Weak one-basedness

Gareth Boxall ^{∗}, David Bradley-Williams ^{†}, Charlotte Kestner ^{‡}, Alexandra Omar Aziz ^{§}, Davide Penazzi ^{¶}

July 13, 2012

Abstract We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterises linearity in the setting of geometric þ-rank 1 structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite dimensional vector spaces over division rings.

1 Introduction

An independence relation | ⌣ is a ternary relation on the set of small subsets of a sufficiently saturated structure M. Roughly speaking, A | ⌣_{C} B is intended to mean “A is independent from B over C”. More precisely, | ⌣ satisfies certain axioms. For convenience, we list the axioms which we shall be using in Definition 1.1. Six of these are taken from Definition 4.1 of [7] and, for notational convenience, we have added the normality axiom (see, for example, [1]). Note that the axioms in [7] are stated to suit the situation where A is a finite tuple. We follow Adler in [1] in expressing them here without that restriction. We have also phrased the invariance axiom without the aid of automorphisms to avoid having to make a strong homogeneity assumption. The penultimate paragraph of this section gives notational conventions which apply throughout this section.

∗Supported by EPSRC grant EP/F009712/1 and the Claude Leon Foundation

†Supported by EPSRC DTG

‡Supported by MALOA PhD Visiting Fellowship and EPSRC DTG

§Supported by the University of Leeds

¶Supported by EPSRC grant EP/F009712/1

Definition 1.1. Let M be a sufficiently saturated structure. Let | ⌣ be a ternary relation on the small subsets of M. Then | ⌣ is an independence relation on M if, for all small A, B, C ⊆ M, the following conditions are satisfied. Invariance: For all A′, B′, C′ ⊆ M such that tp(A, B, C) = tp(A′, B′, C′), if A | ⌣_{C} B then A′ | ⌣_{C′} B′. Symmetry: If A | ⌣_{C} B then B | ⌣_{C} A. Transitivity: For all D ⊆ C, if C ⊆ B then A | ⌣_{D} B if and only if both A | ⌣_{D} C and A | ⌣_{C} B. Extension: There exists A′ ⊧ tp(A/C) such that A′ | ⌣_{C} B. Normality: A | ⌣_{C} B if and only if A | ⌣_{C} B ∪ C. Finite character: A | ⌣_{C} B if and only if, for all finite B′ ⊆ B, A | ⌣_{C} B′. Local character: There exists D ⊆ B such that |D| ≤ |T h(M)| · |A| and A | ⌣_{D} B.

Good examples of independence relations include the case where M is a vector space and | ⌣ is linear independence or where M is an algebraically closed field and | ⌣ is algebraic independence. An important difference between these two examples is that linear independence is linear while algebraic independence (in an algebraically closed field) is not. Various conditions, which characterise linearity of an independence relation | ⌣ in certain situations, have been defined. For the rest of this section we fix a sufficiently saturated infinite structure M and an independence relation | ⌣ on M. So “A ⊆ M” means “A ⊆ M and A is small”. An important role is played by the notion of modularity which we now recall.

Definition 1.2. (M, | ⌣) is modular if, for all A, B ⊆ M, A | ⌣_{acl(A)∩acl(B)} B.

Here acl is the model-theoretic algebraic closure operator in M. There is a much studied weaker notion called local modularity. A further weakening is given by Berenstein and Vassiliev in [3] and is as follows.

Definition 1.3. (M, | ⌣) is weakly locally modular if, for all A, B ⊆ M, there exists C ⊆ M such that C | ⌣_{∅} AB and A | ⌣_{acl(AC)∩acl(BC)} B.

There is a related notion called one-basedness which Berenstein and Vassiliev weaken in [4] to obtain the following (in which we treat the finite tuple ā as a small set by forgetting the order of its entries).

Definition 1.4. (M, | ⌣) is weakly one-based if, for all ā ∈ M^{n} and B ⊆ M, there exists C ⊆ M such that B ⊆ C, ā | ⌣_{B} C and, for all ā′ ⊧ tp(ā/C), if ā | ⌣_{C} ā′ then ā | ⌣_{ā′} C.

There are several other relevant properties to consider including one which is actually called linearity. We recall its definition in §3. Some background is given at the beginning of [4] and we recall some of that now. For the remainder of this paragraph, the independence relation in question is always | ⌣^{acl} (see Definition 2.3) which is known to coincide with | ⌣^{þ} , the independence relation which comes from þ-forking (see [11]), for the structures under consideration. When M is strongly minimal, it is known that local modularity, one-basedness and linearity all coincide. In the more general setting where M is simple with SU-rank 1, one-basedness and linearity are known to coincide and to be strictly weaker than local modularity. In this setting it is proved in [3] (using a result from [12]) that weak local modularity is equivalent to one-basedness and linearity. In the even more general setting where M is geometric and has þ-rank 1, it is shown in [4] that weak local modularity is equivalent to weak one-basedness. Also in this setting it is shown in [4] that weak one-basedness is equivalent to a notion called generic linearity and implies linearity. We prove in §3, in this geometric þ-rank 1 setting, that linearity implies generic linearity and therefore that weak one-basedness is equivalent to linearity. This is proved in [4] under the assumption that M is dense o-minimal.

The notion of þ-rank for a formula is given in Definition 4.3 of [9]. We follow the convention that M has þ-rank 1 if and only if the formula “x = x” has þ-rank 1 in M. We recall what it means for M to be (pre)geometric in Definition 2.3.

In §2 we observe that the equivalence between weak one-basedness and weak local modularity, which is proved in [4] in the case where M is pregeometric and | ⌣ = | ⌣^{acl}, extends to the general setting of an arbitrary sufficiently saturated infinite structure M and an arbitrary independence relation | ⌣ on M, provided one uses an appropriately modified definition of weak local modularity.

The notion of a lovely pair (N, P (N)) of geometric structures has been extensively studied (see [3] and [5]). It consists of a geometric structure N expanded by a unary predicate P which names a well-behaved elementary substructure P (N). A nice example is given by the real field together with a predicate for the subfield of all real algebraic numbers (see [8]). Lovely pairs of geometric structures play an interesting role in the history of the topics we are considering here. In [3] weak local modularity of | ⌣^{acl} in a sufficiently saturated geometric structure is characterised in terms of the modularity of an independence relation in a corresponding lovely pair. Lovely pairs, being an especially well-behaved kind of expansion, also provide a test of the robustness of the notion of weak one-basedness. It is proved in [5] that if M is geometric with þ-rank 1 then the corresponding theory of lovely pairs is rosy and so sufficiently saturated models of it are equipped with the independence relation | ⌣^{þ}. We prove in §4 that if M is geometric with þ-rank 1 and (M, | ⌣^{acl}) is weakly one-based then | ⌣^{þ} in a sufficiently saturated model of the corresponding theory of lovely pairs will also be weakly one-based. Berenstein and Vassiliev prove this in [4] under an additional assumption.

An earlier version of [4] had contained several questions about the theory of the projective geometry of an infinite-dimensional vector space over a division ring. See Section 4 of [4] to understand the relevance of these geometries. In the Appendix in §5 we address the issue of stability, showing that the theory of the projective geometry of an infinite-dimensional vector space over an infinite division ring is stable if and only if the theory of the division ring is stable. We do this by proving a quantifier elimination result for the vector space in an appropriate language. We do not claim that these results are new. Indeed they seem to be essentially well-known. However, we are not aware of a suitable reference for them and so thought we would include them here.

Our terminology and notation are fairly standard. The following applies to the first four sections. Parameter sets (as opposed to definable sets) are denoted by the letters A, B, C or D or by variants of them such as A′. We always work in a sufficiently saturated infinite structure and so all such sets are automatically assumed to be small. When we say that two tuples of parameter sets have the same type (possibly over some other parameter set), for example tp(A, B, C/D) = tp(A′, B′, C′/D), we mean that this is true for some well-ordering of each of these parameter sets. Elements of M^{n}, for some finite n, are denoted by ā, b̄, c̄ or d̄ (or ā′ etc.). We use e to denote an imaginary element (an element of M^{eq}). We use x, y, z as real variables, x̄, ȳ, z̄ as finite tuples of real variables and w as an imaginary variable. We usually just write sets or tuples next to each other to indicate their union or the tuple obtained by writing one before the other. The conventions in use in §5 are made clear in §5.

We would like to thank Alexander Berenstein and Evgueni Vassiliev for some useful conversations and especially for sharing their work with us and so allowing us to contribute to it. We would like to thank Rizos Sklinos who was involved in discussions at an early stage and Anand Pillay for many useful ideas and pieces of advice. The five of us were all based at Leeds when we began work on this project and four of us are or were PhD students of Anand Pillay. In this capacity and on this occasion, we would like to thank him for the enormous impact he has had on our lives.

2 Some equivalent notions

Throughout this section, M is a sufficiently saturated infinite structure and | ⌣ is an independence relation on M. We recalled one formulation of weak one-basedness in Definition 1.4. The following alternative version is also given in [4] (but here we give it a slightly different name for ease of reference).

Definition 2.1. (M, | ⌣) is very weakly one-based if, for all ā ∈ M^{n} and B ⊆ M, there exists ā′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′ and ā | ⌣_{ā′} B.

Remark 2.2. (M, | ⌣) is very weakly one-based if and only if, for all ā ∈ M^{n} and B ⊆ M, there exists ā′′ ⊧ tp(ā/B) such that ā′′ | ⌣_{B} ā and ā′′ | ⌣ ā B.

Proof. Suppose for ā ∈ M^{n} and B ⊆ M we have ā′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′ and ā | ⌣_{ā′} B. Then as ā ⊧ tp(ā′/B) we have an ā′′ such that tp(āā′/B) = tp(ā′′ā/B). The result follows by invariance.

We use acl to denote model-theoretic algebraic closure in the structure M. Recall the following standard notions.

Definition 2.3. M is pregeometric if acl has the Steinitz exchange property (in which case we say (M, acl) is a pregeometry). In this case, for ā ∈ M^{n} and B, C ⊆ M, we say ā | ⌣_{B}^{acl} C if dim(ā/B) = dim(ā/BC), where this notion of dimension is obtained from acl analogously to the way that transcendence degree is obtained from the algebraic closure operator in an algebraically closed field. For A, B, C ⊆ M, we say A | ⌣_{C}^{acl} B if ā | ⌣_{C}^{acl} B for all finite tuples ā from A. If in addition T h(M) eliminates the quantifier ∃^{∞}, we say that M is geometric.

It is well known that | ⌣^{acl} is an independence relation on M when M is pregeometric. It is proved in [4] that weak local modularity, weak onebasedness and very weak one-basedness are all equivalent when M is pregeometric and | ⌣ = | ⌣^{acl}. We would like to extend this equivalence to the general setting which we are considering here, that of an arbitrary sufficiently saturated infinite structure M and an arbitrary independence relation | ⌣ on M. However we do not see how to make this work with weak local modularity as in Definition 1.3 and so we consider the following version instead.

Definition 2.4. (M, | ⌣) is very weakly locally modular if, for all ā ∈ M^{n} and B ⊆ M, there exists C ⊆ M such that C | ⌣ ā B, C | ⌣_{B} ā and ā | ⌣_{acl(āC)∩acl(BC)} B.

It is easy to check that very weak local modularity coincides with weak local modularity when M is pregeometric and | ⌣ = | ⌣^{acl}. After replacing weak local modularity with very weak local modularity, Berenstein and Vassiliev’s equivalence extends to our general setting. However, some preliminary work is required. The argument in [4] makes use of the following property which we do not claim to be true generally.

Property 2.5. Let ā ∈ M^{n} and B ⊆ M. Let ā′ ⊧ tp(ā/B). If ā | ⌣_{B} ā′ and ā | ⌣_{ā′} B then ā′ | ⌣ ā B.

It is proved in [4] that (M, | ⌣) has Property 2.5 when M is pregeometric and | ⌣ = | ⌣^{acl}. The argument may be phrased in terms of additivity of U-rank and so extended to apply to any independence relation which has a U-rank which is finite on every type. However we do not see how to stretch it to the general setting of an arbitrary sufficiently saturated infinite structure M and an arbitrary independence relation | ⌣ on M. This does not matter since we require only the following strengthening of very weak one-basedness which can be proved, in this general setting, by other means.

Lemma 2.6. Suppose (M, | ⌣) is very weakly one-based. Let ā ∈ M^{n} and B ⊆ M. Then there exists ā′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′, ā | ⌣_{ā′} B and ā′ | ⌣ ā B.

Proof. By very weak one-basedness and Remark 2.2, there exists ā_{1} ⊧ tp(ā/B) such that ā_{1} | ⌣_{B} ā and ā_{1} | ⌣ ā B. Using very weak one-basedness and Remark 2.2 again, there exists ā_{2} ⊧ tp(ā_{1}/Bā) such that ā_{2} | ⌣_{Bā} ā_{1} and ā_{2} | ⌣ ā_{1} Bā. Continuing in this way, for i = 2, 3, 4, ..., we obtain ā_{i+1} ⊧ tp(ā_{i}/Bāā_{1}...ā_{i−1}) such that ā_{i+1} | ⌣_{Bāā_{1}...ā_{i−1}} ā_{i} and ā_{i+1} | ⌣ ā_{i} Bāā_{1}...ā_{i−1}. Let ā_{0} = ā.

Having constructed the sequence (ā_{i})_{i<ω}, we could continue the process and extend it to a sequence (ā_{i})_{i<α}, for any small infinite ordinal α. Then, for each n < ω and i_{0} < i_{1} < ... < i_{n} < i_{n+1} < α, we would have ā_{i_{n+1}} | ⌣_{Bā_{i_{0}} ā_{i_{1}}...ā_{i_{n−1}}} ā_{i_{n}} and ā_{i_{n+1}} | ⌣ ā_{in} Bā_{i_{0}} ā_{i_{1}}...ā_{i_{n−1}}. (Both of these independences follow, by transitivity, normality and invariance, from the construction of (ā_{i})_{i<α}.)

By choosing α large enough and then applying a well-known consequence of the Erdős-Rado theorem (which is stated as Theorem 1 in [2] where references are also given), we obtain a sequence indexed by ω with all the properties stated for our original sequence (ā_{i})_{i<ω} and with the additional property of being indiscernible over B. Therefore we may assume that our original sequence (ā_{i})_{i<ω} is indiscernible over B and from now on we do so.

For all n < ω, by transitivity and normality, we have ā_{1} | ⌣ ā_{0} B, ..., ā_{n} | ⌣_{ā_{0}...ā_{n−1}} B and so we get ā_{1}...ā_{n} | ⌣ ā_{0} B, using transitivity, normality and symmetry.

Let λ be (|T h(M)| · |B|)^{+} considered as an ordinal. Then there is an indiscernible sequence (c̄_{i})_{i<λ} of tuples from M such that tp(c̄_{i_{1}}...c̄_{i_{n}}/B) = tp(ā_{j_{1}}...ā_{j_{n}}/B) for all natural numbers n and all i_{1} < i_{2} < ... < i_{n} < λ and j_{n} < j_{n−1} < ... < j_{1} < ω. Given that λ is a limit ordinal with cofinality greater than |T h(M)| · |B|, it is a well-known consequence of the local character axiom (in conjunction with transitivity, symmetry and normality) that there cannot exist a sequence (d̄_{i})_{i<λ} of finite tuples from M such that d̄_{i+1} |̸ ⌣_{d̄_{0}...d̄_{i}} B for all i < λ. Therefore there will be some i < λ such that c̄_{i+1} | ⌣_{c̄_{0}...c̄_{i}} B and c̄_{i+2} | ⌣_{c̄_{0}...c̄_{i+1}} B. We also have c̄_{0}...c̄_{i} | ⌣ c̄_{i+1} B. We then get c̄_{i+2} | ⌣ c̄_{i+1} B. We also have c̄_{i+1} | ⌣ c̄_{i+2} B and c̄_{i+1} | ⌣_{B} c̄_{i+2}. Since ā ⊧ tp(c̄_{i+1}/B), there exists ā′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′, ā | ⌣_{ā′} B and ā′ | ⌣ ā B.

We are now in a position to prove the main result of this section which extends Berenstein and Vassiliev’s equivalence to the setting where M is an arbitrary sufficiently saturated infinite structure and | ⌣ is an arbitrary independence relation on M.

Theorem 2.7. The following are equivalent:

(1) (M, | ⌣) is very weakly locally modular,

(2) (M, | ⌣) is weakly one-based,

(3) (M, | ⌣) is very weakly one-based.

Proof. After Lemma 2.6, the rest of the proof of Theorem 2.7 is essentially given in [4]. For convenience we write it out here. We use the axioms of an independence relation, as stated in Definition 1.1, and well-known consequences of them freely and without specific reference.

Assume (1). Let ā ∈ M^{n} and B ⊆ M. Let C ⊆ M be such that C | ⌣_{B} ā, C | ⌣ ā B and ā | ⌣_{acl(āC)∩acl(BC)} B. Let ā′ ⊧ tp(ā/acl(BC)) such that ā | ⌣_{BC} ā′. Then C | ⌣_{ā′} B and so acl(ā′C) ∩ acl(BC) | ⌣_{ā′} B. But acl(ā′C) ∩ acl(BC) = acl(āC) ∩ acl(BC). So acl(āC) ∩ acl(BC) | ⌣_{ā′} B. We also have ā | ⌣_{C} Bā′ and so ā | ⌣_{acl(āC)∩acl(BC)} Bā′ and so ā | ⌣_{(acl(āC)∩acl(BC))ā′} B. Therefore ā | ⌣_{ā′} B. We also have ā′ ⊧ tp(ā/B) and ā | ⌣_{B} ā′. Therefore (3).

Assume (3). Let ā ∈ M^{n} and B ⊆ M. By assumption and Lemma 2.6, there exists ā′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′, ā | ⌣_{ā′} B and ā′ | ⌣ ā B. Let C = ā′. Then ā | ⌣_{acl(āC)∩acl(BC)} B. Therefore (1).

Assume (3). Let ā ∈ M^{n} and B ⊆ M. By assumption and Lemma 2.6, there exists ā′′ ⊧ tp(ā/B) such that ā | ⌣_{B} ā′′, ā | ⌣_{ā′′} B and ā′′ | ⌣ ā B. Let C = Bā′′. Let ā′ ⊧ tp(ā/C) such that ā | ⌣_{C} ā′. Then ā′′ | ⌣_{ā′} B. Also ā | ⌣_{_{ā′′}_{ā′}} B, since ā | ⌣_{ā′′} B and ā | ⌣_{Bā′′} ā′. So ā | ⌣_{ā′} B. From ā | ⌣_{B} C and ā | ⌣_{C} ā′ we get ā | ⌣_{Bā′} Cā′. Therefore ā | ⌣_{ā′} C. Therefore (2).

Assume (2). Let ā ∈ M^{n} and B ⊆ M. Let C ⊆ M be such that B ⊆ C, ā | ⌣_{B} C and, for all ā′ ⊧ tp(ā/C), if ā | ⌣_{C} ā′ then ā | ⌣_{ā′} C. There is some ā′ ⊧ tp(ā/C) such that ā | ⌣_{C} ā′. We then have ā | ⌣_{B} ā′ and ā | ⌣_{ā′} B. Clearly ā′ ⊧ tp(ā/B). Therefore (3).

3 Notions of Linearity

For this section we continue to assume that M is a sufficiently saturated infinite structure and | ⌣ is an independence relation on M. In addition, throughout this section, we assume M is geometric and | ⌣ = | ⌣^{acl}. Recall that a family of plane curves is given by a pair of formulas φ(x, y, w) and ψ(w), possibly with parameters from M, such that x and y are variables of the home sort (real variables) but w possibly belongs to an imaginary sort and, for each e ∈ M^{eq} such that M^{eq} ⊧ ψ(e), the subset of M^{2} defined by φ(x, y, e) has acl-dimension 1 (i.e. it is infinite and no element ab of it is acl-independent over e together with the parameters in φ). Recall too that a family of plane curves is said to be normal if, for any two distinct e, e′ ∈ M^{eq} such that M^{eq} ⊧ ψ(e) and M^{eq} ⊧ ψ(e′), the set defined by φ(x, y, e) ∧ φ(x, y, e′) is finite. In [4] (which refers back to [10]) a family of plane curves given by φ(x, y, z̄) and ψ(z̄), where z̄ is a tuple of real variables, is said to be almost normal if, for all c̄ ⊧ ψ(z̄), there exist only finitely many c̄′ ⊧ ψ(z̄) such that the set defined by φ(x, y, c̄) ∧ φ(x, y, c̄′) is infinite.

In the following definition, linearity is a standard notion and generic linearity is defined in [4] (which refers back to [10]). The dimension referred to in the definition of linearity is some extension of acl-dimension from M to M^{eq}. We only consider linearity in situations where a well-behaved such extension is known to exist such as when M has þ-rank 1.

Definition 3.1. (1) (M, | ⌣^{acl}) is linear if, for every normal family φ(x, y, w) and ψ(w) of plane curves, the set defined by ψ(w) has dimension < 2.

(2) (M, | ⌣^{acl}) is generically linear if, for every almost normal family φ(x, y, z̄) and ψ(z̄) of plane curves (where z̄ is a tuple of real variables), the set defined by ψ(z̄) has acl-dimension < 2 (i.e. dim(c̄/d̄) < 2 for all c̄ ⊧ ψ(z̄), where d̄ are the parameters in ψ and dim is as in Definition 2.3).

Berenstein and Vassiliev prove in [4] that (M, | ⌣^{acl}) is weakly one-based if and only if it is generically linear. Assuming M has þ-rank 1, they also prove that (M, | ⌣^{acl}) is linear if it is weakly one-based. We reverse this implication using the following variation on the theme of linearity.

Definition 3.2. (1) Let e ∈ M^{eq}. Let E be the equivalence relation on M^{n} such that e belongs to the sort of M^{eq} which is defined as being the set of all equivalence classes of E. We call e a finite set imaginary if every equivalence class of E is finite. In this case we call the sort to which e belongs an FSI-sort.

(2) We say that a normal family φ(x, y, w) and ψ(w) of plane curves is FSI-normal if the variable w ranges over an FSI-sort.

(3) We say that (M, | ⌣^{acl}) is FSI-linear if, for every FSI-normal family φ(x, y, w) and ψ(w) of plane curves, the set defined by ψ(w) has acldimension < 2.

Here we speak of acl-dimension even though we are talking about a set of imaginary elements. On this occasion it is perfectly safe to do so because there will be a definable set Z ⊆ M^{n} and a ∅-definable (in the sense of M^{eq}) function f (the function which quotients out by E) such that f(Z) is the set defined by ψ(w) and each fibre of f is finite. We take the acl-dimension of the set defined by ψ(w) to be equal to the acl-dimension of Z. As is well known, this agrees with the extension of acl-dimension to M^{eq} that we get using | ⌣^{þ} when M has þ-rank 1.

The “only if” part of the following result is essentially proved in [4].

Theorem 3.3. (M, | ⌣^{acl}) is generically linear if and only if (M, | ⌣^{acl}) is FSI-linear.

Proof. (⇒) Suppose (M, | ⌣^{acl}) is not FSI-linear. Let φ(x, y, w) and ψ(w) be an FSI-normal family of plane curves which witnesses this. Let Z and f be as in the paragraph after Definition 3.2. Let ψ′(z̄) define Z. Let φ′(x, y, z̄) be such that M^{eq} ⊧ (∀xyz̄)[ψ′(z̄) → (φ′(x, y, z̄) ↔ φ(x, y, f(z̄)))]. Then φ′(x, y, z̄) and ψ′(z̄) form an almost normal family of plane curves (since, for c̄, c̄′ ⊧ ψ′(z̄), either the set defined by φ′(x, y, c̄) ∧ φ′(x, y, c̄′) is finite or f(c̄) = f(c̄′) and recall that the fibres of f are finite). The set defined by ψ(w) has acl-dimension ≥ 2, by assumption, and so the set defined by ψ′(z̄) has acl-dimension ≥ 2. Therefore (M, | ⌣^{acl}) is not generically linear.

(⇐) Suppose (M, | ⌣^{acl}) is not generically linear. Let φ(x, y, z̄) and ψ(z̄) be an almost normal family of plane curves which witnesses this. Fix some c̄ ⊧ ψ(z̄) such that dim(c̄/d̄) ≥ 2, where dim is as in Definition 2.3 and d̄ are the parameters in ψ and (we may assume also) in φ. Let m < ω be maximal subject to there being distinct c̄_{1}, ..., c̄_{m} ⊧ ψ(z̄) such that c̄ = c̄_{1} and the formula φ(x, y, c̄_{1}) ∧ ... ∧ φ(x, y, c̄_{m}) defines an infinite set. For each k < ω, let ψ_{k}(z̄_{1}, ..., z̄_{k}) be the formula (∃^{∞}xy)[φ(x, y, z̄_{1}) ∧ ... ∧ φ(x, y, z̄_{k}) ∧ ψ(z̄_{1}) ∧ ... ∧ ψ(z̄_{k}) ∧ ⋀_{i≠j} z̄_{i} ≠ z̄_{j}]. Let ψ′(z̄_{1}, ..., z̄_{m}) be ψ_{m}(z̄_{1}, ..., z̄_{m}) ∧ (∀z̄_{m+1})[¬ψ_{m+1}(z̄_{1}, ..., z̄_{m+1})]. Let E be the equivalence relation which says that the order of the z̄_{i}’s in z̄_{1}...z̄_{m} does not matter. Then E is ∅-definable in M. Let f be the function which quotients out by E. Then f is ∅-definable in M^{eq}. Let ψ′′(w) be such that M^{eq} ⊧ (∀z̄_{1}...z̄_{m})[ψ′′(f(z̄_{1}, ..., z̄_{m})) ↔ ψ′(z̄_{1}, ..., z̄_{m})]. Let φ′(x, y, w) be such that M^{eq} ⊧ (∀xyz̄_{1}...z̄_{m})[φ′(x, y, f(z̄_{1}, ..., z̄_{m})) ↔ φ(x, y, z̄_{1}) ∧ ... ∧ φ(x, y, z̄_{m})]. Then φ′(x, y, w) and ψ′′(w) form a normal family of plane curves. Since the fibres of f are finite, φ′(x, y, w) and ψ′′(w) form an FSI-normal family of plane curves. Let Z be the set defined by ψ′(z̄_{1}, ..., z̄_{m}). We have c̄_{1}...c̄_{m} ∈ Z such that c̄_{1} = c̄. Clearly Z is definable over the parameters in φ and ψ. Therefore Z has acl-dimension ≥ 2. The set defined by ψ′′(w) is f(Z) and so it too has acl-dimension ≥ 2. Therefore (M, | ⌣^{acl}) is not FSI-linear.

It is clear that linearity implies FSI-linearity when M has þ-rank 1. So, combining Theorem 3.3 with the results from [4] referred to above, we get the following.

Corollary 3.4. Suppose the geometric structure M has þ-rank 1. Then (M, | ⌣^{acl}) is weakly one-based if and only if (M, | ⌣^{acl}) is linear.

This was proved in [4] under the assumption that M is dense o-minimal, via an argument which overlaps to some extent with our proof of the “if” part of Theorem 3.3.

4 Lovely pairs

Throughout this section we assume that M is a sufficiently saturated infinite structure which is also geometric and has þ-rank 1. Furthermore, P is a new unary predicate which is interpreted in M so that the expansion N = (M, P (M)) is a sufficiently saturated model of the theory of lovely pairs of models of T h(M). See [3] for the relevant definitions. We shall use | ⌣^{þ} to denote þ-independence in the structure N. It is known to follow from what is proved in [5] that | ⌣^{þ} is an independence relation on N. We continue to use | ⌣^{acl} to denote acl-independence in the structure M. One might wonder whether weak one-basedness of (M, | ⌣^{acl}) would imply weak one-basedness of (N, | ⌣^{þ}). This is proved in [4] under an additional assumption (namely their Assumption 5.8). In this section we show that this additional assumption is not needed.

For the rest of this section we assume that (M, | ⌣^{acl}) is weakly one-based. Recall from [4] that it is then also weakly locally modular. It is proved in [3] that then the algebraic closure operator in N coincides with acl in M and so | ⌣^{acl} is also an independence relation on N.

In addition to the independence relations | ⌣^{þ} on N and | ⌣^{acl} on either M or N, we shall also want to use the independence relation | ⌣^{scl} on N. The closure operator scl is defined in [3] as follows. Given a ∈ M and B ⊆ M, we have a ∈ scl(B) if and only if a ∈ acl(B ∪ P (M)). Then | ⌣^{scl} is obtained from scl analogously to the way that | ⌣^{acl} is obtained from acl.

Let ā ∈ M^{n} and B, C ⊆ M. It is proved in [3] that ā | ⌣^{þ}_{C} B if both ā | ⌣_{C}^{acl} B and ā | ⌣_{C}^{scl} B. It is well-known, and clear from the definition of | ⌣^{þ} , that ā | ⌣_{C}^{acl} B if ā | ⌣^{þ}_{C} B. So the relation | ⌣^{þ} lies, in strength, somewhere between | ⌣^{acl} ∧ | ⌣^{scl} and | ⌣^{acl}.

The following fact is an immediate consequence of the definition of the theory of lovely pairs of models of T h(M), together with the saturation assumption (see [3]).

Fact 4.1. Let a ∈ M and B ⊆ M. Suppose a ∉ acl(B). Then tp_{M}(a/B) has a realisation in P (M).

We now prove the main theorem of this section. There is some overlap between the argument presented here and that used to obtain the corresponding result, Proposition 5.9, in [4] (where their Assumption 5.8 was used). We use the axioms of an independence relation, as stated in Definition 1.1, and well-known consequences of them freely and without specific reference.

Theorem 4.2. Suppose (M, | ⌣^{acl}) is weakly one-based. Then (N, | ⌣^{þ}) is weakly one-based.

Proof. Let ā ∈ M^{n} and B ⊆ M. Using the weak one-basedness of (M, | ⌣^{acl}), let D′ ⊇ B be such that ā | ⌣_{B}^{acl} D′ and, for all ā′ ⊧ tp_{M}(ā/D′), if ā | ⌣_{D′}^{acl} ā′ then ā | ⌣_{ā′}^{acl} D′. Using Fact 4.1, let D ⊧ tp_{M}(D′/Bā) such that D ⊆ scl(B). We may assume ā = ā_{0}ā_{1}, where ā_{0} is scl-independent over D and ā_{1} ⊆ scl(Dā_{0}). Let p̄ ∈ P (M)^{m} be such that ā_{1} ⊆ acl(Dā_{0}p̄). Using the weak one-basedness of (M, | ⌣^{acl}) again, let C′ ⊇ D be such that āp̄ | ⌣_{D}^{acl} C′ and, for all ā′p̄′ ⊧ tp_{M}(āp̄/C′), if āp̄ | ⌣_{C′}^{acl} ā′p̄′ then āp̄ | ⌣_{ā′p̄′}^{acl} C′. Using Fact 4.1

again, let C ⊧ tp_{M}(C′/Dāp̄) be such that C ⊆ scl(D). Then ā | ⌣_{B}^{acl} C and, since C ⊆ scl(B), ā | ⌣_{B}^{scl} C. Therefore ā | ⌣^{þ}_{B} C.

Let ā′ ⊧ tp_{N}(ā/C) be such that ā | ⌣^{þ}_{C} ā′. We may assume p̄ was chosen so that āp̄ | ⌣^{þ}_{C} ā′. Let p̄′ be such that ā′p̄′ ⊧ tp_{N}(āp̄/C). We may assume āp̄ | ⌣^{þ}_{C} ā′p̄′. Then āp̄ | ⌣_{C}^{acl} ā′p̄′ and so āp̄ | ⌣_{ā′p̄′}^{acl} C. We also have ā | ⌣_{C}^{acl} ā′.

We then get ā | ⌣_{D}^{acl} ā′ and so also ā | ⌣_{ā′}^{acl} D and then ā | ⌣_{ā′}^{acl} C.

Let ā_{2} be a maximal subtuple of ā_{0} such that ā_{2} is scl-independent over ā′. Using āp̄ | ⌣_{ā′p̄′}^{acl} C we then get ā ⊆ scl(ā_{2}ā′). So we have ā | ⌣_{ā_{2}ā′}^{scl} C. We also have ā | ⌣_{ā_{2}ā′}^{acl} C. Therefore ā | ⌣^{þ}_{ā_{2}_{ā′}} C. Since ā_{2} is a subtuple of ā_{0}, we have ā_{2} | ⌣_{∅}^{scl} C and ā_{2} | ⌣_{∅}^{acl} C. Therefore ā_{2} | ⌣^{þ}_{∅} C. Using ā_{2} | ⌣^{þ}_{C} ā′ we then get ā_{2} | ⌣^{þ}_{∅} Cā′ and so ā_{2} | ⌣^{þ}_{ā′} C.

From ā | ⌣^{þ}_{ā_{2}_{ā′}} C and ā_{2} | ⌣^{þ}_{ā′} C we get ā | ⌣^{þ}_{ā′} C. Therefore (N, | ⌣^{þ}) is weakly one-based.

5 Appendix: Infinite-dimensional projective geometries

Let F be an infinite division ring and let V be an infinite-dimensional vector space over F. We use Geom(V) to refer to the structure (G, (‘x ∈ cl(y_{1}, …, y_{n})’)_{n≥1}) which is the geometry associated with the pregeometry (V, span). It is a classical result in projective geometry that F is definable in Geom(V). See, for example, Part 5 of the notes [6].

We aim to show that Th(Geom(V)) is stable if and only if Th(F) is stable. To that end, we consider the two-sorted structure (V, F). We refer to the sort of V as the vector sort and to the sort of F as the field sort. A natural choice of language L for the structure (V, F) consists of the ring language on the field sort for the division ring structure on F, the abelian group language on the vector sort for vector addition on V, and a function from the cartesian product of the field sort and the vector sort to the vector sort for scalar multiplication on V. We shall use variables x, y, z for the field sort and variables u, v, w for the vector sort. Clearly, Geom(V) is interpretable in (V, F).

We will expand the structure (V, F) in a natural way to prove a quantifier elimination result for Th(V, F) which we can then apply to count the number of types.

We extend L as follows:

• For every formula φ(x̄) in the ring language, we add a predicate P_{φ}(x̄) on the field sort.

• For every n ≥ 1, 1 ≤ i ≤ n, we add a new n + 1-ary function symbol λ_{i}^{n}(u_{1}, …, u_{n}, v) from the appropriate Cartesian product of the vector sort to the field sort.

Call this language L_{F,λ}.

We make (V, F) into an L_{F,λ}-structure as follows:

• For every formula φ(x̄) in the ring language, we interpret P_{φ} as the solution set of φ(x̄) in F.

• For every n ≥ 1, 1 ≤ i ≤ n and s_{1}, …, s_{n}, r ∈ V,

(λ_{i}^{n})^{(V,F)}(s_{1}, …, s_{n}, r) = \begin{cases}0 & if s_{1}, …, s_{n} are not linearly independent, \\ 0 & if s_{1}, …, s_{n}, r are linearly independent, \\ a_{i} & if s_{1}, …, s_{n} are linearly independent and \\ & r = ∑_{i=1}^{n} a_{i}s_{i}. \end{cases}

Let T_{F,λ} be the theory of (V, F) as an L_{F,λ}-structure. Note that T_{F,λ} is a definitional expansion of the L-theory of (V, F). In particular, every L-structure (U, K) with (U, K) ≡ (V, F) can be uniquely expanded to an L_{F,λ}-structure satisfying T_{F,λ}.

Theorem 5.1. T_{F,λ} has quantifer elimination.

Proof. We shall use the well-known fact that a complete (first-order) theory T has quantifier elimination if, whenever M and N are ω-saturated models of T, the collection of finite partial isomorphisms between M and N has the back-and-forth property.

Note first that because of the functions λ_{i}^{n}, finitely generated substructures of a model (U, K) of T_{F,λ} consist exactly of pairs (S, A) where either A is a finitely generated subring of K and S = {0} or A is a finitely generated division subring of K and S is a finite-dimensional A-subspace of U such that any A-linearly independent subset of S is also K-linearly independent. (Note that if 0 ≠ s ∈ U, then a^{−1} = λ_{1}^{1}(as, s) for any non-zero a ∈ K, and if s_{1}, …, s_{n} ∈ S \ {0} are not linearly independent over K, then, after reordering if necessary, we may assume that for some 1 ≤ m < n, s_{1}, …, s_{m} are linearly independent over K and s_{m+1} = ∑_{i=1}^{m} a_{i}s_{i} for some a_{1}, …, a_{m} ∈ K not all zero. But then a_{i} = (λ_{i}^{n})^{(U,K)}(s_{1}, …, s_{m}, s_{m+1}) ∈ A and so s_{1}, …, s_{n} are not linearly independent over A either.)

Now let (U, K) and (W, J) be ω-saturated models of T_{F,λ}, and let f : (S, A) → (R, B) be a finite partial isomorphism from (U, K) to (W, J). We may assume that A is a division subring of K. Then we can find ā ∈ A^{n} such that A is the division subring of K generated by ā and s̄ ∈ S^{m} such that S is the A-subspace of U generated by s̄ and s̄ is linearly independent over K. We need to show that whenever a ∈ K \ A or s ∈ U \ S, there is a finite partial isomorphism g extending f with a ∈ dom(g) or s ∈ dom(g), respectively. (Similarly, we can then also extend f so that its image contains any given element from (W, J).)

First, let a ∈ K be given. By Morleyisation of the field sort and ωsaturation, we can find b ∈ J such that āa and f(ā)b have the same division ring type. Let A′ be the division subring of K generated by āa, S′ be the A′-submodule of U generated by s̄, and let B′ be the division subring of J generated by f(ā)b, R′ be the B′-submodule of W generated by f(s̄). Then (S′, A′) and (R′, B′) are obviously again closed under all the functions λ_{i}^{n}, and are thus again (finitely generated) substructures of (U, K) and (W, J), respectively, and f ∪ {(a, b)} extends to a partial isomorphism g : (S′, A′) → (R′, B′).

Now let s ∈ U be given. There are two cases:

Case 1: s is K-linearly independent from S. Since W is infinite-dimensional over J we can choose r ∈ W J-linearly independent from R. Let S′ be the A-submodule of U generated by S ∪ {s}, and let R′ be the B-submodule of W generated by R ∪ {r}. Then obviously, (S′, A) and (R′, B) are closed under all the functions λ_{i}^{n}, and thus are again (finitely generated) substructures of (U, K) and (W, J), respectively, and f ∪ {(s, r)} extends to a partial isomorphism g from (S′, A) to (R′, B).

Case 2: s is not K-linearly independent from S. Let c_{i} = (λ_{i}^{m})^{(U,K)}(s̄, s), i = 1, …, m. By Morleyisation of the field sort and ω-saturation of (W, J), there are d_{1}, …, d_{m} ∈ J such that ā, c_{1}, …, c_{m} and f(ā), d_{1}, …, d_{m} have the same division ring type. Let r = ∑_{i=1}^{m} d_{i}f(s_{i}). Then obviously, f ∪ {(s, r)} extends to a partial isomorphism between the substructure of (U, K) generated by (s̄s, ā) and the substructure of (W, J) generated by (f(s̄)r, f(ā)). (Note that the substructure generated by (s̄s, ā) is exactly (S′, A′) where A′ is the division subring generated by ā, c_{1}, …, c_{m} and S′ is the A′-subspace of U generated by s̄s, and similarly for the substructure generated by (f(s̄)r, f(ā)).)

Proposition 5.2. If Th(F) is stable (superstable, totally transcendental) then Th(V, F) is stable (superstable, totally transcendental).

Proof. Clearly it is enough to prove, for every infinite cardinal κ, that T_{F,λ} is κ-stable if Th(F) is κ-stable.

Let κ be an infinite cardinal such that Th(F) is κ-stable. Let (U, K) be a model of T_{F,λ} of cardinality κ (i.e., both U and K have cardinality κ). We count the number of 1-types in each sort over (U, K).

Let a be an element of the field sort of an elementary extension (U′, K′) of (U, K). We may assume a ∈ K′ \ K. By Theorem 5.1, the type of a over (U, K) is then already determined by the division ring type of a over K since all the solutions of λ_{i}^{n}(s_{1}, …, s_{n}, r) = x with s_{1}, …, s_{n}, r ∈ U lie already within K so that the vector sort cannot contribute any new information about a. Thus, the number of 1-types in the field sort is κ by κ-stability of Th(F).

Now let s be an element of the vector sort of an elementary extension (U′, K′) of (U, K). Again we may assume s ∉ U. Then there are two cases:

Case 1: s is K′-linearly independent from U. Then by Theorem 5.1, the type of s over (U, K) is uniquely determined by this. So in this case, we have a unique type.

Case 2: s is not K′-linearly independent from U. Then there exist linearly independent s_{1}, …, s_{n} ∈ U such that s is not linearly independent from s_{1}, …, s_{n}. Then again by Theorem 5.1, it is easily seen that tp(s/(U, K)) is fully determined by this information (including the choice of s_{1}, ..., s_{n}) together with the division ring type of λ_{1}^{n}(s_{1}, …, s_{n}, s), …, λ_{n}^{n}(s_{1}, …, s_{n}, s) over K, for which there are, by the assumption of κ-stability of Th(F), only κ many possibilities. Therefore, we count a total of ℵ_{0}κκ, where the first two factors ℵ_{0}κ correspond to the number of choices of finite linearly independent tuples from U and the last κ to the choice of the division ring type over K of the coefficients in the linear combination.

Thus, we count a total of

κ + ℵ_{0}κκ + 1 = κ

many 1-types in the vector space sort over (U, K). (The first summand κ counts the realized types.)

This also shows that T_{F,λ} is superstable (totally transcendental) if Th(F) is superstable (totally transcendental).

Proposition 5.3. Th(Geom(V)) is stable (superstable, totally transcendental) if and only if Th(F) is stable (superstable, totally transcendental).

Proof. The left-to-right direction follows immediately from definability of F in Geom(V). The right-to-left direction follows immediately from interpretability of Geom(V) in (V, F) and Proposition 5.2.

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