FORKING GEOMETRY ON THEORIES WITH AN INDEPENDENT PREDICATE

JUAN FELIPE CARMONA

Abstract. We prove that a simple geometric theory of SU-rank 1 is n-ample if and only if the associated theory equipped with an predicate for an independent dense subset is n-ample for n at least 2.

1. Introduction

The notion of n-ampleness, introduced by Pillay in [5], roughly measures the complexity of the forking geometry: the Hrushovski construction ab-initio exhibits that, in strongly minimal theories, forking geometry can be more complicated than the geometry of vector spaces (1-based theories) and yet less than algebraic geometry (theories interpreting a field). The property that holds in Hrushovski’s example is called CM-triviality and it fits in this hierarchy of n-amples. In fact, (see [5] and [6]) a theory T is 1-based if and only if is not 1-ample and is CM-trivial if and only if is not 2-ample. Furthermore if T interprets a field, then T is n-ample for all n. However, the converse is not true: Evans [4] constructs a 1-based theory with a reduct which is n-ample for every n but does not interpret an infinite group.

It is a major problem to find theories of finite rank that are not CM-trivial but do not interpret a field. Baudisch and Pillay ([1]) obtained a 2-ample theory which is of infinite rank not interpreting any infinite group.

On the other hand, Berenstein and Vassiliev in [2] exhibit a 1-based (not 1ample) theory T such that T^{ind} is not 1-based, where T^{ind} stands for the theory of the pair (M, H), where M ⊧ T and H is an independent dense subset of M. We prove in this paper that in this case T^{ind} is CM-trivial. Moreover,we prove that for n ≥ 2, T is not n-ample if and only if T^{ind} is not n-ample.

2. Independent predicates in geometric theories

In this section we write down the principal definitions and results on geometric theories with an independent predicate that we will use in this paper. All proofs can be found in [2] and [3].

Definition 2.1. A complete theory T is geometric if eliminates ∃^{∞} and algebraic closure satisfies the exchange property in every model of T.

2000 Mathematics Subject Classification. 03C45.

Key words and phrases. geometric structures, U-rank one theories, n-ample theories.

I would like to thank Alexander Berenstein and Amador Martin-Pizarro for some useful remarks.

The author was supported by a grant from Mazda Fundation and by Colfuturo-Ascun-Embajada Francesa.

Let T be a complete geometric theory in a language L and let L_{H} = L ∪ {H} where H is a new unary predicate. T^{ind} is the L_{H} theory extending T together with the axioms:

(1) for all L-formulas φ(x, ȳ)

∀ȳ(φ(x, ȳ)nonalgebraic → ∃x ∈ Hφ(x, ȳ)) (Density property)

(2) for all L-formulas φ(x, ȳ), for all n ∈ ω and for all ψ(x, ȳ, z̄)

∀ȳ(φ(x, ȳ)nonalgebraic ∧ ∀ȳz̄∃^{≤n}xψ(x, ȳ, z̄)

→ (∃x ∉ H∀z̄ ∈ H(φ(x, ȳ) ∧ ¬ψ(x, ȳ, z̄))) (Extension property)

In these axioms, “non algebraicity" can be expressed in a first order way due to the elimination of ∃^{∞}.

From now on by acl() and | ⌣ we mean algebraic closure and algebraic independence in the sense of T.

Proposition 2.1. (Berenstein,Vassiliev [2]) If T is a geometric theory and

(M, H) ⊧ T^{ind}

is ℵ_{0}-saturated, then:

(1) If A ⊂ M is finite dimensional and q ∈ S_{n}(A) has dimension n, then there is ā ∈ H(M)^{n} such that ā ⊧ q (Generalized density property).

(2) If A ⊂ M is finite dimensional and q ∈ S_{n}(A) then there is ā ⊧ q such that ā | ⌣_{A}H (Generalized extension property).

Definition 2.2. An H-structure is an ℵ_{0}-saturated model of T^{ind} .

Definition 2.3. Let (M, H) be an H-structure and c a tuple in M. We denote by HB(c), the H-basis of c, the smallest tuple h ⊆ H such that c | ⌣_{h}H.

Also for A ⊆ M, A algebraically closed (in the sense of T^{ind}), the H-basis of c relative to A, denoted by HB(c/A), will stand for the smallest tuple h_{A} ∈ H such

that c | ⌣_{h_{A}A}H.

Proposition 2.2. For every c, the basis HB(c) exists.

Proof. Let h and h′ be tuples of H such that c | ⌣_{h}H and c | ⌣_{h′}H. It suffices to prove

that if h′′ = h ∩ h′ then c | ⌣_{h′′}H.

We can write c as c_{1}c_{2} where c_{1} is independent over H and c_{2} ⊆ acl(c_{1}H). So by definition of h and h′ we know that c_{2} ⊆ acl(c_{1}h) and c_{2} ⊆ acl(c_{1}h′). If c_{2} ⊈ acl(c_{1}h′′) then, by exchange property, there is an element in g in h \ h′ (or in h′ \ h), such that g ∈ acl(c_{1}h′). But c_{1} was chosen to be independent from H so actually g ∈ acl(h′). This yields a contradiction as H is an independent subset. □

Proposition 2.3. Let (M, H) an H-structure, let c and A be subsets of M and assume that A = acl(A) and HB(A) ⊆ A, then HB(c/A) exists.

Proof. Again, let h and h′ be minimal such that c | ⌣_{hA}H and c | ⌣_{h′A}H. In particular

we have that hh′ ∩ A = ∅.

Write c as c_{1}c_{2} where c_{1} is independent over AH and c_{2} ⊆ acl(c_{1}AH). Then c_{2} ⊆ acl(c_{1}Ah) and c_{2} ⊆ acl(c_{1}Ah′). Let h′′ = h ∩ h′, if c_{2} ⊈ acl(c_{1}Ah′′) then by exchange there is an element g ∈ h \ h′ (or viceversa) such that g ∈ acl(c_{1}Ah′).

Claim: we have that g ∉ acl(Ah′).

If not, as g ∉ h′ then by exchange there is an element a′ and a subset A′ of A such that a ∉ acl(A′) and a′ ∈ acl(A′gh′), then some (non empty) subset of gh′ must be contained in HB(A), HB(A) ⊆ A and h′g∩A ⊆ h′h∩A = ∅. Contradiction.

Therefore, as g ∈ acl(c_{1}Ah′) \ acl(Ah′), c_{1} is not independent over AH.

□

We defined the relative H-basis over algebraic closed sets A with HB(A) ⊆ A. The next theorem shows that actually these hypothesis impose that the set over which the H-base is defined must be algebraically closed in T^{ind}.

Theorem 2.1. (Berenstein,Vassiliev [2]) If (M, H(M)) is an H-structure and A is a subset of M then the algebraic closure of A in the sense of L_{H} (that we will denote by acl_{H}(A)) is the algebraic closure in the sense of L of A ∪ HB(A).

From now on, by HB(A/B) we mean HB(A/ acl_{H}(B)).

The next theorem provides a characterization of the canonical bases in T^{ind} in terms of H-basis and algebraic closure.

Theorem 2.2. (Berenstein, Vassiliev [2]) Let T an SU-rank 1 geometric theory and (M, H) be an H-structure (sufficiently saturated), a a tuple of M and B ⊂ M acl_{H}-closed. Then the canonical base cb_{H}(a/B) of stp_{H}(a/B), is interalgebraic (in the sense of L_{H}) with cb(a HB(a/B)/B).

Example 2.1. Let V a vector space over ℚ such that |V | > ℵ_{0} and let H = {h_{0}, h_{1}, ...} be a countable independent subset of V. Then (V, H) is an H-structure.

Moreover, if t is a vector independent of H and t_{0} = t + v_{0} then cb_{H}(t/t_{0}) is interalgebraic with cb(tv_{0}/t_{0}) = t_{0}. So t |̸ ⌣ t_{0}, but acl_{H}(t) ∩ acl_{H}(t_{0}) = ∅ hence T h(V, H) is not 1-based.

This example shows that 1-basedness is not preserved in T^{ind}.

We will see in the next section that if T is a SU-rank 1 geometric theory, then T^{ind} is 1-based iff T is trivial.

3. Ampleness

Definition 3.1. A simple theory T is not n-ample if for every sets a_{0}, ..., a_{n} of M^{eq} which satisfy the next conditions:

For all 1 ≤ i ≤ n − 1.

(1) a_{i+1} | ⌣_{a_{i}}a_{i−1}...a_{0},

(2) acl^{eq}(a_{0}...a_{i−1}a_{i+1}) ∩ acl^{eq}(a_{0}...a_{i−1}a_{i}) = acl^{eq}(a_{0}...a_{i−1}).

We have a_{n}  | ⌣_{acl^{eq}(a_{1})∩acl^{eq}(a_{0})}a_{0}.

Here we use the definition given by Evans in [4]. This definition seems more natural that the one given by Pillay in [5]. Nevertheless all the results that we present here work for both definitions.

From now on we will assume that T is a SU-rank 1 geometric theory eliminating imaginaries. By Theorem 1.2. canonical basis are interalgebraic with a tuple of elements, so T^{ind} has geometric elimination of imaginaries. Then, for the definition of n-ampleness, it suffices to work with real elements in an H-structure (M, H).

Proposition 3.1. The H-basis are transitive in the sense that

HB(c/B) ∪ HB(B) = HB(cB).

In particular, if A ⊆ B and acl_{H}(cA) ∩ acl_{H}(B) = acl_{H}(A) then

HB(c/A) ⊆ HB(c/B).

Proof. It’s clear that HB(c/B) ∪ HB(B) ⊆ HB(cB). On the other hand,

c | ⌣_{B∪HB(c/B)}H

and

B  | ⌣_{HB(c/B) HB(B)}H

then

cB  | ⌣_{HB(c/B) HB(B)}H.
□

Lemma 3.1. If T is trivial, then for every set A, acl(A) = acl_{H}(A).

Proof. Let h = acl(A) ∩ H. If x ∈ acl(A) ∩ acl(H), then by triviality x ∈ acl(h′) for some h′ ∈ H ∩ acl(A) = h. Hence HB(A) ⊆ h ⊆ acl(A). □

The previous lemma and Proposition 2.1 implies that, in a trivial theory, for every set A and every B = acl_{H}(B) we have HB(A/B) ⊆ acl(A). Because

HB(A/B) ⊆ acl_{H}(AB) \ B = acl(AB) \ B ⊆ acl(A).

Proposition 3.2. T is trivial iff T^{ind} is 1-based.

Proof. If T is trivial then for every a and b with b = acl_{H}(b) we have

h = HB(a/b) ⊆ acl(a),

so acl_{H}(cb_{H}(a/b)) = acl_{H}(cb(ah/b)) ⊆ acl_{H}(a).

Suppose now T^{ind} is 1-based and assume that T is not trivial, then there exists a tuple a and elements b and h such that b ∈ acl(ah) and b ∉ acl(a) ∪ acl(h). We can assume that a is an independent tuple minimal with this property and therefore that a | ⌣ H (by the Generalized Extension Property). Moreover, as tp(h/a) is not algebraic, we can assume that h belongs to H by density. It is clear that h = HB(b/a) so cb_{H}(b/a) is interalgebraic (in T^{H}) with cb(bh/a). Now, the theory T^{ind} is 1-based, hence acl_{H}(cb_{H}(b/a)) = acl_{H}(b) ∩ acl_{H}(a). On the other hand acl_{H}(a) = acl(a) as a | ⌣ H and also acl_{H}(b) = acl(b) because a | ⌣_{h}H and b ∈ acl(ah),

then b | ⌣_{h}H. But b ∉ acl(h) so HB(b) = ∅.

However, minimality of a yields acl(cb(bh/a)) = acl(a), hence acl(a) ⊆ acl(b) and h ∈ acl(ab) ⊆ acl(b). This is a contradiction. □

Theorem 3.1. For n ≥ 2 T is n-ample iff T^{ind} is n-ample.

Proof. (⇒) Assume T is n-ample, then there are sets a_{0}, ..., a_{n} such that:

(1) a_{i+1} | ⌣_{a_{i}}a_{i−1}...a_{0},

(2) acl(a_{0}...a_{i−1}a_{i+1}) ∩ acl(a_{0}...a_{i−1}a_{i}) = acl(a_{0}...a_{i−1}).

(3) a_{n} |̸ ⌣_{acl(a_{1})∩acl(a_{0})}a_{0}

By the generalized extension property, there are a′_{0}...a′_{n} such that tp(a′_{0}...a′_{n}) = tp(a_{0}...a_{n}) and a′_{0}...a′_{n} | ⌣ H.

As the H-bases of any subset of {a′_{0}..., a′_{n}} are empty, algebraic closure in T^{ind} is the same as in T. So condition (2) holds in T^{ind}.

By the characterization of the canonical bases (since H-bases are empty), condition (1) holds also in T^{ind}. But if

a′_{n} {|^{H} \atop ⌣}_{acl_{H}(a′_{1})∩acl_{H}(a′_{0})} a′_{0}

then

a′_{n}  | ⌣_{acl(a′_{1})∩acl(a′_{0})}a′_{0}

This is a contradiction.

(⇐)Assume T is not n-ample. Let a_{0}, ..., a_{n} be such that for all 1 ≤ i ≤ n − 1

(1) a_{i+1} | ⌣_{a_{i}}^{H}a_{i−1}...a_{0}

(2) acl_{H}(a_{0}...a_{i−1}a_{i+1}) ∩ acl_{H}(a_{0}...a_{i−1}a_{i}) = acl_{H}(a_{0}...a_{i−1}).

We may assume that a_{i} = acl_{H}(a_{i}) for every i ≤ n.

Claim 1. In these conditions we have the following chain:

HB(a_{n}/a_{0}) ⊆ HB(a_{n}/a_{0}a_{1}) ⊆ ... ⊆ HB(a_{n}/a_{0}...a_{n−1}).

Because

a_{n} {|^{H} \atop ⌣}_{a_{i+1}}a_{i}...a_{0}

therefore

acl_{H}(a_{n}a_{i−1}...a_{0}) ∩ acl_{H}(a_{i}, ..., a_{0}) ⊆ acl_{H}(a_{i+1}a_{i−1}...a_{0}),

hence, by (2),

acl_{H}(a_{n}a_{i−1}...a_{0}) ∩ acl_{H}(a_{i}a_{i−1}...a_{0}) = acl_{H}(a_{i−1}...a_{0}).

The conclusion follows from Proposition[2.1]. Note that this only make sense if n ≥ 2.

Let’s call h = HB(a_{n}/a_{0}) and h′ = HB(a_{n}/a_{0}, ..., a_{n−1}). Hence h ⊆ h′ by the previous claim. As the canonical basis cb_{H}(a_{n}/acl_{H}(a_{0}...a_{n−1})) is interalgebraic (in T^{H}) with cb(a_{n}h′/acl_{H}(a_{0}...a_{n−1})), then

a_{n}h | ⌣_{a_{n−1}} acl_{H}(a_{n−1}a_{n−2}...a_{0}).

Define recursively tuples a′_{i}, b_{i} for 0 ≤ i ≤ n − 1 in the following way: For the case i = 0 let a′_{0} = ∅ and b_{0} = a_{0}. For i > 0 let a′_{i} ⊆ acl_{H}(a_{i}, b_{i−1}, .., b_{0}) be a maximal tuple independent over

acl(a_{i}b_{i−1}, ..., b_{0}) (in the sense of T), and b_{i} = acl(a_{i}a′_{i}). In particular we have that

acl(b_{i}, ..., b_{0}) = acl_{H}(a_{i}, ..., a_{0}).

Note that we can take a′_{i} = HB(a_{i}, b_{i−1}, ..., b_{0}). Define also b_{n} as a_{n}h′.

Claim 2. For i ≤ n − 1 we have b_{i} | ⌣_{b_{i−1}}b_{i−2}...b_{0}:

By definition a′_{i} | ⌣ a_{i}b_{i−1}..b_{0}, hence a′_{i} | ⌣_{a_{i}}b_{i−1}...b_{0}. On the other hand, as a_{i} {|^{H} \atop ⌣}_{a_{i−1}}a_{i−1}...a_{0}

(by (1)) and a_{i−1} ⊂ b_{i−1} ⊆ acl_{H}(a_{i−1}, ..., a_{0}), then a_{i} {|^{H} \atop ⌣}_{b_{i−1}}b_{i−1}...b_{0}. This implies by

transitivity that b_{i} | ⌣_{b_{i−1}}b_{i−2}...b_{0} for i ≤ n − 1.

Note also that b_{n} | ⌣_{b_{n−1}}b_{n−2}...b_{0} by definition of h′ and the characterization of

canonical bases in T^{ind}.

Claim 3. For i ≤ n−1 we have acl(b_{i+1}b_{i−1}...b_{0})∩acl(b_{i}b_{i−1}...b_{0}) = acl(b_{i−1}...b_{0}). Because, for every i ≤ n,

acl(b_{i}...b_{0}) = acl_{H}(a_{i}...a_{0}),

then by (2)

acl(a_{i+1}b_{i−1}...b_{0}) ∩ acl(b_{i}b_{i−1}...b_{0}) = acl(b_{i−1}...b_{0}).

So, if acl(a_{i+1}b_{i−1}...b_{0}) ∩ acl(b_{i}b_{i−1}...b_{0}) ⊊ acl(b_{i+1}b_{i−1}...b_{0}) ∩ (b_{i}b_{i−1}...b_{0}), then by exchange there exists a ∈ acl(a′_{i+1}) ∩ acl(a_{i+1}b_{i}...b_{0}). Contradiction.

Therefore, Claims 2, 3 and non n-ampleness of T imply that b_{n} | ⌣_{b_{1}∩b_{0}}b_{0}, moreover

we get that a_{n}h | ⌣_{a_{1}∩a_{0}}a_{0} because h ⊆ h′ and b_{1} ∩ b_{0} = a_{1} ∩ b_{0}.

Hence, again by definition of h and characterization of canonical bases, a_{n} {|^{H} \atop ⌣}_{a_{1}∩a_{0}}a_{0}.

□

References

[1] Baudisch A., Pillay A., A free pseudospace, The Journal of Symbolic Logic 65: 443-660, 2000.

[2] Berenstein A., Vassiliev E., Geometric structures with an independent subset, (Preprint).

[3] Berenstein A., Vassiliev E., On lovely pairs of geometric structures, Annals of Pure and Applied Logic. 161(7),866-878, 2010.

[4] Evans D., Ample dividing, The Journal of Symbolic Logic 68:1385-1402, 2003.

[5] Pillay A., The geometry of forking and groups of finite Morley rank, The Journal of Symbolic Logic. 60(4):1251-1259, 1995.

[6] Pillay A., A note on CM-triviality and the geometry of forking The Journal of Symbolic Logic. 65(1):474-480, 2000.

Universidad de los Andes, Cra 1 No 18A-10, Bogotá, Colombia