} ⊧ θ and ḡ generic in Vb̄_{1} over B_{1} ∋ b_{1}. Condition (1) in θ(z̄) yields δ(ḡ/B_{1}) = 0. We need only show that δ(ḡ/[ḡT], B_{1}) < 0 for every torus T with 1 ≤ d := dim(T) ≤ n − 1. Set ḡ′ := [ḡT] and let W be the locus of ḡ over acl(B_{1}ḡ′). We consider three cases: Case 1: ḡ ∈ acl(B_{1}ḡ′), i.e. W is a point. Then

δ(ḡ/B_{1}ḡ′) = − l. dim_{ℚ}(ḡ/B_{1}ḡ′) = l. dim_{ℚ}(ḡ′/B_{1}) − l. dim_{ℚ}(ḡḡ′/B_{1})
= l. dim_{ℚ}(ḡ′/B_{1}) − l. dim_{ℚ}(ḡ/B_{1}) = (n − d) − n = −d < 0.

Case 2: cd(W) = cd(Vb̄_{1}). Since W ⊊ Vb̄_{1} and by Lemma 3.2

δ(ḡ/B_{1}ḡ′) ≤ dim(W) − cd(W) < dim(Vb̄_{1}) − cd(Vb̄_{1}) = 0.

Case 3: W is infinite and cd(W) < cd(Vb̄_{1}). Choose some W ⊆ W̃ ⊆ Vb̄_{1} irreducible maximal such that cd(W̃) ≤ cd(W). Then, W̃ is cd-maximal with minimal Torus T_{i} in the above collection. Note that i ≠ 0 because cd(W) < cd(Vb̄_{1}), and i ≠ 1 because W is infinite. So, W̃ ⊆ V ∩ ḡT_{i}. Take now some irreducible component W̃ ⊂ W̃′ of V ∩ ḡT_{i} of maximal dimension. By cd-maximality cd(W̃′) > dim(W̃′). So

δ(ḡ/B_{1}ḡ′) ≤ dim(W) − cd(W) ≤ dim(W̃) − cd(W̃) ≤ dim(W̃′) − cd(W̃′) < 0.

□

The above proof points out how to find — uniformly — a generic (definable) subset φ_{1}(x̄, b̄) of a code variety V (x̄, b̄) such that φ_{1}(x̄, b̄) is strongly minimal in the theory T_{ω} of the generic (non-collapsed) green field in case we color x̄ green. Given a uniformly definable family V (x̄, z̄) of code varieties whose associated tori are {T_{0}, …, T_{r}} as in 2.1, we can define φ_{1}(x̄, z̄) ⊆ V (x̄, z̄) as follows:

• ā ∈ Vb̄ realizes φ_{1}(ā, b̄) if and only if for i = 2, …, r the following condition holds: In case Vb̄ ∩ āT_{i} is infinite then cd(W) > dim(W) for all irreducible components W of Vb̄ ∩ āT_{i} of maximal dimension.

The above condition is definable by Corollary 2.3.

Lemma 4.4. Let (Vz̄(x̄) : z̄ ⊧ θ) be a family of code varieties and φ_{1}(x̄, z̄) be as above. Then φ_{1}(x̄, b̄) is generic in Vb̄ for b̄ ⊧ θ. Moreover, for all ā ⊧ φ_{1}(x̄, b̄) and B ∋ b̄ the following holds:

(1) δ(ā/B) ≤ 0.

(2) If δ(ā/B) = 0, then either ā ∈ ⟨B⟩ or ā is generic in Vb̄ over B.

Proof. The proof of Lemma 4.3 shows that φ_{1}(x̄, b̄) is generic in Vb̄. Suppose now that ā is neither generic in Vb̄ nor contained in ⟨B⟩. We need to show that δ(ā/B) < 0. Let W be its locus over acl(B). As in the previous proof, we consider three cases: Case 1: ā ∈ acl(B) (equivalently, W is a single point). Then,

δ(ā/B) = − l. dim_{ℚ}(ā/B) < 0.

Case 2: W is infinite with cd(W) = cd(Vb̄). As above W ⊊ Vb̄, so δ(ā/B) ≤ dim(W) − cd(W) < dim(Vb̄) − cd(Vb̄_{1}) = 0. Case 3: W is infinite with cd(W) < cd(Vb̄). A similar argument as in 4.3 yields the claim. Remark that the conditions on genericity and θ are now part of φ_{1}(x̄, b̄). □

Given two formulae φ(x̄) and ψ(x̄) we write φ(x̄) ∼ ψ(x̄) if the Morley rank of their symmetric difference is less than MR(φ(x̄)). Therefore MR(φ) = MR(ψ) and ∼ ist symmetric. Let p be a minimal prealgebraic (strong) type, B some set of parameters and ḡ ⊧ p|B. Then ḡ is a L_{mult}-basis of the minimal prealgebraic extension ⟨B⟩ ⊆ ⟨Bḡ⟩. Moreover, dim(ḡ/B) = k = \frac{n}{2} and δ(ḡ/[ḡT], B) < 0 for every torus T different from T_{0} and T_{1}. Minimal prealgebraic extensions are preserved under affine transformations (which correspond to bases change in L_{mult}). Same holds for strong types as well as degree 1 formulae. Let us explain this carefully: consider ℂ^{∗} as a ℚ-vector space modulo torsion. Then, the group GL_{n}(ℚ) acts on the set of strong types modulo torsion. If X_{1} and X_{2} are two definable sets in (ℂ^{∗})^{n} of degree 1 and T ⊆ (ℂ^{∗})^{n} × (ℂ^{∗})^{n} is an n-dimensional torus such that (X_{1} × X_{2}) ∩ T projects generically onto both X_{1} and X_{2}, then T induces a toric correspondence between X_{1} and X_{2}. The following holds:

Lemma 4.5. Let φ(x̄) be minimal prealgebraic.

• If there is a toric correspondence between φ(x̄) and some formula ψ(x̄) of Morley degree 1, then ψ(x̄) is also minimal prealgebraic.

• Let m̄ ∈ (ℂ^{∗})^{n}. Then φ(x̄ · m̄) is minimal prealgebraic. □

Definition 4.6. Let X ⊆ (ℂ^{∗})^{n} be definable set of degree 1. A formula φ(x̄, z̄) and a torus T encode X if there is some b̄ such that T induces a toric correspondence between φ(x̄, b̄) and X. We say that φ encodes X if the above correspondence is the identity (i.e. φ(x̄, b̄) ∼ X).

Definition 4.7. A code α are integers n_{α}, k_{α} and a ∅-definable formula φ_{α}(x̄, z̄) satisfying the following:

(a) The length of x̄ is n_{α} = 2k_{α}.

(b) φ_{α}(x̄, b̄) is a subset of (ℂ^{∗})^{n_{α}}.

(c) φ_{α}(x̄, b̄) is either empty or has Morley rank k_{α} and Morley degree 1.

(d) If φ_{α}(x̄, b̄) ≠ ∅, then φ_{α}(x̄, b̄) is minimal prealgebraic with irreducible Zariski closure V_{α}(x̄, b̄).

(e) Suppose φ_{α}(x̄, b̄) ≠ ∅. Then δ(ā/B) ≤ 0 for every b̄ ∈ B and ā ⊧ φ_{α}(x̄, b̄). Moreover, δ(ā/B) = 0 if and only if ā ∈ ⟨B⟩ or ā is B-generic in φ_{α}(x̄, b̄).

(f) φ_{α}(x̄, z̄) encodes every multiplicative translate of φ_{α}(x̄, b̄).

(g) If ∅ ≠ φ_{α}(x̄, b̄) ∼ φ_{α}(x̄, b̄′), then b̄ = b̄′.

If follows from (g) that b̄ is the canonical basis of the minimal prealgebraic type determined by φ_{α}(x̄, b̄).

Lemma 4.8. Every minimal prealgebraic definable set X can be encoded by some code α.

Proof. Let V_{α}(x̄, b̄) be the variety associated to X. Then V_{α}(x̄, b̄) is a code variety. By Lemma 4.3 there is some formula θ(z̄) such that V_{α}(x̄, b̄′) — and all its multiplicative translates — is a code variety for b̄′ ⊧ θ. Let now φ_{1}(x̄, z̄) ⊆ V_{α}(x̄, z̄) be as in in Lemma 4.4. Note that for every multiplicative translate V_{α}(x̄ · m̄, z̄) the corresponding translate φ_{1}(x̄ · m̄, z̄) ⊆ V_{α}(x̄ · m̄, z̄) yields the claim in Lemma 4.4. Set now

φ_{α}(x̄, z̄z̄′) := V (x̄ · z̄′, z̄) ∧ φ_{1}(x̄ · z̄′, z̄) ∧ θ(z̄).

Therefore, φ encodes X and satisfies properties (a)-(f).

By definability of ∼-equivalence and elimination of imaginaries we may assume that φ also satisfies (g). □

Set now θ_{α}(z̄) := ∃x̄ φ_{α}(x̄, z̄).

Lemma 4.9. Let α and β be codes. Then there is a finite set G(α, β) of tori in (ℂ^{∗})^{2n_{α}} such that if φ_{α}(x̄, b̄) ≠ ∅ and T induces a toric correspondence between φ_{α}(x̄, b̄) and φ_{β}(x̄, b̄′), then T ∈ G(α, β).

Proof. If there is no such toric correspondence between any instances of α and β, then set G(α, β) := ∅. Otherwise, let T, b̄ and b̄′ as above. Let V_{α} (resp. V_{β}) be the family of code varieties associated to α (resp. β). Moreover, let {T_{0}, …, T_{ν}} be the finite collection of tori as in 2.1 for V_{α} × V_{β}. Set B := acl(b̄b̄′). Choose some B–generic point (ā, ā′) in (V_{α}(x̄, b̄) × V_{β}(x̄′, b̄′)) ∩ T.

Let W ⊆ (V_{α} × V_{β}) ∩ T be the locus of (ā, ā′) over B. Then T is the minimal torus of W, and dim(W) = cd(W) = k_{α}.

If we show that W ⊆ (V_{α}×V_{β}) is cd-maximal, then T lies in {T_{0}, …, T_{ν}}. Choose now some variety W ⊊ W′ ⊆ (V_{α} × V_{β}). We may assume that W′ is B–definable. If (ḡ, ḡ′) is B–generic in W′, then cd(W′) = l. dim_{ℚ}(ḡ, ḡ′/B)−dim(ḡ, ḡ′/B). Hence,

cd(W′) = [l. dim_{ℚ}(ḡ/B) − dim(ḡ/B)] + [l. dim_{ℚ}(ḡ′/Bḡ) − dim(ḡ′/Bḡ)]

= cd(W) + l. dim_{ℚ}(ḡ′/Bḡ) − dim(ḡ′/Bḡ) > cd(W) − δ(ḡ′/Bḡ) ≥ cd(W).

The proper inequality follows from dim(ḡ′/Bḡ) > 0 because W ⊊ W′. The last inequality follows from the fact that ḡ′ is B–generic in V_{β}, so it also realizes φ_{β}(x̄, b̄′).

□ Theorem 4.10. There exists a collection C of codes such that every minimal prealgebraic definable set can be encoded by a unique element α in C and finitely many tori.

Proof. The collection C will be obtained as an increasing union of finite sets constructed by recursion. Encode first all minimal prealgebraic subsets of (ℂ^{∗})^{n} for every n. Fix some n ≥ 2 and list all minimal prealgebraic subsets (X_{i} : i < ω) of (ℂ^{∗})^{n} up to isomorphism. Let α_{0} encode X_{0} as in 4.8. Define C_{0} = {α_{0}}.

Suppose by induction that C_{i} has been already defined encoding all X_{j}’s with j ≤ i. If X_{i+1} can be encoded by some element in C_{i} and some torus T, then set C_{i+1} = C_{i}. Otherwise find some code α_{i+1} and b̄_{0} as in 4.8 encoding X_{i+1}. Define:

ρ(z̄) := ∀ȳ( ⋀_{k=0}^{i} ⋀_{T ∈G(α_{k},α_{i+1})} ¬χ_{α_{k},α_{i+1}}^{T}(ȳ, z̄)) ,

where χ_{α,β}^{T}(b̄, b̄′) expresses that T induces a toric correspondence between φ_{α}(x̄, b̄) and φ_{β}(x̄, b̄′) (this is a definable condition).

Now, φα̂(x̄, z̄) := φ_{α_{i+1}}(x̄, z̄) ∧ ρ(z̄) satisfies properties (a)–(g) in Definition 4.7. We claim that it also satisfies property (f). Let m̄ be in (ℂ^{∗})^{n} such that φα̂(x̄·m̄, b̄) cannot be encoded by φα̂. Equivalently, there are some k ≤ i and some torus T ∈ G(α_{k}, α_{i+1}) such that T induces a toric correspondence between φ_{α_{k}}(x̄, b̄_{1}) and φ_{α_{i+1}}(x̄ · m̄, b̄). Find m̄_{1} ∈ (ℂ^{∗})^{n_{α}} with (m̄_{1}, m̄) ∈ T. Then T induces a toric correspondence between φ_{α_{k}}(x̄ · m̄_{1}^{−1}, b̄_{1}) and φ_{α_{i+1}}(x̄, b̄). By property (f) the set φ_{α_{k}}(x̄ · m̄_{1}^{−1}, b̄_{1}) can be encoded by φ_{α_{k}}, which yields a contradiction. Hence, set C_{i+1} = C_{i} ∪ {α̂}.

Given some minimal prealgebraic definable set X, there is some X_{i} such that X and X_{i} are isomorphic. By construction, there exists a unique code α ∈ C and finitely many tori which encode X_{i} (and hence X): finiteness of the set of tori follows from G(α, α) being finite (see Lemma 4.9). □

Definition 4.11. A good code is an element of C.

5. Difference sequences

Recall the following result due to M. Ziegler (in a more general setting) in unpublished work [18].

Lemma 5.1. Let A be an algebraically closed subset. If the tuples ā, b̄ and ā · b̄ are pairwise independent over A, then tp(ā/A) is the generic type of an A-definable coset of a torus.

This result is extremely relevant for our purposes due to an observation by Mustafin [12], who noticed that no code variety could be a coset of a torus.

Lemma 5.2. Let V be a code variety. Then its multiplicative stabiliser is finite. In particular, V is no coset of a torus.

Proof. For V = V (x̄, b̄) as above, choose T = stab(V)^{0} the connected component of its multiplicative stabiliser. By construction T is a torus. Choose now generic b̄-independent elements ḡ in V and ā in T. By definition, the element ā · ḡ is generic in V over b̄. Therefore δ(ā/ḡ, b̄) = δ(ā/b̄) = dim(T). On the other hand, δ(ā · ḡ/ḡ, b̄) ≤ 0, since V is a code variety. Since ā · ḡ and ā are ḡb̄-interdefinable (multiplicatively), it follows that dim(T) = 0. □

Definition 5.3. Let (ē_{0}, …, ē_{λ}) a sequence of length λ + 1. The i^{th} derivation ∂_{i} maps

(ē_{0}, …, ē_{λ})

to

(ē_{0} · ē_{i}^{−1}, …, ē_{i−1} · ē_{i}^{−1}, ē_{i}^{−1}, ē_{i+1} · ē_{i}^{−1}, …, ē_{λ} · ē_{i}^{−1}).

A sequence obtained after composing the operators (∂_{i})_{i≤λ} finitely many times is a difference sequence. If ν < λ and we only consider operators (∂_{i})_{i≤ν}, then we call the resulting sequence a ν–difference sequence.

Note 5.4. For every λ there exist only finitely many different derivations (precisely (λ + 2)! many). Moreover, the set of derivations of a given fixed sequence is closed under permutations, since the transposition (ij) equals ∂_{j} ∘ ∂_{i} ∘ ∂_{j}.

Fix for each α ∈ C a positive integer m_{α} such that every b̄ ⊧ θ_{α} lies in the definable closure of some (any) Morley sequence of φ_{α}(x̄, b̄) of length m_{α}.

Theorem 5.5. For every α in C and λ ≥ m_{α} there is some formula ψ_{α}(x̄_{0}, …, x̄_{λ}) (whose realizations will be called difference sequences) satisfying the following:

(h) If ⊧ ψ_{α}(ē_{0}, …, ē_{λ}), then ē_{i} ≠ ē_{j} for i ≠ j.

(i) Given b̄ ⊧ θ_{α} and a Morley sequence {ē_{0}, …, ē_{λ}, f̄} for φ_{α}(x̄, b̄), then

⊧ ψ_{α}(ē_{0} · f̄^{−1}, …, ē_{λ} · f̄^{−1}).

(j) For any realization (ē_{0}, …, ē_{λ}) von ψ_{α} there exists a unique b̄ with ⊧ φ_{α}(ē_{i}, b̄) for i = 0, …, λ. Moreover, b̄ lies in the definable closure of any segment of length m_{α} of the ē_{i}’s. Hence, b̄ is called the canonical parameter of the sequence ē_{0}, …, ē_{λ}.

(k) If ⊧ ψ_{α}(ē_{0}, …, ē_{λ}), then ⊧ ψ_{α}(ē_{0}, …, ē_{λ′}) for each m_{α} ≤ λ′ < λ.

(l) Let i ≠ j and (ē_{0}, …, ē_{λ}) be a realization of ψ_{α} with canonical parameter b̄ as in (j). If there is some T in G(α, α) and ē′_{j} with (ē_{j}, ē′_{j}) ∈ T, then ē_{i} |̸ ⌣b̄ ē′_{j} ·ē_{i}^{−1} in case ē_{i} is a generic realization of φ_{α}(x̄, b̄).

(m) If ⊧ ψ_{α}(ē_{0}, …, ē_{λ}), then ⊧ ψ_{α}(∂_{i}(ē_{0}, …, ē_{λ})) for i ∈ {0, …, λ}.

Proof. We find ψ_{α}(x̄_{0}, …, x̄_{λ}) inductively in λ. Consider the following type-definable property Σ(ē_{0}, …, ē_{λ}):

there exist some b̄′ and a Morley sequence ē′_{0}, …, ē′_{λ}, f̄ of φ_{α}(x̄, b̄′)

with ē_{i} = ē′_{i} · f̄^{−1}.

It is easy to see that Σ has properties (h)–(k) and (m). Note that (ē_{i} : i ≤ λ) is a Morley sequence over b̄′f̄. In particular, its canonical parameter b̄ lies in dcl(b̄′f̄). Let T ∈ G(α, α) and (ē_{j}, ē_{j}^{∗}) ∈ T. Then ē_{j}^{∗} ∈ acl(ē_{j}), so ē_{j}^{∗} | ⌣b̄ ē_{i} for i ≠ j. If ē_{i} | ⌣b̄ ē_{j}^{∗} · ē_{i}^{−1}, then ē_{i}^{−1}, ē_{j}^{∗} and ē_{j}^{∗} · ē_{i}^{−1} will determine a pairwise b̄–independent triple, since

MR(ē_{j}^{∗}/b̄, ē_{j}^{∗} · ē_{i}^{−1}) = MR(ē_{i}^{−1}/b̄, ē_{j}^{∗} · ē_{i}^{−1}) = MR(ē_{i}^{−1}/b̄) = MR(ē_{j}/b̄) = MR(ē_{j}^{∗}/b̄)

so ē_{j}^{∗} | ⌣b̄ ē_{j}^{∗} ·ē_{i}^{−1}. By Lemma 5.1 the type tp(ē_{i}^{−1}/b̄) will be the generic type of some torus coset. Likewise for tp(ē_{i}/b̄). This contradicts Lemma 5.2 since ē_{i} is generic in φ_{α}(x̄, b̄). Therefore, property (l) also holds.

Find now by compactness a finite set ψ_{0} in Σ which implies (h)-(l). Define

ψ_{α}(x̄_{0}, …, x̄_{λ}) := ⋀_{∂ derivation} ψ_{0}(∂(x̄_{0}, …, x̄_{λ})).

□

6. Green up!

From now on we will consider an extension L^{∗} := L_{Morley} ∪ {Ü}, where Ü is a new unary predicate which determines the green coloring (from the german word grün). An L^{∗}-structure is a pair (A, Ü(A)) consisting of a structure A in D (i.e. A = ⟨A⟩ ⊆ ℂ) and a divisible torsion-free subgroup Ü(A) of A \ {0}, that is, a ℚ-vector space. Given two L^{∗}-structures B and A, we write B ⊆ A in case B ⊂ A as elements of D and Ü(A) ∩ B = Ü(B).

The δ-function introduced in Section 3 will be modified accordingly: Given an L^{∗}-structure A finitely generated with respect to ⟨·⟩, set

δ(A) := 2 dim(A) − l. dim_{ℚ}(Ü(A)).

If B ⊆ A and A is f.g. over B, or more generally both l. dim_{ℚ}(Ü(A)/ Ü(B)) and dim(A/B) are finite, define

(6.1)  δ(A/B) := 2 dim(A/B) − l. dim_{ℚ}(Ü(A)/ Ü(B)).

This agrees with Poizat’s context (cf. [15, Section 3]). Hence,bases, generated, linearly (in)dependent refer to the underlying L_{mult}-structure. Similarly, generic, Morley sequence, (in)dependent, transcendental or algebraic refer to the theory ACF_{0}, unless otherwise specified. From now no, acl(M) denotes the field theoretical algebraic closure of M.

Note 6.1. Let B ⊆ A as above. If A has a green basis over B, that is A = ⟨Ü(A)B⟩, then the previous definition agrees with the definition of δ(A/B) in Section 3. In particular, property (e) in Definition 4.7 holds for green realizations of a code instance φ_{α}(x̄, b̄).

Given B ⊆ A, we say that B is strong in A, if δ(A′/B) ≥ 0 for every A′ = ⟨A′⟩ ⊂ A f.g over B. We denote it by B ≤ A. If there is some b̄ ∈ B generating B, then we write b̄ ≤ A in case B ≤ A. Similarly, set δ(ā/b̄) := δ(⟨āb̄⟩/⟨b̄⟩).

Lemma 6.2. Suppose all structures can be embedded into a common L^{∗}-structure M. Then:

(1) For B ⊆ C ⊆ A, we have δ(A/B) = δ(C/B) + δ(A/C).

(2) δ(⟨AB⟩/B) ≤ δ(A/A ∩ B). (Submodularity)

(3) If C ≤ M and C′ ≤ M, then C ∩ C′ ≤ M.

(4) For every A ⊆ M there exists a unique A ⊆ C = ⟨C⟩ ≤ M minimal such. We call such a set the strong closure of A (in M) and denote it by cl_{M}(A).

(5) If (A_{i})_{i<α} is an increasing sequence with A_{i} ≤ K for all i, then ⋃_{i} A_{i} ≤ M.

Now consider the class of L^{∗}-Structures K := {M | ∅ ≤ M}. Unlike in [15] we are not interested in L^{∗}-structures whose underlying L_{Ring}-structure is an algebraically closed field but mere expansions of structures in D with hereditarily non-negative predimension function δ.

Assumption 6.3. From now on, δ will be as in in (6.1). A realization of a code or a difference sequence will consist exclusively of green elements, unless otherwise specified. Likewise, a minimal prealgebraic extension M ≤ N in K is a minimal prealgebraic extension of structures N/M in D such that N has a green basis over M.

Given a strong extension B ≤ A in K with l. dim_{ℚ}(A/B) < ∞, we can find a decomposition B = A_{0} ≤ A_{1} ≤ … ≤ A_{n−1} ≤ A_{n} = A such that A_{i+1}/A_{i} is minimal strong for all i < n. Note that a strong extension M ≤ N in K (M ⊊ N) is minimal (strong) if there is no M′ = ⟨M′⟩ with M < M′ ≤ N. The following result is easy to prove.

Lemma 6.4. Let B ≤ A be a minimal extension. One of the following cases holds:

(1) algebraic: Ü(A) = Ü(B) and A = ⟨Ba⟩ for some a ∈ acl(B) \ B. Then, δ(A/B) = 0.

(2) white generic: Ü(A) = Ü(B) and A = ⟨Ba⟩ for some element a transcendental over B. Then, δ(A/B) = 2.

(3) green generic: A contains a basis consisting of a green singleton a over B. Moreover, a is transcendental over B and δ(A/B) = 1.

(4) minimal prealgebraic: B ≤ A is minimal prealgebraic as in 6.3, that is, A contains a green basis ā over B such that the (strong) type of ā over B is minimal prealgebraic. In this case δ(A/B) = 0.

The class K can be easily axiomatized as shown in [15]:

Theorem 6.5. Let (M, Ü(M)) be an L^{∗}-expansion of an algebraically closed field of characteristic 0. Then M ∈ K if and only if the following (definable) conditions hold:

(1) Ü(M) is a trosion-free divisible multiplicative subgroup of M.

(2) Ü(M) has no non-trivial algebraic points.

(3) For every ∅-definable variety V (x̄) ⊆ (ℂ^{∗})^{2n+1} of dimension n with associated tori {T_{0}, …, T_{r}} as in 2.1, then ā ∈ T_{i} for some i > 0 for all ā ∈ V ∩ Ü(M).

Note that (2) follows from (1) and (3).

The class K has the amalgamation property with respect to strong embeddings. Moreover, it has the JEP and contains only countably many f.g. structures up to isomorphism. Hence, the Fraïssé-Hrushovski limit M_{ω} of the subcollection of f.g. structures in (K, ≤) exists. We call M_{ω} the generic model of K. Let T_{ω} be the L^{∗}-theory of M_{ω}. Recall the following result from [15]:

Theorem 6.6. The generic model M_{ω} is ω-saturated. Its theory T_{ω} is ω-stable of Morley rank ω · 2. Moreover, Ü(M_{ω}) has Morley rank ω.

7. Green counts

This section contains the main result of a combinatorial flavour, which will be extremely useful in order to show that the generic model M_{ω} can be collapsed into a finite rank one.

Definition 7.1. Let A and M be elements in K with a common strong substructure B. An L^{∗}-structure M′ in K is an amalgam of A and M over B if A and M are strongly embedded in M′ over B and M′ = ⟨M, A⟩ (after identification of A and M with their images under their respective embeddings). In case M and A (or rather, their images in M′) are algebraically independent over B and M ∩ A = B, then M′ is a free amalgam.

Following [15] we obtain the following:

Lemma 7.2. Given M, A and B in K with B ≤ A and B ≤ M. Then there is an amalgam M′ in K of A and M over B such that A and M are algebraically independent over B. If B is algebraically closed in A or in M, then the amalgam can chosen to be a free amalgam.

The following Lemma yields a lower bound for the length of a difference sequence for a good code in order to recover a Morley segment inside the sequence over a strong subset of parameters.

Lemma 7.3. For every code α and every natural number n there exists a positive integer λ_{α}(n) = λ ≥ 0 such that given any strong extension M ≤ N in K and a difference sequence (ē_{0}, …, ē_{μ}) for α in N with canonical parameter b̄, we have that if μ ≥ λ then either the canonical parameter of a λ-derived sequence of (ē_{0}, …, ē_{μ}) lies in acl(M), or the sequence (ē_{0}, …, ē_{μ}) contains a Morley subsequence for φ_{α}(x̄, b̄) over M of length n.

Proof. Given (ē_{0}, …, ē_{μ}) as above such that first part of the statement does not hold, define:

X_{1} = {i ∈ [m_{α}, μ] : ē_{i} generic over M ∪ {ē_{0}, …, ē_{i−1}}},
X_{2} = {i ∈ [m_{α}, μ] : ē_{i} ⊆ ⟨M ∪ {ē_{0}, …, ē_{i−1}}⟩},
X_{3} = [m_{α}, μ] \ (X_{1} ∪ X_{2}).

After possible permutation of the set of indices, we may assume that X_{1} < X_{3} < X_{2} (note that we may have indices of X_{2} go to X_{3} and indices of X_{3} land in X_{1}). Since b̄ ∈ dcl(ē_{0}, …, ē_{m_{α}−1}), then by Property (e)

δ(ē_{i}/M, ē_{0}, …, ē_{i−1}) ≤ −1 for i ∈ X_{3} and
δ(ē_{i}/M, ē_{0}, …, ē_{i−1}) = 0 for i ∈ X_{1} ∪ X_{2}.

It follows from M ≤ N that

0 ≤ δ(ē_{0}, …, ē_{μ}/M) ≤ δ(ē_{0}, …, ē_{m_{α}−1}/M) + ∑_{i=m_{α}}^{μ} δ(ē_{i}/M, ē_{0}, …, ē_{i−1})
≤ m_{α}n_{α} + (−1)|X_{3}| ,

therefore |X_{3}| ≤ m_{α}n_{α}.

Let now r = m_{α} +|X_{1}|+|X_{3}|, and s = r(n_{α} +1). Take I ⊆ {ē_{r}, …, ē_{μ}} of cardinality |I| = rn_{α} + 1. To simplify the notation, assume that I = {r, …, s}. Choose varieties W_{0} ⊂ V_{0}, …, W_{t} ⊂ V_{t} (with the V_{i}’s irreducible) such that ψ_{α}(x̄_{0}, …, x̄_{s}) equals ⋃_{i≤t}(V_{i} \W_{i}). Let T_{0}, …, T_{ℓ} be the associated tori to the V_{i}’s as in 2.1. The point (ē_{0}, …, ē_{s}) lies in some V_{i_{0}} \ W_{i_{0}} for some i_{0} ≤ t. Let W be its locus over acl(M). Choose W ⊆ W′ ⊆ V_{i_{0}} maximal such that cd(W′) ≤ cd(W). By construction W′ is cd-maximal, so there is some j ∈ {0, … ℓ} such that T_{j} is its minimal torus. Fix some m̄ in W′ ⊆ m̄T_{j} ∩ V_{i_{0}}. We may assume that m̄ ∈ acl(M), since W ⊆ m̄T_{j} by acl(M)-definability of W′. Choose (ā_{0}, …, ā_{s}) a generic point of W′ over acl(M) and paint it green. Then, the point lies in V_{i_{0}} \ W_{i_{0}}, since (ē_{0}, …, ē_{s}) was an specialization of (ā_{0}, …, ā_{s}) and V_{i_{0}} \ W_{i_{0}} is Zariski open in its closure.

Therefore ψ_{α}(ā_{0}, …, ā_{s}) holds. Note that

r · n_{α} ≥ l. dim_{ℚ}(ē_{0}, …, ē_{r−1}/M) = l. dim_{ℚ}(ē_{0}, …, ē_{s}/M) ≥ cd(W) ≥ cd(W′)

= ∑_{i≤s} l. dim_{ℚ}(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) − dim(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1})

≥ ∑_{r≤i≤s} l. dim_{ℚ}(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) − dim(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}).

Property (e) yields that δ(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) ≤ 0 for i ≥ r ≥ m_{α}, that is,

2 dim(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) ≤ l. dim_{ℚ}(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}).

Hence, if ā_{i} ∉ ⟨acl(M), ā_{0}, …, ā_{i−1}⟩ then

l. dim_{ℚ}(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) − dim(ā_{i}/ acl(M), ā_{0}, …, ā_{i−1}) ≥ 1.

Therefore, there is some t ∈ {r, …, s} with ā_{t} ∈ ⟨acl(M), ā_{0}, …, ā_{t−1}⟩. The linear dependence will be determined by the coset m̄T_{j}. So m̄T_{j} also determines that ē_{t} ∈ ⟨acl(M), ē_{0}, …, ē_{t−1}⟩.

Consider now all possible pairs (t, j) as above. This determines a (rn_{α}+1)(ℓ+1)- coloring of all (rn_{α} + 1)-subsets of {r, …, μ}. By (finite) Ramsey’s theorem, there is some number μ_{0}, such that for μ ≥ μ_{0} there is a monochromatic subset I ⊆ {r, …, μ} of cardinality |I| ≥ m_{α}+rn_{α}+1. Equivalently, there is some t ∈ {r, …, s} and some j ≤ ℓ with

ē_{i_{t}} ∈ ⟨acl(M), ē_{0}, …, ē_{r−1}, ē_{i_{r}}, ē_{i_{r+1}}, …, ē_{i_{t−1}}⟩,

for all i_{r} < ⋯ < i_{s} in I. Moreover, the linear dependence comes from some m̄T_{j} with m̄ in acl(M) (note that the tuple m̄ ∈ acl(M) may change). Let γ_{i} be the (t + i)^{th} element in I. For i > 0 we have that ē_{γ_{i}}ē_{γ_{0}}^{−1} lies in acl(M). Hence, the canonical parameter of the difference sequence lies in

acl(∂_{γ_{0}}(ē_{0}, …, ē_{μ})) ⊆ acl(M),

which contradicts our assumption.

Hence, there is an upper bound for μ depending on r, hence a lower bound for X_{1} depending on μ. □

8. Prealgebraicity go home!

Choose now finite-to-one functions μ^{∗}, μ : C → ℕ such that:

μ^{∗}(α) ≥ n_{α}k_{α} + 1,
μ^{∗}(α) ≥ λ_{α}(m_{α} + 1),
μ(α) ≥ λ_{α}(μ^{∗}(α)).

where λ_{α} is the function obtained in Lemma 7.3.

Definition 8.1. Let K^{μ} be the subcollection of elements M in K such that no good code α has a (green) difference sequence in M of length at least μ(α) + 1.

The class K^{μ} is universally axiomatizable relative to ACF_{0}. We want now to obtain a theory T^{μ} whose models lie in K^{μ}. Moreover, green prealgebraic extensions of strong subsets will become algebraic in T^{μ}. In order to ensure that T^{μ} is complete, we will impose that every good code attains the maximal number of realizations and moreover describe all this in an elementary way.

Lemma 8.2. Let M ∈ K^{μ} and M′ ∈ K be a minimal prealgebraic extension of M not in K^{μ}. Given a good code α and a difference sequence (ē_{0}, …, ē_{μ(α)}) in M′ with canonical parameter b̄ in acl(M), then there exists some i such that all ē_{j}’s with i ≠ j lie in M and ē_{i} is an M-generic realization of φ_{α}(x̄, b̄) which generates M′ over M.

Proof. We may assume that M is algebraically close, for otherwise we may define an L^{∗}-structure on acl(M),by setting Ü(acl(M)) = Ü(M); minimality of the extension M′/M yields that M is algebraically closed in M′ so we could replace M′ by ⟨M′ acl(M)⟩. Since M ≤ M′, then it follows from property (e) that for each j either ē_{j} lies in M or is a generic realization of φ_{α}(x̄, b̄) over M. Since M ∈ K^{μ}, there must be some generic ē_{i}. By minimality of the extension, it follows that M′ = ⟨Mē_{i}⟩. Suppose there is another M-generic realization ē_{j} different from ē_{i}. Then M′ = ⟨Mē_{i}⟩ = ⟨Mē_{j}⟩, so there is some tuple m̄ in M and a toric correspondence induced by some T ∈ G(α, α) with

(ē_{i} · m̄, ē_{j}) ∈ T.

Let ē′_{j} := ē_{i} · m̄. In particular, ē′_{j} · ē_{i}^{−1} ∈ M. Since ē_{i} is M-generic, then

ē_{i} | ⌣_{b̄} ē′_{j} · ē_{i}^{−1},

which contradicts property (l) of a difference sequence. □

Corollary 8.3. Let M ∈ K^{μ} and M′ ∈ K be a minimal extension of M. If l. dim_{ℚ}(M′/M) = 1, then M′ lies also in K^{μ}. Otherwise, M′/M is minimal prealgebraic, and M′ does not lie in K^{μ} if and only if there is some good α ∈ C and a difference sequence (ē_{0}, …, ē_{μ(α)}) for α in M′ such that one of the following holds:

a) ē_{0}, …, ē_{μ(α)−1} lie in M and ⟨M, ē_{μ(α)}⟩ = M′. Moreover, α is the unique good code describing the extension M′ ≥ M.

b) There is some subsequence of length μ^{∗}(α) which is a Morley sequence for φ_{α}(x̄, b̄) over Mb̄, where b̄ is the canonical parameter of (ē_{0}, …, ē_{μ(α)}).

Proof. Consider first the case l. dim_{ℚ}(M′/M) = 1. If Ü(M′) = Ü(M), then there are no new green difference sequences in M′, so M′ ∈ K^{μ}. Otherwise, l. dim_{ℚ}(Ü(M′)/M) = 1 and M′ = ⟨Ü(M′), M⟩ (the green generic case). Suppose M′ is not in K^{μ}, then there is a good code α and a difference sequence (ē_{0}, …, ē_{μ(α)}) for α in M′ witnessing this fact. Let b̄ be the canonical parameter of some derived sequence and ē be a generic element over Mb̄. Then

l. dim_{ℚ}(M′/M) ≥ l. dim_{ℚ}(ē/Mb̄) ≥ 2,

which contradicts our assumption. By Lemma 8.2 there is no such derived sequence. However, Lemma 7.3 yields a contradiction since μ(α) ≥ λ_{α}(μ^{∗}(α)) and μ^{∗}(α) ≥ 1.

Let now M′ be a minimal prealgebraic extension of M. If a) or b) hold, then clearly M′ does not belong to K^{μ}. Conversely, if M′ is not in K^{μ}, then there is some good code α and a difference sequence (ē_{0}, …, ē_{μ(α)}) for α in M′ witnessing this fact. Let b̄ be its canonical parameter. If we may choose the difference sequence such that b̄ lies in acl(M), then case a) holds by Lemma 8.2. Otherwise no difference sequence has canonical parameter in acl(M), which yields case b) because μ(α) ≥ λ_{α}(μ^{∗}(α)) by Lemma 7.3.

In order to show that α is uniquely determined, consider another good code α′ different from α with M′ = ⟨M, ē_{μ(α′)}⟩. Then n_{α} = n_{α′} = l. dim_{ℚ}(M′/M) and the locus of (ē_{μ(α)}, ē_{μ(α′)}) over M determines a coset of a torus in G(α, α′). Since both α and α′ are good, G(α, α′) = ∅ by construction, obtaining hence the desired contradiction. □

Corollary 8.4. Given a good code α there is a ∀∃-sentence χ_{α} such that every structure M in K^{μ} satisfies χ_{α} if and only if it contains no minimal prealgebraic extension in K^{μ} given by α.

Proof. Let α ∈ C, M in K^{μ} and b̄ ∈ M such that a generic realization ā of φ_{α}(x̄, b̄) generates a minimal prealgebraic extension M[ā] := ⟨Mā⟩ over M. If M[ā] does not belong to K^{μ}, either case a) or b) in 8.3 hold.

a) is equivalent to the existence of some good code α′ and a difference sequence (ē_{0}, …, ē_{μ(α′)}) in M′, whose first μ(α′) many elements (and hence its canonical parameter) lie in M and M[ā] = ⟨Mē_{μ(α′)}⟩. By uniqueness, α = α′. Since M ≤ M[ā], it follows that ē_{μ(α)} is M-generic. Therefore, ā can be mapped to ē_{μ(α)} over M by some ℚ–basis change. Equivalently, there is some green tuple m̄ ∈ M and some torus T ∈ G(α, α) such that T induces a toric correspondence between ψ_{α}(ē_{0}, …, ē_{μ(α)−1}, x̄) and φ_{α}(x̄ · m̄, b̄). Finiteness of G(α, α) allows us to express this fact by an existential formula over b̄.

b) implies that there is some good code β and a difference sequence (ē_{0}, …, ē_{μ(β)}) in M[ā] with μ^{∗}(β) many M-linearly independent elements. We need only consider finitely many such β′s since

n_{β}μ^{∗}(β) ≤ l. dim_{ℚ}(M[ā]/M) = n_{α}.

Assume first the following

Assumption: ψ_{β} equals V_{1} \ W_{1}, where V_{1} is a irreducible variety and W_{1} ⊊ V_{1} is a proper subvariety, both acl(∅)-definable.

Let V_{0} = V_{α}(x̄, b̄) be the corresponding code variety for φ_{α}(x̄, b̄), that is, the locus of ā over acl(M). Set V = V_{0} × V_{1} and let {T_{0}, …, T_{r}} be the tori associated to V as in 2.1. Take W ⊆ V to be the locus of (ā, ē_{0}, …, ē_{μ(β)}) over acl(M). Note that cd(W) = cd(V_{0}) = k_{α} by 3.2. Since W projects generically onto V_{0}, then cd(W′) ≥ cd(W) for all W′ ⊇ W. Let now T be the minimal torus of W and m̄ ∈ M such that W ⊆ m̄T. The coset m̄T contains hence the green tuple (ā, ē_{0}, …, ē_{μ(β)}).

We call a torus coset c̄T gay (as in colorful, let the distinction be made) if it contains a green tuple . In case T is given by the equations ∏ x_{i}^{λ_{ij}} = 1 (j = 1, …, d), it is easy to see that c̄T is gay if and only if c′_{j} := ∏ c_{i}^{λ_{ij}} is green for j = 1, …, d. If c̄T is gay and T ⊆ T′ (that is, T′ lies on top of T), then c̄T′ is also gay.

Choose now some M-definable W′ maximal such that cd(W′) = cd(W) containing W. Hence, W′ is cd-maximal and its minimal torus equals some T_{i}. Moreover, W′ ⊆ m̄T_{i}. Since m̄T ⊆ m̄T_{i} and m̄T is gay, so is m̄T_{i}. Let (ā^{∗}, ē_{0}^{∗}, …, ē_{μ(β)}^{∗}) be generic in W′ over M. We may assume that ā^{∗} = ā since they have the same M-type. It follows from cd(W′) = cd(W) = cd(V_{0}) that

l. dim_{ℚ}(ā, ē_{0}^{∗}, …, ē_{μ(β)}^{∗}/ acl(M))− dim(ā, ē_{0}^{∗}, …, ē_{μ(β)}^{∗}/ acl(M))
= l. dim_{ℚ}(ā/ acl(M)) − dim(ā/ acl(M))

so

l. dim_{ℚ}(ē_{0}^{∗}, …, ē_{μ(β)}^{∗}/Mā) = dim(ē_{0}^{∗}, …, ē_{μ(β)}^{∗}/Mā) =: ℓ.

Choose now an L_{mult}-basis f_{0}, …, f_{ℓ−1} in (ē_{0}^{∗}, …, ē_{μ(β)}^{∗}) over Mā. The elements (f_{0}, …, f_{ℓ−1}) are hence algebraically independent over Mā. Gayness of m̄T_{i} yields a structure N in K if we paint (ā^{∗}, ē_{≤μ(β)}^{∗}) green (after closing it under ⟨·⟩). Note that

N = ⟨Māē_{0}^{∗}, …, ē_{μ(β)}^{∗}⟩ = ⟨M[ā]f_{0}, …, f_{ℓ−1}⟩,

where f_{0}, …, f_{ℓ−1} is a tuple of green independent generic elements. Set F_{i} := ⟨Māf_{0}, …, f_{i−1}⟩ and observe that F_{i} ≤ F_{i+1} gives a tower of green generic extensions for 0 ≤ i ≤ ℓ − 1. By Collorary 8.3 (repeatedly) we have that:

(*) M[ā] ∈ K^{μ} if and only if N ∈ K^{μ}. Now, (ē_{0}, …, ē_{μ(β)}) is a specialization of (ē_{0}^{∗}, …, ē_{μ(β)}^{∗}), both lying in F. By assumption ψ_{β} is Zariski open in F, so ⊧ ψ_{β}(ē_{0}^{∗}, …, ē_{μ(β)}^{∗}) since (ē_{0}, …, ē_{μ(β)}) realizes ψ_{β}. Therefore, the existence of a green difference sequence for β (of length μ(β)+1) implies the existence of another one in N = M[ā][f̄], which may be obtained by only finitely many possibilities. Conversely, it suffices to ensure the existence of (ē^{∗}, …, ē_{μ(β)}^{∗}) ∈ N to conclude that M[ā] ∉ K^{μ} by (*). Consider the following definable conditions:

There is a tuple m̄ ∈ M and an irreducible component W′ of V ∩ m̄T_{i} (where V = V_{0} × V_{1} and V_{0} = V_{α}(x̄, b̄) is as above) such that:

(1) The coset m̄T_{i} is gay.

(2) W′ projects generically onto V_{0}.

(3) cd(W′) = cd(V_{0}).

(4) ⊧ ψ_{β}(ē_{0}^{∗}, …, ē_{μ(β)}^{∗}) for generic (ā^{∗}, ē_{0}^{∗}, …, ē_{μ(β)}^{∗}) in W′.

Therefore, we obtain an existential sentence over b̄ for each T_{i}. The disjunction of all these formulae yields the desired sentence.

For the general case, decompose ψ_{β} into a finite union of locally closed sets V_{i} \ W_{i} (for 1 ≤ i ≤ t). We proceed as above for each i and form the disjunction of all the sentences so obtained. □

9. Fraïssé à la verte

This section shows that K^{μ} has the amalgamation property with respect to strong embeddings. Hence, we obtain a rich field as in [14]. Work done in previous sections yields now the following key result:

Lemma 9.1. Let A, B and M be structures in K^{μ}, where B is a common strong substructure of both A and M. Let M′ be their free amalgam over B and consider a difference sequence (ē_{0}, …, ē_{μ(α)}) in M′ for some good code α. Then there is some derived sequence whose canonical parameter lies either in acl(M) or in acl(A).

Proof. Suppose the statement does not hold. Then, find ē_{0}, ē_{1}, …, ē_{m_{α}} of length m_{α} + 1 which is a Morley sequence over both M and A after applying Lemma 7.3 twice by choice of μ(α) and μ^{∗}(α). Let E = {ē_{0}, …, ē_{m_{α−1}}}. The canonical parameter b̄ lies in dcl(E) and

ē_{m_{α}} | ⌣_{b̄} ME  und  ē_{m_{α}} | ⌣_{b̄} AE .

Decompose each tuple in E as a product of a green tuple in M and one in A and define E_{M} (resp. E_{A}) to be the collection of these factors in M (resp. A). Set E′ = E_{M} ∪ E_{A}. Then b̄ ∈ dcl(E′) and by interdefinability of E und E′ over M (resp. over A), we conclude that

ē_{m_{α}} | ⌣_{b̄} ME′  und  ē_{m_{α}} | ⌣_{b̄} AE′ ,

and

ē_{m_{α}} | ⌣_{BE′} M  und  ē_{m_{α}} | ⌣_{BE′} A .

Let ē_{m_{α}} = m̄ · ā with m̄ in M and ā in A. Since M | ⌣_{B} A then M | ⌣_{BE′} A, so {ē_{m_{α}}, m̄, ā} is a pairwise BE′-independent triple. This contradicts Lemma 5.2 by Lemma 5.1, since stp(ē_{m_{α}}/BE′) will be the generic type of a torus coset. □

An embedding B in A is strong if the image of B in A is a strong substructure.

Theorem 9.2. K^{μ} has the amalgamation property with respect to strong embeddings.

Proof. let B ≤ M and B ≤ A be structures in K^{μ}. We need to find a strong extension M′ of M in K^{μ} with B ≤ A′ ≤ M′, where A and A′ are B-isomorphic. Decomposing both B ≤ A and B ≤ M in minimal extensions, we may reduce it to the case where both A and M are minimal extensions of B. If any of them is algebraic, add the corresponding elements to the ⟨·⟩-closure (we obtain a new structure in K^{μ} since there are no new green points).

Otherwise, we may consider the free amalgam M′ of M and A over B by Lemma 7.2. If M′ ∈ K^{μ}, we are done. Otherwise, we need only show that M and A are B-isomorphic. It follows from Corollary 8.3 that both M and A are minimal prealgebraic over B. Note that only the first case in Lemma 9.1 may occur, so there is a good code α and a difference sequence (ē_{0}, …, ē_{μ(α)}) in M′ with canonical parameter b̄. By symmetry we may assume that b̄ lies in acl(M). After possible permutations we may assume by Lemma 8.2 that ē_{0}, …, ē_{μ(α)−1} lie in M and ē_{μ(α)} is an M-generic realization of α.

Case 1. There is some (μ(α) − 1)-derived difference sequence with canonical parameter in acl(B).

Work hence with the above sequence, which we will still denote by (ē_{0}, …, ē_{μ(α)}). It usffices to show that ē_{μ(α)} lies in A. Otherwise, ē_{μ(α)} is generic in φ_{α}(x̄, b̄) over A and M, so independent from A and from M over B. Find two green tuples ā ∈ A and m̄ ∈ M with ē_{μ(α)} = m̄ · ā. Observe that ē_{μ(α)}, ā and m̄ are pairwise Bindependent, whihc contradicts Lemma 5.2 by Lemma 5.1. Minimal prealgebraicity of A over B implies that A = ⟨B, ē_{μ(α)}⟩. Since A ∈ K^{μ}, there is some ē_{i} in M \ B. Since B ≤ M, then ē_{i} is B-generic by property (e). Hence, the map ē_{i} ↦ ē_{μ(α)} induces a B-isomorphism between A and M.

Case 2. No (μ(α) − 1)-derivation has canonical parameter in acl(B).

As above decompose ē_{μ(α)} = m̄ · ā with m̄ ∈ M and ā ∈ A both green tuples. Since ē_{μ(α)} is M-generic, then 0 = δ(ē_{μ(α)}/M) = δ(ā/M), so ā generates A over B. Apply now Lemma 7.3 to B ≤ M′ and (ē_{0}, …, ē_{μ(α)}). There is some Morley segment of length μ^{∗}(α) over Bb̄.

Since

μ^{∗}(α) ≥ n_{α}k_{α} + 1 > n_{α} ≥ MR(m̄/Bb̄),

there is some ē_{i} in M with m̄ | ⌣_{Bb̄} ē_{i}. In particular, ē_{μ(α)} and ē_{i} have the same type over Bb̄m̄, and so do ā = m̄ · ē_{μ(α)}^{−1} and m̄ · ē_{i}^{−1}. By minimality, ā ↦ m̄ · ē_{i}^{−1} induces a B-isomorphism between A and M. □

Using notation developed by Poizat [14] we say that an L^{∗}-structure M in K^{μ} is rich if for every f.g. B ≤ M and every f.g. strong extension B ≤ A in K^{μ} there is some strong substructure A′ ≤ M with A ≃_{B} A′. Since every algebraic strong extension of an element in K^{μ} lies again in K^{μ}, it follows that rich structures are algebraically closed fields.

Corollary 9.3. There is a (unique upto isomorphism) countable rich structure in K^{μ}. Moreover, all rich structures are L_{∞,ω}-equivalent.

10. Axioms for T^{μ}

Recall that given ā, B ⊆ M ∈ K we say that B is strong in M if ⟨B⟩ ≤ M and we denote by δ(ā/B) the quantity δ(⟨Bā⟩/⟨B⟩). Let T^{μ} be the elementary theory of rich fields in K^{μ}. We will show in this section that T^{μ} is axiomatizable and model-complete.

Definition 10.1. Let M ⊨ T^{μ} and B be some subset of M. Denote by cl_{d}^{M}(B) the collection of all f.g. A ⊆ M with δ(A/ cl(B)) = 0. Set d_{M}(A/B) := d(A/B) := δ(cl(⟨A, B⟩)/ cl(B)).

Note that cl_{d}^{M}(B) = {a ∈ M : d(a/B) = 0}.

It is easy to see [15] that:

Lemma 10.2. For any structure in K the following holds:

(1) d(āc̄/B) = d(ā/Bc̄) + d(c̄/B).

(2) The closure operator cl_{d} defines a pregeometry over the set Ü of green points whose associated dimension function is d.

Lemma 10.3. Let e ≥ 0, a subset B = ⟨B⟩ ≤ M ∈ K and a tuple ā ∈ M. Then:

(1) If δ(ā/B) = e, then there is some existential L^{∗}-formula τ_{δ}(x̄, z̄) and a

tuple b̄ ∈ B such that: • ⊧ τ_{δ}(ā, b̄),

• for every ā′ and b̄′ ∈ B′ ⊆ M′ ∈ K with ⊧ τ_{δ}(ā′, b̄′), then δ(ā′/B′) ≤ e.

(2) If d(ā/B) = e, then there is an existential L^{∗}-formula τ_{d}(x̄, z̄) and a b̄ ∈ B

such that:

• ⊧ τ_{d}(ā, b̄),

• for every ā′ and alle b̄′ ∈ M′ ∈ K with ⊧ τ_{d}(ā′, b̄′), then d(ā′/b̄′) ≤ e.

Proof. We need only prove part (1). Hence, choose ā ∈ M and B ≤ M as above. Let B = A_{0} ≤ A_{1} ≤ … ≤ A_{n} = ⟨Bā⟩ =: A be the decomposition of B ≤ A into minimal strong embeddings. Then e = δ(ā/B) = ∑_{i=1}^{n} δ(A_{i}/A_{i−1}). Therefore we may assume that n = 1, that is, B ≤ A is minimal strong. Four cases may occur by Lemma 6.4. Cases (1)–(3) are easy so we may hence consider case (4), that is, a minimal prealgebraic extension. Let c̄ be a green basis for A/B and b̄ ∈ B with l. dim_{ℚ}(āc̄/B) = l. dim_{ℚ}(āc̄/b̄) and āc̄ | ⌣b̄ B. Let α ∈ C be unique encoding a/B. Choose some quantifier-free L^{∗}-formula τ̃(x̄, ȳ, z̄) with ⊧ τ̃(ā, c̄, b̄) such that:

• The tuples ā and c̄ are interdefinable over b̄ (explicitly given).

• ⊧ τ̃(x̄, ȳ, z̄) → ⋀_{i} Ü(y_{i}).

• c̄ realizes some φ_{α}(ȳ, b̄_{1}), where b̄_{1} lies in acl(b̄) (explicitly given).

By property (e) the formula τ_{δ}(x̄, z̄) := ∃ȳτ̃(x̄, ȳ, z̄) satisfies the required conditions, because δ(ā′/b̄′) = δ(c̄′/b̄′) ≤ δ(c̄′/ acl(b̄′) ≤ 0 if ⊧ τ̃(ā′, c̄′, b̄′).

The general case may be reduced to the minimal one by considering some tuple to express the decomposition into minimal extensions. □

Given M ⊆ N both elements in K such that M is L^{∗}-existentially closed in N, it follows from 10.3 that M ≤ N: otherwise, there is a tuple ā ∈ M with d_{M}(ā) > d_{N}(ā) = e. Choose some τ_{d} as in 10.3 such that N ⊧ τ_{d}(ā, b̄) for some b̄ ∈ ⟨∅⟩ ⊆ M. Hence, M ⊧ τ_{d}(ā, b̄) (since M is existentially closed in N), which contradicts d_{M}(ā/b̄) > e. Hence, the following holds:

Lemma 10.4. If M is an elementary submodel of N ⊨ T^{μ}, then M is strong in N. □

Define now an L^{∗}-theory T̃^{μ} as follows:

(1) Every model lies in K^{μ}.

(2) Every model is an algebraically closed field of characteristic 0.

(3) Given a model M, a good code α and a parameter b̄ in M, then there is no extension of M in K^{μ} given by a green M-generic realization of φ_{α}(x̄, b̄).

(4) In some ω-saturated model there are infinitely many d-independent green generic elements.

Poizat [15] axiomatized universally condition “∅ ≤ M”: since ψ_{α}(x̄_{0}, …, x̄_{ν}) are quantifier-free, then axiom (1) is universal. Both ACF_{0} and hence (2) are inductively axiomatizable. Corollary 8.4 yields the inductively axiomatization of (3), and the ∃∀-axiomatization of (4) follows from Lemma 10.3.

A key result is the following:

Theorem 10.5. An L^{∗}-structure M is rich if and only if it is an ω-saturated model of T̃^{μ}. In particular, T̃^{μ} = T^{μ} and T̃^{μ} is complete.

Proof. The proof is divided into two parts: first, we show that every ω-saturated model of T̃^{μ} is rich in K^{μ}. Then show that all rich structures are models of T̃^{μ}, which yields ω-saturation of rich structures because they are all ∞-equivalent by Corollary 9.3.

Hence, let M be an ω-saturated model of T̃^{μ}, a finite subset B ≤ M and A ≥ B a f.g. structure in K^{μ}. We need to embedd A in M strong over B. We may assume that A/B is minimal strong. By Lemma 6.4 there are four possibilities:

A/B is algebraic. we are done by axiom (2).

A/B is minimal prealgebraic. Consider the free amalgam M′ of M and A over B. Moreover, let α be the good code encoding A/B. Axiom (3) implies that M′ is not in K^{μ}. Since K^{μ} has the amalgamation property with respect to strong embeddings by Theorem 9.2, then A must be already embeddable in M over B.

A/B is green generic, i.e. generated by a green transcendental element a. Axiom (4) and ω-saturation imply that M contains infinitely many green d-independent generic elements (g_{i})_{i∈ℕ}. Since d(B) = e < ∞, there is some i ∈ ℕ with d(g_{i}/B) = 1. Hence, ⟨Bg_{i}⟩/B is a green generic extension (which lies in M), and the map a ↦ e_{i} yields a strong embedding of A in M over B.

A/B is white generic, i.e. generated by a white generic element w over B and Ü(A) = Ü(B). It is easy to see that the sum w′ of two B-generic d-independent green elements g_{1} and g_{2} is white B-generic, that is, d(w′/B) = 2. As above we can find such elements g_{1}, g_{2} in M, so we are done.

Now, let M be a rich structure in K^{μ}. In order to show that M ⊧ T^{μ}, we first show that M is an algebraically closed field. Let a ∈ acl(M) and B ≤ M f.g. with a ∈ acl(B). If a is green, then a in B, because B ≤ M. Otherwise, a is white and Ü(B) = Ü(⟨Ba⟩). Hence, the extension ⟨Ba⟩ ≥ B is in K^{μ}, so we can realize it in M over B.

For axiom (3), consider a good code α and some parameter b̄ ∈ M. Let ā be an M-generic realization of φ_{α}(x̄, b̄). If ⟨Mā⟩ ∈ K^{μ}, then choose some B_{0} ≤ M containing b̄. Therefore, ⟨B_{0}ā⟩ ∈ K^{μ} and by richness of M, we can find some ā_{0} ∈ M with ⟨B_{0}ā⟩ ≅ ⟨B_{0}ā_{0}⟩ =: B_{1} ≤ M. Iterating this process with B_{1}, we can find a sequence B_{0} ≤ B_{1} ≤ B_{2} ≤ … in M, such that B_{i+1} := ⟨B_{i}ā_{i}⟩ ≅ ⟨B_{i}ā⟩. Then ā_{1}, ā_{2}, … is a sufficiently long green Morley sequence for φ_{α}(x̄, b̄), whose difference realizes ψ_{α}, contradicting M ∈ K^{μ}.

Axiom (4) is satisfied by M by richness. □

Remark that given A ⊆ M ∈ K, we define cl(A) as the smallest ⟨·⟩-closed subset containing A with cl(A) ≤ M. If A is f.g., so is cl(A).

Corollary 10.6. T^{μ} is complete. Two tuples ā and ā′ in two models M and M′ have the same type if and only if there is some L^{∗}-isomorphism f from cl_{M}(ā) to cl_{M′}(ā′), mapping ā to ā′.

Proof. The rich field obtained in Section 9 is a model of T^{μ} by Theorem 10.5. Given two models M and M′ of T^{μ}, we may replace them by elementary extension and assume that they are ω-saturated. Hence they are rich by Theorem 10.5, hence elementarily equivalent by Corollary 9.3. So are M and M′.

In order to prove the second statement, suppose that both M and M′ are ωsaturated, since by Lemma 10.4 the closure cl_{M}(ā) does not change. Hence, M and M′ are rich structures and the map f : cl_{M}(ā) → cl_{M′}(ā′) induces a back-and-forth system, so f is elementary.

Suppose now that ā in M has the same type as ā′ in M′. Since cl(ā) is in the algebraic closure (in T^{μ}) of ā, there is an elementary map f from cl(ā) to M′ by ω-saturation, mapping ā onto ā′. Let A′ = f(cl(ā)). Hence A′ ≤ M′ because A′ has the same type as cl(ā′). Therefore, A′ = cl(ā′). □

Corollary 10.7. The theory T^{μ} is model-complete.

Proof. We will give a straight-foward proof due to M. Ziegler. We need only show that given any two models M and N of T^{μ} with M ⊆ N then M is strong in N. This implies that cl_{M}(ā) = cl_{N}(ā) for any ā in M. So the inclusion is elementary by Corollary 10.6. In particular, we will show the following:

Claim. If M ⊧ T^{μ} and M ⊆ N ∈ K^{μ}, then M is strong in N.

Otherwise, choose some M ⊆ N_{1} with minimal l. dim_{ℚ}(N_{1}/M) = e. Since M = acl(M), then e ≥ 2. Minimality of e implies that δ(N_{0}/M) ≥ 0 for every N_{0} = ⟨N_{0}⟩ with M ⊆ N_{0} ⊊ N_{1}. In particular, M ≤ N_{0}. Choose now some N_{0} with l. dim_{ℚ}(N_{0}/M) = e − 1.

−1 ≥ δ(N_{1}/M) = δ(N_{1}/N_{0}) + δ(N_{0}/M)

and δ(N_{1}/N_{0}) ≥ −1 imply that δ(N_{0}/M) ≤ 0. Since M ≤ N_{0} , the extension N_{0}/M is prealgebraic in K^{μ} (i.e. a tower of algebraic and minimal prealgebraic extensions). Hence, find N′/M minimal prealgebraic with M ≤ N′ ≤ N_{0}, which contradicts axiom (3). □

Note 10.8. One can show that axiom (4) follows already from (1)–(3), by aproximating a green generic extension by suitable prealgebraic extensions. On the other hand, the ∀∃-axiomatization follows from model-completeness (Corollary 10.7) by general model-theoretical arguments.

11. Ranks

We show in this section that T^{μ} has Morley rank 2. Let acl^{μ} denote the algebraic closure in models of T^{μ}. All model-theoretical notions refer exclusively to T^{μ}, which we will emphazise with μ if necessary. We will show that acl^{μ}(ā) is the union of all minimal prealgebraic extensions of cl(ā).

Lemma 11.1. Both closures acl^{μ} and cl_{d}^{M} agree in models of T^{μ}.

Proof. If B is f.g., so is cl(B) and contained in acl^{μ}(B). Hence, we may assume that B is f.g. and strong in M ⊧ T^{μ}.

First, show that cl_{d}^{M}(B) ⊆ acl^{μ}(B). Let A ⊂ M be f.g. with δ(A/B) = 0. Since l. dim_{ℚ}(A/B) is finite, then we may decompose A/B into a finite sequence of minimal extensions. If A′ ⊇ B is such that δ(A′/B) = 0, then A′ ≤ M because B ≤ M. Hence, we may assume that A/B is minimal. By Lemma 6.4 one of the following holds:

i) A is algebraic over B (in the field reduct). Hence, A ⊆ acl^{μ}(B).

ii) A is minimal prealgebraic over B. Choose a good code α and some parameter b̄ in acl(B) encoding A/B by Theorem 4.10. Then, A = ⟨Bā⟩ for some generic green realization ā ⊧ φ_{α}(x̄, b̄). We need only show that all green realizations of φ_{α}(x̄, b̄) lie already in M, for otherwise there is some M ≼ N and ā′ ∈ N not completely contained in M. Hence, ā′ is generic over M cotradicting axiom (3).

For the other inclusion, choose some a ∈ M \ cl_{d}^{M}(B). Set A = cl(B, a) and observe that δ(A/B) > 0. Decompose now A/B in minimal extensions B ≤ A_{0} ≤ A_{1} ≤ … ≤ A_{n} = A. Then there is some i < n with δ(A_{i+1}/A_{i}) > 0. By Lemma 6.4 we obtain l. dim_{ℚ}(A_{i+1}/A_{i}) = 1 so the extension A_{i+1}/A_{i} is either white or green generic. Corollary 8.3 implies that the free amalgam of A_{i+1} and every strong extension of A_{i} lies in K^{μ}. Richness of M implies that there are infinitely many A′ ≤ M, isomorphic to A_{i+1} over A_{i}. Moreover, they all have the same type over A_{i} by Corollary 10.6. So, A_{i+1} ⊈ acl^{μ}(A_{i}), hence a ∉ acl^{μ}(B), because A_{i+1} is algebraic over ⟨B, a⟩ and B ⊆ A_{i}. □

Theorem 11.2. T^{μ} has Morley rank 2 and is uncountably categorical. The white generic type has Morley rank 2 and the green generic one has Morley rank 1. Algebraic closure is determined by cl_{d}-closure in any model of T^{μ}. Moreover, for all ā and B we have that :

MR(ā/B) = U(ā/B) = d(ā/B).

Proof. Let M be an ω-saturated model of T^{μ}, seen as subset of the monster model of T^{μ}. We compute MR(a/M) for elements a coming from the monster model. Since

0 ≤ d(a/M) ≤ δ(a/M) ≤ 2.

there are four cases to consider:

d(a/M) = 0: By Lemma 10.4, we have that a ∈ acl^{μ}(M) = M. So MR(a/M) = 0.

d(a/M) = 1 and a is green: Then ⟨Ma⟩ is strong in the monster model and tp(a/M) is the type of the green generic element by Corollary 10.6. Since all other green generic types are algebraic, then MR(a/M) = 1, so Ü defines a strongly minimal set.

d(a/M) = 1 with a white: There must be some green point c ∈ cl(⟨M, a⟩) \ M. Hence, ⟨Mc⟩ ≤ cl(Ma) and d(a/Mc) = 0. Therefore, a and c are T^{μ}-interalgebraic over M and by the above case, MR(a/M) = MR(c/M) = 1.

d(a/M) = 2: Then Ü(⟨Ma⟩) = Ü(M) and ⟨Ma⟩ is strong in the monster model. Corollary 10.6 implies that tp(a/M) is the type of the white generic element, that is, the generic type of the field. Since all other types have Morley rank at most 1, then MR(a/M) ≤ 2. Now, Ü(M) is an infinite group of infinite index in M. Therefore, MR(a/M) = 2 and T^{μ} has Morley rank 2.

For the last statement, MR(ā/B) = U(āB) = d(ā/B), recall that MR = U holds on all ℵ_{1}-categorical theories. So Morley rank is additive and the above shows that MR(a/B) = d(a/B) for elements. Since d is also additive, then we are done. □

Note 11.3. For every natural number r ≥ 2 there is some field of rank r with a multiplicative green subgroup of rank r − 1. Work with the following predimension function (see [6]):

δ(A) = r dim(A) − l. dim_{ℚ}(Ü(A)).

Note 11.4. Rigidity results due to J. Kirby [10] may be useful to prove that there is a field of finite Morley rank of characteristic 0 equipped with a non-algebraic subgroup of a semiabelian variety.

References

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[2] J. Baldwin, K. Holland, Constructing ω-stable structures: fields of rank 2, JSL 65 (2000), 371–391.

[3] A. Baudisch, A. Martin-Pizarro, M. Ziegler, Hruhovski’s Fusion, submitted, (2006).

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[17] F. O. Wagner, Bad fields in positive characteristic, Bull. London Math. Soc., 35 (2003), 499–502.

[18] M. Ziegler, Lemma für Daniels beschränkte Automorphismen, preprint (2004).

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany. baudisch@mathematik.hu-berlin.de

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany. Equipe de Logique Mathématique, Université Denis-Diderot Paris VII, 75251 Paris

Cedex 05, France.

Institut Camille Jordan, Université Lyon 1, 43 bd 11 du Nov. 1918, 69622 Villeurbanne, France.

hils@mathematik.hu-berlin.de

Institut Camille Jordan, Université Lyon 1, 43 bd 11 du Nov. 1918, 69622 Villeurbanne, France.

Institut für Mathematik, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany. pizarro@math.univ-lyon1.fr

Institut Camille Jordan, Université Lyon 1, 43 bd 11 du Nov. 1918, 69622 Villeurbanne, France.

wagner@math.univ-lyon1.fr