THE GEOMETRY OF 1-BASED MINIMAL TYPES

TRISTRAM DE PIRO AND BYUNGHAN KIM

Abstract. In this paper, we study the geometry of a (non-trivial) 1-based SU rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type which is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any ω-categorical, non-trivial, 1-based theory, a vector space over a finite field is interpretable.

Geometric stability theory is one of the most important themes in model theory. Originally developing as a pure subject, it has turned out to be the major technical bridge connecting pure model theory and its applications to algebraic geometry and number theory. It mainly focuses on rank-1 types/structures or regular types where a canonical combinatorial geometry can be assigned. In other words, it is primarily concerned with geometric aspects of Shelah’s stability theory [19], the study of stable structures. Arguably, the first major achievements of geometric stability theory are Zilber’s results from the early 1980s (the translated version is [23]) on a strongly minimal ω-categorical structure. He showed that the geometry assigned to the structure is locally modular, hence, if non-trivial, must either be affine or projective over a finite field. A different proof was discovered independently by Cherlin, Harrington and Lachlan [6]. The results imply both the finiteness in rank of ω-categorical, ω-stable structures [6] and the non-finite axiomatizability of totally categorical theories. [23, 6] Since then, a number of prime researchers have obtained new enlightening results such as Buechler’s dichotomy theorem [3], Hrushovski’s group configuration theorems in several contexts [10, 11], his classification of a locally modular type [11], and other generalizations in the context of superstable theories [10, 11, 14]. Refined notions such as 1-basedness, regular types and p-weight have also been introduced. Pillay, in his book [17], makes a complete exposition of the subject. Hrushovski has now shifted the direction of research towards applications in algebraic geometry and number theory, which has deepened and broadened the subject. It is well known that, using geometric stability theory and in particular Zilber’s principle on ‘Zariski structures’, he solved the Mordell-Lang conjecture [12] and other problems in number theory.

From the mid 1990s, after the initial papers [15], [16] of Kim and Pillay, simplicity theory, introduced by Shelah [18], has developed rapidly and extensively. Simplicity theory is the study of a class of structures, called simple, properly containing that of stable structures. The central organising principle has been the Independence Theorem (type amalgamation), leading to the introduction of Lascar strong types, canonical bases and hyperimaginaries. Definability and the interplay with local stability have also played an important developing role. An account on the subject is written by Wagner [22], one of the major contributors. However, there are a number of obstacles to initiating an analogous geometric simplicity theory. This is mainly due to the lack of stationarity. Hrushovski’s counterexample shows that Zilber’s result on ω-categorical strongly minimal structures cannot be generalized in the context of ω-categorical rank-1 structures. Namely, there is a non 1-based, rank-1 ωcategorical structure [13].

1991 Mathematics Subject Classification. Primary: 03C45.

The second author was supported by an NSF grant.

In this paper we attempt to develop initial geometric simplicity theory using classical results in geometry as essential tools. We succeed in obtaining fruitful positive results and examples. We mainly study the solution set D of a non-trivial 1-based (simple) SU rank-1 Lascar strong type. Any finite tuple (not in the algebraic closure of the empty set) in a simple structure realizes a rank-1 Lascar strong type over some parameters. In particular, in any 1-based or supersimple theory, we can find a rank-1 type over a finite parameter. Moreover, if the 1-based theory is ω-categorical, then the theory must have finite SU-rank (and is therefore supersimple), so every type is coordinatized by rank-1 types [9]. Hence studying such a 1-based D is important in understanding ω-categorical or even general 1-based theories. (As is well known, any ω-categorical, ω-stable theory is 1-based. The class of ω-categorical 1-based structures properly contains that of smoothly approximable structures, too.) Furthermore, in the case of a supersimple rank-1 theory, since any model is the disjoint union of acl(∅) and the solution sets of rank-1 Lascar strong types (over ∅), we can restrict our attention to the solution set of a Lascar strong type. In [21], using his notion of ‘generic pair’, Vassiliev investigated the rank-1 theory. Here we study rank-1 types using a straightforward argument so that we can provide, for example, a direct proof of the fact that linearity implies 1-basedness. We also obtain the following more general results.

Local modularity of D implies 1-basedness (however, the converse is not true in the general simple theory context unlike the stable case). We first prove that the (local, resp.) modularity of the pregeometry of D implies that the (localized, resp.) geometry is isomorphic to a projective geometry over a division ring (*). These facts give the impression that, as in the stable case, the geometry of locally modular D should be affine over a division ring. However, we construct an example refuting this surmise.

On the other hand, it follows from (*) that if D is (locally) modular and ω-categorical then, after possibly naming a point, a sort of D^{eq} representing the geometry of D has a (definitional) stable reduct preserving independence. The reduct must be strongly minimal having a projective geometry over a finite field. Then, by the classical result on strongly minimal structures, a definable vector space over a finite field can be recovered inside the reduct. In particular, an infinite group is interpretable. For the more general case of a 1-based ω-categorical D, we also show that a sort representing the geometry on some finite union of sorts in D^{eq} has a generic (i.e. independence-preserving) strongly minimal stable reduct whose geometry is projective over a finite field. Again, a projective space over a finite field is interpretable. In fact we obtain a more general result. Namely, for any 1-based D, we prove that the canonical geometry, extending (or completing) the geometry of D, on the set of canonical bases of surfaces in D^{3}, is projective over a division ring (Theorem 3.9) (assuming non-triviality). Here, by a surface in D^{3}, we mean a rank 2 Lascar strong type of a tuple in D^{3}. It follows that in any ω-categorical, non-trivial, 1-based theory, a vector space over a finite field is interpretable over a finite parameter.

In this paper we assume that the underlying theory T is simple, having, for convenience, elimination of hyperimaginaries. This assumption can be removed if T is supersimple [4], or by working in M̄^{heq} and replacing notions in M̄^{eq} with corresponding notions in M̄^{heq} (e.g. acl by bdd, and so on).

1. Locally modular type

We start this section by recalling the definitions of (pre)geometry, modularity and local modularity ([17, 2.1]).

Definition 1.1. Let S be a set. If an operation cl : P(S) → P(S) satisfies the following properties, then we say that (S, cl) is a pregeometry.

(1) For A ⊆ S, A ⊆ cl(A) = cl(cl(A)).

(2) For A ⊆ B ⊆ S, cl(A) ⊆ cl(B).

(3) For A ⊆ S and a, b ∈ S, if a ∈ cl(Ab) ∖ cl(A), then b ∈ cl(Aa).

(4) If a ∈ cl(A), then a ∈ cl(A_{0}), for some finite A_{0} ⊆ A.

The pregeometry (S, cl) is said to be homogeneous if for any closure X of a finite set and a, b ∈ S ∖ X, there is an automorphism of (S, cl) which fixes X pointwise and moves a to b. We say the pregeometry (S, cl) is a geometry if cl(∅) = ∅ and cl(a) = {a} for all a ∈ S. If (S, cl) is a pregeometry, then we can associate a canonical geometry (Ŝ, ĉl), where Ŝ = {cl(a)|a ∈ S ∖ cl(∅)} and for A ⊆ S we put Â = {cl(a)|a ∈ A}, and ĉl(Â) = {cl(b)|b ∈ cl(A)}.

Let (S, cl) be a pregeometry. We say that A(⊆ S) is independent if a ∉ cl((A ∖ {a}) for all a ∈ A. Given B, a subset B_{0} ⊆ B is said to be a basis for B if B ⊆ cl(B_{0}) and B_{0} is independent. It follows that any two bases for B have the same cardinality, denoted by dim(B). Any A ⊆ S gives a localized pregeometry on S defined by cl_{A}(B) = cl(A ∪ B), and a notion of dimension over A (dim(−/A)). For A, B, C ⊆ S, if dim(A′/C) = dim(A′/B ∪ C) for any finite A′ ⊆ A, then we say that A is independent from B over C.

Definition 1.2. Let (S, cl) be a pregeometry.

(1) (S, cl) is trivial if cl(A) = ⋃{cl({a}) : a ∈ A} for all A ⊆ S.

(2) (S, cl) is modular if X is independent from Y over X ∩ Y for all closed X, Y, or equivalently, if dim(X) + dim(Y) = dim(X ∪ Y) + dim(X ∩ Y) for finite dimensional closed X and Y.

(3) (S, cl) is locally modular if it is modular over some point in S.

(4) (S, cl) is locally finite if the closure of a finite set is finite.

(5) A 3-tuple (ca_{1}a_{2}) ∈ S is a non-modular triangle if dim(ca_{1}a_{2}) = 3, cl(a_{1}a_{2}) = cl(a_{1}) ∪ cl(a_{2}) and there are u_{i} ∈ cl(ca_{i}) ∖ (cl(c) ∪ cl(a_{i})) for i = 1, 2.

Let S′ be the geometry of a pregeometry S. S is trivial (modular, or locally modular, resp.) if and only if S′ is (resp.). Moreover, each of the properties is preserved under localization. Thus modularity implies local modularity.

Fact 1.3. (S, cl) is modular iff for any subset A and b ∈ S, whenever c ∈ cl(Ab), then c ∈ cl(ab) for some a ∈ cl(A).

Remark 1.4. (1) Suppose that (S, cl) has a non-modular triangle (ca_{1}a_{2}). Then (S, cl) is not modular: Let u_{i} be as in 1.2.5. First, suppose cl(u_{1}) = cl(u_{2}). Since a_{1}, a_{2} ∈ cl(cu_{1}) ∪ cl(cu_{2}) = cl(cu_{1}), we get 2 = dim(cu_{1}) = dim(ca_{1}a_{2}u_{1}) = 3, a contradiction. Hence dim(u_{1}u_{2}) = 2. Now a_{1} ∉ cl(u_{1}u_{2}), as otherwise, since c ∈ cl(a_{1}u_{1}u_{2}) = cl(u_{1}u_{2}) and a_{2} ∈ cl(cu_{2}), dim(u_{1}u_{2}) = dim(ca_{1}a_{2}u_{1}u_{2}) = 3, a contradiction. Similarly, a_{2} ∉ cl(u_{1}u_{2}). Recall that cl(a_{1}a_{2}) = cl(a_{1}) ∪ cl(a_{2}). Hence cl(a_{1}a_{2}) ∩ cl(u_{1}u_{2}) = cl(∅). Thus 4 = dim(a_{1}a_{2}) + dim(u_{1}u_{2}) ≠ dim(a_{1}a_{2}u_{1}u_{2}) + dim(cl(a_{1}a_{2}) ∩ cl(u_{1}u_{2})) = 3. This shows that S is not modular.

(2) If (S, cl) is a modular geometry and cl(cd) = {c, d} for any two distinct points c, d ∈ S then S is trivial: By (4) of 1.1, it is sufficient to show that the closure of any finite set is itself. Suppose not, say there exists d ∉ A = {a_{1}, ..., a_{n}} such that d ∈ cl(A). Then cl(a_{n}d) = {a_{n}, d} and cl(a_{1}...a_{n−1}) = {a_{1}, ..., a_{n−1}} by the induction hypothesis. By modularity n = dim(a_{1}...a_{n}) = dim(a_{1}...a_{n−1}) + dim(a_{n}d) = n + 1, a contradiction.

Let us first state two important classical results on the geometry which will be used later.

Fact 1.5. [1] A projective (i.e. non-trivial and modular) geometry of dimension ≥ 4 in which each closed set of dimension 2 contains at least 3 elements, is isomorphic to a projective geometry over a division ring.

Fact 1.6. [7] A locally projective (i.e. non-trivial and locally modular), locally-finite geometry of dimension > 4, in which all closed sets of dimension 2 have the same size, is affine or projective geometry over a finite field.

(For some reason, in the above Fact 1.6, the dimension condition is erroneously known to be ≥ 4 to model theorists. This may be due to the fact that, in Doyen and Hubaut’s original paper, they counted their dimension 1 less than what is commonly defined by model theorists. They even constructed a counterexample in dimension 4.)

From now on, we consider a pregeometry given on a type. As usual, the saturated model M̄ of the simple theory T is fixed. We work in M̄^{eq}. (Tuples and sets are small from M̄^{eq}.) We lso fix a set D, which is a solution set (in the saturated model) of a Lascar strong type. For notational simplicity, we suppose D is over ∅. Let us further suppose that (D, cl) forms a pregeometry where cl is an Aut(M̄)-invariant closure relation on D (i.e. a ∈ cl(B) if and only if f(a) ∈ cl(f(B)) for each f ∈ Aut(M̄) ). For safety, we assume that dim(D) = Card(M̄). The typical examples of D are, of course, rank-1 types with cl(−) = acl(−) ∩ D, or regular types. D′ denotes the geometry of D. We shall use the same closure notation for the pregeometry and its geometry.

Remark 1.7. The closure of ∅ is ∅. The size of cl(p) does not depend on p ∈ D, hence (D′, cl) is homogeneous iff (D, cl) is.

Now, we observe that a modular (D, cl) is well behaved.^{1}

Lemma 1.8. (D modular.) Suppose there exist a, b ∈ D such that dim{a, b} = 2 and cl(ab) = cl(a) ∪ cl(b). Then D is trivial and homogeneous.

1There may exist a reasonable demand for distinguishing between the terminology ‘modular type’ and ‘type having a modular geometry’. Here, we consider the two terms to be equal.

Proof. It suffices to show triviality. Moreover, we can assume that D forms a geometry. Then the condition in the lemma simply says that cl(ab) = {a, b}.

Now, by Remark 1.4.2, it is enough to show that cl(cd) = {c, d} for any two distinct points c, d: Suppose not, say there is a third point u′ ∈ cl(cd). Let q(x, d) = tp(c/d). As D is the solution set of a Lascar strong type, by the Independence Theorem for Lascar strong types, we can amalgamate q(x, a) and q(x, b). Let c′ be a realization of a common nonforking extension of q(x, a) and q(x, b). Then dim(a, b, c′) = 3, and there are u, v such that tp(cdu′) = tp(c′bu) = tp(c′av). Hence the tuple (c′ab) forms a non-modular triangle. By Remark 1.4.1, this contradicts the modularity of D and finishes the proof. □

In the above lemma, the assumption that D is modular is necessary. We shall see in section 2 that even local modularity of D is insufficient to imply the conclusion. There are other 1-based (3.7) and even a non 1-based example [13], where the closure of some pair of points can be 2 or 3 elements.

By now, we have the following theorems due to Fact 1.5 and the previous lemma.

Theorem 1.9. If D is modular, then either D is trivial, or the geometry D′ of D is projective over a division ring. In either case, (D, cl) is homogeneous.

Suppose that D is ω-categorical. Then D′ is a sort in D^{eq}. Let R_{n} be an n+1-ary relation on D′ such that (a_{0}, a_{1}, ..., a_{n}) ∈ R_{n} if and only if a_{0} ∈ cl(a_{1}, ..., a_{n}). Then R_{n} is ∅-definable in D′ (so in D^{eq}), and (D′, R_{n})_{n∈ω} is a (definitional) reduct of D′. Hence, if (D, cl) is non-trivial and modular, then by the previous theorem, (D′, R_{n})_{n∈ω} is strongly minimal and quantifier eliminable, with canonical pregeometry (D′, cl), that is projective over a finite field F. Then by the general result on strongly minimal structures, a vector space (V, +, λ)_{λ∈F} over F is definable in (D′, R_{n})_{n∈ω}^{eq}, possibly over a finite parameter. We summarize this in the following theorem.

Theorem 1.10. Let D be modular and ω-categorical. Then the sort S_{E} in D^{eq}, where E(x, y) ≡ x ∈ cl(y), has a (definitional) stable reduct preserving independence. Hence, in a non-trivial D, a vector space over a finite field is interpretable (i.e. definable in D^{eq} over some finite set.) In particular, the infinite additive group is definable in D^{eq}.

If, additionally, D has SU-rank 1 and the usual algebraic closure relation, then D itself has a generic stable reduct.

Now we investigate the locally modular case. We fix some notation. Choose p ∈ D, and let D_{0} denote the pregeometry D \ cl(p) with closure operation cl_{p}. D′_{0} denotes the geometry of D_{0}. Recall that we use the same closure notation for the pregeometry and its geometry. Note that x ∈ cl_{p}(y) is an equivalence relation on D_{0}. We obtain the following results (1.11,12) analogous to the previous ones (1.8,9). However, (1.11,12) are not automatic consequences of (1.8,9), since in the simple context D_{0} need not be the solution set of a unique type over p.

For the rest of this section, D is locally modular.

Lemma 1.11. Suppose that there are distinct a′, b′ ∈ D′_{0} such that cl_{p}(a′b′) = {a′b′}. Then cl(c′d′) = {c′, d′} for any two points c′, d′ in D′_{0}. Hence, D′_{0} is trivial.

Proof. Triviality will follow from the main assertion by Remark 1.4.2. Fix points a, b in D from a′, b′, respectively. Due to the limitation on type amalgamation, we need more complicated arguments to prove the lemma. We shall calculate dimension in D, unless stated otherwise.

Suppose that the lemma is not true. Then there is u′ ∈ D′_{0} such that u′ ∈ cl_{p}(c′d′)\{c′, d′}. Now choose (in D) points c, d, u from c′, d′, u′ respectively. Then clearly dim(pcdu) = 3, and dim(pX) = 3 for each 2-subset X of {c, d, u} (*).

Case I) u ∈ cl(cd): Then the closure of any two of the u, c, d contains the third point (**). Now, we claim the following.

Claim 1) There are v_{i} ∈ D with v_{i} ∈ v′_{i} ∈ D′_{0} (for i = 1, 2, 3) such that, tp(cdu) = tp(v_{1}bv_{2}) = tp(v_{1}av_{3}), and dim(pabv_{1}) = 4.

Proof of claim: Let q(x, d) = tp(c/d). Then we can amalgamate q(x, b) and q(x, a). Let v_{1} be a realization of a common nonforking extension. Then clearly there are v_{2}, v_{3}

such that tp(cdu) = tp(v_{1}bv_{2}) = tp(v_{1}av_{3}). Moreover, we can assume \begin{aligned}v_{1} & ⌣ | & p \\ & ab \end{aligned} , so that

dim(pabv_{1}) = 4. In particular, p ∉ cl(v_{1}b). Now by (**), v_{2} ∈ cl(v_{1}b). Hence cl(p) ≠ cl(v_{2}), and v′_{2} ∈ D_{0}. Similarly v′_{3} ∈ D_{0}. Therefore the claim is proved.

As u ∈ cl(cd) \ {c, d}, we have v_{3} ∈ cl(v, a) \ {v, a} and v_{2} ∈ cl(v, b) \ {v, b}. If v_{3} ∈ cl(ap), then p ∈ cl(av_{3}) ⊆ cl(av_{1}), contradicting dim(pabv_{1}) = 4. Similarly, v_{3} ∉ cl(v_{1}p) and v_{2} ∉ cl(bp) ∪ cl(v_{1}p). Thus (v′_{1}a′b′) forms a non-modular triangle in D′_{0}, contradicting modularity. Hence case I does not happen.

We consider the remaining case.

Case II) u ∉ cl(cd): Then by (*), each 3 subset of {p, c, d, u} has dimension 3 (***). This time, let q(x, cd) = tp(p/cd) . There are u_{2}, u_{4} such that tp(cd) = tp(au_{2}) = tp(bu_{4}) and dim(abu_{2}u_{4}) = 4. Amalgamate q(x, au_{2}), q(x, bu_{4}) and let u_{1} ∈ D be a realization of a common nonforking extension, independent of p_{0}. Then dim(pu_{1}abu_{2}u_{4}) = 6. Now there are u_{3} ∈ u′_{3} ∈ D′_{0} and u_{5} ∈ u′_{5} ∈ D′_{0} such that tp(pcdu) = tp(u_{1}au_{2}u_{3}) = tp(u_{1}bu_{4}u_{5}). By use of (***), one can see that dim(pu_{2}u_{3}u_{4}u_{5}) = 5 and a′, b′ ∉ cl_{p}(u′_{2}u′_{3}u′_{4}u′_{5}). Thus cl_{p}(a′b′) ∩ cl_{p}(u′_{2}u′_{3}u′_{4}u′_{5}) = ∅. This again violates the modularity of D′_{0}. Hence Case II does not happen either.

Therefore Lemma 1.11 is proved. □

Theorem 1.12. D′_{0} is either trivial or projective over a division ring.

Lemma 1.13. D is trivial if and only if D_{0} is trivial.

Proof. We can safely assume that D forms a geometry. It is enough to show that triviality of D_{0} implies that of D. Let D_{0} be trivial.

Claim) The closure of any two distinct points {p, q} in D is itself

Proof of claim: Suppose not, say r ∈ cl(pq)\{p, q}. Choose s ∉ cl(pq) = cl(pqr) (*). Since D_{0} is trivial, cl(pqs) = cl(ps) ∪ cl(pq) and dim(pqs) = 3. Hence q, r ∉ cl(ps). Now, since the localized pregeometry of D at s is equivalent to D_{0} and D_{0} is trivial, cl(qs)∪cl(ps) = cl(pqs). In particular, r ∈ cl(qs)∪cl(ps). Hence, as r ∉ cl(ps), r ∈ cl(qs). As dim(qs) = 2, s ∈ cl(qr), this contradicts (*). Thus the claim is verified.

Now, by the claim, for n + 1 points p, q_{1}, ..., q_{n}, since D_{0} is trivial, cl(pq_{1}...q_{n}) = cl(pq_{1}) ∪ ... ∪ cl(pq_{n}) = {p, q_{1}, ..., q_{n}}. Hence D is also trivial. □

By the same argument as for Theorem 1.10 above, if D is locally modular ω-categorical, after naming a point p in D, a sort in (D_{p})^{eq} representing D′_{0} has a generic (independencepreserving) stable reduct (over p). Hence, if D is non-trivial, a vector space over a finite field is again interpretable over a finite parameter. We shall in fact prove the interpretability of a vector space in the more general 1-based context (3.23).

By now, one would naturally expect that, as in stable theories, a (non-trivial, non-modular) locally modular D should be affine. However, the example in the following section shows that this is not the case.

2. Non affine locally modular type

First, we fix an extended language L′ = L_{A} ∪ {P } where P is a unary predicate and L_{A} is the language of the theory of affine space over a finite field F of cardinality at least 5. (We shall see below a number of reasons why this condition is imposed. In fact, one can do the same construction with a projective space over a finite field of cardinality ≥ 4.) Now, suppose V_{1} is a finite affine space over F generated by independent elements {x_{0}, ..., x_{n}}. Let P colour some of the elements in V_{1}. We call the colouring good if for every line l in V_{1}, we can find at least 2 points in l coloured by P, and let gc(n) > 0 be the finite number of good colourings, for n ≥ 1, described by L′-formulas Col_{r,n}(x_{0}, x_{1}, …, x_{n}). Now suppose V_{2} ⊃ V_{1} is an affine space with basis {x_{0}, ..., x_{n+1}}. Then, given a good colouring Col_{r,n}(x_{0}, …, x_{n}) of V_{1}, there exists a good colouring extending it to V_{2}. (For example, colour every point in V_{2} \ V_{1} with P while leaving the colouring on V_{1}. Since every line in V_{2} is either already in V_{1} or hits V_{1} once, the condition q ≥ 5 implies that this is a good colouring.) We denote the finite number of all possible such colourings by gc(r, n) and let them be described by L′-formulas Col_{r,n}^{s,n+1}(x_{0}, x_{1}, …, x_{n}, x_{n+1}), for 1 ≤ s ≤ gc(r, n). For n = 0, we interpret Col_{0,0}^{s,1}(x_{0}, x) and Col_{1,0}^{s,1}(x_{0}, x) to mean that P gives a good colouring on the line generated by {x_{0}, x} extending the colourings ¬P and P respectively on x_{0}, and we let gc(0, 0) and gc(1, 0) denote the number of such colourings.

Now let T_{A} denote the theory of affine space over F. Then the L_{A}-formulas x ∈ cl(y_{1}, ..., y_{n}) and dim(y_{1}...y_{n}) = m(≤ n) make sense. (We note here that the dimension defined throughout the paper is one more than the usual affine subspace dimension.) The theory T_{LM} is given in the language L′ by the following axioms:

T_{LM} ≡ T_{A} ∪ {LM} ∪ {F_{n} : n ≥ 0},

where LM(local modularity) is the following sentence in L′:

LM ≡ ∀x∀y(x ≠ y → ∃z_{1}∃z_{2}(z_{1} ≠ z_{2} ∧ P (z_{1}) ∧ P (z_{2}) ∧ z_{1}, z_{2} ∈ cl(x, y))),

F_{n}(colour extension), for n ≥ 1, is the L′-sentence:

F_{n} ≡ ⋀_{1≤r≤gc(n)} ∀y_{0} … y_{n}(dim(y_{0} … y_{n}) = n + 1 ∧ Col_{r,n}(y_{0}, y_{1}, …, y_{n}) → ⋀_{1≤s≤gc(r,n)} ∃x(dim(y_{0} … y_{n}x) = n + 2 ∧ Col_{r,n}^{s,n+1}(y_{0}, y_{1}, …, y_{n}, x))), and F_{0} is the sentence:

F_{0} ≡ ∀y(¬P (y) → ⋀_{1≤s≤gc(0,0)} ∃x(dim(yx) = 2 ∧ Col_{0,0}^{s,1}(y, x))) ∧ . ∀y(P (y) → ⋀_{1≤s≤gc(1,0)} ∃x(dim cl(yx) = 2 ∧ Col_{1,0}^{s,1}(y, x))).

We first show that the theory T_{LM} is consistent. This can be done by a union of chains argument inside a model V of T_{A}. We first point out the following on any geometry.

Remark 2.1. Suppose that (S, cl) is a geometry. Let {A_{i} ∈ S|i ∈ I} be independent over B ⊆ S, i.e. for finite ā_{i} ∈ A_{i}, dim(ā_{i}/B) = dim(ā_{i}/B ∪ ⋃_{j≠i} A_{j}).

(1) cl(B) = cl(BA_{i_{1}} … A_{i_{n}}) ∩ cl(BA_{j_{1}} … A_{j_{m}}) for i_{1}, …, i_{n}, j_{1}, … j_{m} distinct.

(2) Each line in cl(B ∪ ⋃_{i} A_{i}) either already lies in one of cl(BA_{i}), or hits ⋃_{i} cl(BA_{i}) at most twice.

Proof. The first statement clearly comes from the fact that

dim(ā_{i_{1}} … ā_{i_{n}}/B) = dim(ā_{i_{1}} … ā_{i_{n}}/B ∪ ⋃_{j≠i_{1},... ,i_{n}} A_{j}).

For the second, assume that a line in cl(B ∪ ⋃_{i} A_{i}) hits ⋃_{i} cl(BA_{i}) 3 times, but does not lie in any cl(BA_{i}). Let {x, y, z} be the hit points. Then clearly x ∈ cl(BA_{k}), y ∈ cl(BA_{m})\cl(BA_{k}), z ∈ cl(BA_{n})\(cl(BA_{m})∪cl(BA_{k})) for distinct k, m, n, whereas z ∈ cl(xy) and z ∈ cl(BA_{n}) ∩ cl(BA_{k}A_{m}), this contradicts (1). □

Now we proceed in a series of steps.

Step 1.

Choose a y_{0} ∈ V and label y_{0} by P. Now add independent elements {x_{1}, …, x_{gc(1,0)}} over y_{0} and colour each cl(y_{0}x_{i}) to witness F_{0} for y_{0}. Let W_{1} = cl(y_{0}, x_{1}, …, x_{gc(1,0)}). Now colour all points in W_{1} ∖ ∪_{i}cl(y_{0}x_{i}) with P. Then W_{1} satisfies LM; if a line in W_{1} already lies in one of cl(y_{0}x_{i}), then we are done by construction. Otherwise by the above remark (2), the assumption Card(F) ≥ 5 ensures that we can pick up at least 3 points on the line labelled with P.

Step 2.

We now want to satisfy F_{0} in W_{1} \ {y_{0}}, so we repeat the method of Step 1 for each point in W_{1} \ {y_{0}}. Namely, for a ∈ W_{1} \ {y_{0}}, find independent {z_{1}, …, z_{gc(k,0)}} which are also independent from W_{1} (k = 0, 1 depends on a ∈ ¬P or a ∈ P). Then colour all the points in cl(W_{1} ∪ {z_{i}}_{i}) with P, except ∪_{i}cl(az_{i}) and W_{1}. W_{1} is already coloured. Colour each cl(az_{i}) to witness F_{0} for a. Then for the same reason as in step 1 ,with remark 2.1, cl(W_{1} ∪ {z_{i}}_{i}) satisfies LM. Now for b ∈ W_{1} \ {y_{0}, a}, do the same thing over cl(W_{1} ∪ {z_{i}}_{i}), and so on. We then obtain a subspace U_{0} ⊃ W_{1} such that U_{0} satisfies LM and for all x ∈ W_{1}, F_{0} is witnessed inside U_{0}.

Step 3.

Now by modifying Step 1 and 2, find U_{1}(⊃ U_{0}) to make F_{1} hold for W_{1} inside U_{1}. Namely, for given distinct x, y ∈ W_{1}, choose independent points {w_{k}}_{k} independent from U_{0} to extend the good colouring on cl(xy) to all possible good colourings on distinct affine planes cl(xyw_{k}). (These colourings are compatible due to the fact that cl(xy) = U_{0} ∩ cl(xyw_{k}) = cl(xyw_{i}) ∩ cl(xyw_{j}) (for i ≠ j by Remark 2.1.1.) Then colour all the points in cl(U_{0} ∪ {w_{k}}_{k}) other than U_{0} and ⋃_{k} cl(xyw_{k}) with P. Again, as in step 1, the fact that Card(F) ≥ 5 and Remark 2.1.2 guarantees that cl(U_{0} ∪ {w_{k}}_{k}) satisfies LM. For the next line from W_{1} other than cl(xy), repeat the process over cl(U_{0} ∪ {w_{k}}_{k}). By iterating this argument, we obtain a chain of spaces W_{1} ⊂ U_{0} ⊂ … ⊂ U_{n} so that U_{n} witnesses {F_{0}, … F_{n}} for W_{1}, and U_{n} satisfies LM. We can choose n large enough (depending on the dimension of W_{1}), so that in fact U_{n} witnesses all F_{i} (for i ≥ 0) for W_{1}.

Rename U_{n} as W_{2}, and repeat the previous steps for W_{2}. Continuing in this way we obtain a chain of spaces y_{0} ⊂ W_{1} ⊂ W_{2} ⊂ … W_{i} ⊂ … such that each W_{i} satisfies LM (by 2.1.2), and all F_{i} (i ≥ 0) are witnessed for W_{i} inside W_{i+1}. Let W = ⋃_{i} W_{i}, then by construction W ⊧ T_{A} ∪ {LM} ∪ {F_{i} : i ≥ 0}. Hence T_{LM} is consistent.

We now have the following theorem.

Theorem 2.2. T_{LM} is ω-categorical, complete and has quantifier elimination. Moreover, T_{LM} is a model companion of T_{A} ∪ {LM}.

The axioms {F_{n} : n ≥ 0} of colour extensions allow a partial isomorphism to be continued. Hence quantifier elimination follows. In particular for an independent tuple (a_{0}, ..., a_{n}), tp(a_{0}, ..., a_{n}) is determined by the colouring on cl(a_{0}, a_{1}, ..., a_{n}). ω-categoricity and hence completeness also follow by a straightforward back and forth partial isomorphism argument. Now given a model M of T_{A} ∪ {LM}, by doing the above process starting with M, one can obtain an extension N of M which is a model of T_{LM}. Hence T_{LM} is a model companion of T_{A} ∪ {LM}.

We now want to show that T_{LM} is a simple theory. As we observed, the condition Card(F) ≥ 4 is enough to obtain the previous theorem. However, we shall see that we need Card(F) ≥ 5 to ensure T_{LM} is simple, particularly for the Independence Theorem. Let us first define a notion of independence on subsets A, B, C of a saturated model V̄ of

T_{LM} by \begin{aligned}A & ⌣ | & B \\ & C \end{aligned} iff A is independent from B over C in the sense of affine space (i.e. in

L_{A}). It follows easily that this notion is automorphism invariant and satisfies symmetry, transitivity, finite character and local character. We need to check the extension axiom and the Independence Theorem. Below we keep using cl to denote affine closure.

Nonforking extension:

So let ā and B, C be a tuple and two small subsets of V̄. Then clearly there is ā′ such that ā′ is (L_{A}-)independent from C over B. Hence, as in 2.1.1, cl(ā′B) ∩ cl(BC) = cl(B). Therefore the colouring of cl(ā′B) according to cl(āB) is compatible with the colouring on cl(BC). Again colour all the points in cl(ā′BC) with P except cl(ā′B) and cl(BC). By a similar argument to that in Remark 2.1.2, cl(ā′BC) with the new colouring satisfies LM. Hence by the axioms {F_{n}} we can embed ā′, while fixing BC with the new colouring, into V̄, and then get the (L′-)nonforking extension of tp(ā/B) over BC.

The Independence Theorem for types:

We will verify this over any closed set B. Here types are L′-types. Let ā and b̄ be indepen-

dent tuples over B, and c̄, d̄ tuples such that \begin{aligned}c̄ & ⌣ | & ā \\ & B \end{aligned} , \begin{aligned}d̄ & ⌣ | & b̄ \\ & B \end{aligned} and tp(c̄/B) = tp(d̄/B).

By the extension property above, we can choose a tuple ē independent from b̄ over Bā such that tp(c̄/Bā) = tp(ē/Bā). Then {ē, ā, b̄} is independent over B. Consider cl(Bēāb̄). We now newly colour cl(Bēāb̄) while leaving the (old) colouring on cl(Bēā) ∪ cl(Bāb̄), as follows. First, colour cl(ē; Bb̄) according to cl(d̄; Bb̄). Since tp(ē/B) = tp(d̄/B), on cl(Bēb̄) ∩ (cl(Bāb̄) ∪ cl(Bāē)) = cl(Bb̄) ∪ cl(Bē) (Remark 2.1.1), the two colourings are the same. Hence the new colouring is compatible with the colouring on cl(Bēā) ∪ cl(Bāb̄). Then colour the rest of the points in cl(Bēāb̄) \ (cl(Bāb̄) ∪ cl(Bēā) ∪ cl(Bēb̄)) with P. Now, since every line in cl(Bēāb̄) either already lies in cl(Bāb̄) ∪ cl(Bēā) ∪ cl(Bēb̄) or hits the set at most 3 times, the condition Card(F) ≥ 5 ensures that this new colouring on cl(Bēāb̄) is a good colouring. As we can embed ē with the new colouring on cl(Bēāb̄) while fixing Bāb̄ the Independence Theorem for types holds. (Hence the notion of type coincides with that of Lascar strong type.)

So T_{LM} is simple. Moreover in V̄ (⊧ T_{LM}), as L′-independence is the same as L_{A}- independence, T_{LM} is a SU-rank 1 theory.

Theorem 2.3. T_{LM} is simple with SU-rank 1 having the same forking independence notion (and so algebraic relations) as T_{A}.

Now we take the solution set V̄ (P) = {a ∈ V̄ |P (a)}. As we checked the Independence Theorem over any closed set, in particular over ∅, V̄ (P) is a 1-(Lascar strong) type. We also have that V̄ (P) is simple, rank-1 and ω−categorical. It remains to show that V̄ (P) is locally modular but not affine.

Local modularity:

Choose d ∈ V̄ (P). Let ef be a pair and B a set from V̄ (P) such that dim_{d}(ef) = 2 and dim_{d}(ef/B) = 1 inside V̄ (P). Working in V̄, we also have dim_{d}(ef) = 2, dim_{d}(ef/B) = 1. As the geometry of V̄ is affine, acl(def) ∩ acl(dB) = acl(dg), where g ≠ d is in V̄. By the axiom LM for V̄ we can assume that g ∈ V̄ (P). Hence, dim_{d}(cl_{1}(def) ∩ cl_{1}(dB)) = 1, where cl_{1}(−) = acl(−) ∩ V̄ (P) is the closure inside V̄ (P). So V̄ (P) is a locally modular geometry.

Not affine:

For this it is sufficient to observe that a good colouring of a line in V̄ can label r points where 2 ≤ r ≤ Card(F).

So we have constructed a non-trivial, locally modular ω-categorical rank-1 structure/type whose geometry is neither affine nor projective over a finite field. Another example of this kind can be constructed, as mentioned previously, by repeating the same process on any projective space over a finite field of cardinality ≥ 4. (This is, in a sense, a necessary requirement. See 3.13.) As is well known, there is no such example in a stable structure, and also by 1.9 there is no example of a modular one.

3. 1-based type

Let us recall some basic facts on 1-basedness. In this section, algebraic closure is taken over M̄^{eq}. The following fact is folklore:

Fact 3.1. (T simple.) The following are equivalent.

(1) T is 1-based, i.e. any sets A and B are independent over acl(A) ∩ acl(B).

(2) For any tuple c̄ and a set A, cb(c̄/A) ⊆ acl(c̄).

(3) \begin{aligned}A & ⌣ | & B \\ & C \end{aligned} iff acl(AC) ∩ acl(BC) = acl(C).

(4) For any indiscernible sequence I = ⟨c̄_{i}|i ∈ ω⟩, I \ {c̄_{0}} is a Morley sequence over c̄_{0}.

Proof. We only sketch the proof. The equivalence of 1 and 2 is well-known. In 3, the implication of the right hand side from the left hand side is true for any simple theory. Easily, the left implication of 3 is equivalent to 1. 4⇒2 follows from the fact that cb(c̄/A) is in the algebraic (bounded) closure of a Morley sequence from Lstp(c̄/A). We supply a rather detailed

proof for 1⇒4. Note first that 1 implies \begin{aligned}c̄_{0} & ⌣ | & c̄_{1}c̄_{2}... \\ & A \end{aligned} where A = acl(c̄_{0}) ∩ acl(c̄_{1}c̄_{2}...).

Then since any indiscernible sequence over B is indiscernible over acl(B) (even over bdd(B)) (*), tp(c̄_{1}c̄_{2}.../acl(c̄_{0})) = tp(c̄_{2}c̄_{3}.../acl(c̄_{0})). Hence A = acl(c̄_{0}) ∩ acl(c̄_{2}c̄_{3}...). Again by (*), tp(c̄_{0}/acl(c̄_{2}c̄_{3}...)) = tp(c̄_{1}/acl(c̄_{2}c̄_{3}...)), and then A = acl(c̄_{1}) ∩ acl(c̄_{2}c_{3}..). Then, us-

ing 1 again, \begin{aligned}c̄_{1} & ⌣ | & c̄_{2}c̄_{3}... \\ & A \end{aligned} . From transitivity, \begin{aligned}c̄_{1} & ⌣ | & c̄_{2}c̄_{3}... \\ & c̄_{0}A \end{aligned} , and c̄_{1} ⌣ | c̄_{2}c̄_{3}... ; c̄_{0} since

A ⊆ acl(c̄_{0}). Thus 4 follows. □

When, for example, a theory T is 1-based of finite SU-rank (e.g. any 1-based ω-categorical T, see remark before 3.23), then as every tuple can be ‘built up’ (coordinatized) from rank-1 tuples, understanding the properties of rank-1 types is essential. This is the topic of the current section.

From now on, we further assume that D is the solution set of a rank-1 Lascar strong type with the usual algebraic closure relation. We shall observe a strong connection between 1-basedness and modularity. We first note the following folklore and also supply a simple proof.

Fact 3.2. If D is locally modular, then D is 1-based.

Proof. (Remark) Suppose that W_{0}, W_{1} ⊆ acl(X)∩acl(Y) and \begin{aligned}X & ⌣ | & Y \\ & W_{0} \end{aligned} and \begin{aligned}X & ⌣ | & Y \\ & W_{1} \end{aligned} .

Then acl(W_{0}) = acl(W_{1}), since \begin{aligned}W_{1} & ⌣ | & Y \\ & W_{0} \end{aligned} and \begin{aligned}W_{0} & ⌣ | & Y \\ & W_{1} \end{aligned} . (This remark will be used

occasionally.)

Claim 1) Choose p ∈ D. For any A ⊆ D^{eq}, there is B ⊆ D containing p such that A and B are interalgebraic over p: There is A_{0} ⊆ D such that A ⊆ dcl(A_{0}). Let A_{1}(⊆ D) realizing tp(A_{0}/Ap) be independent from A_{0} over Ap. Now, since D is modular over p, if we let B = acl(A_{0}p)∩acl(A_{1}p)∩D, then A_{0}, A_{1} are independent over B(= Bp). By the remark above, acl(Ap) = acl(Bp). Hence the claim is proved.

Claim 2) Given B_{0}, B_{1} ⊆ D^{eq}, B_{0} and B_{1} are independent over B_{2} = acl(B_{0}p) ∩ acl(B_{1}p) for given p ∈ D. (i.e. D is 1-based over p): Suppose not, say B_{0} forks with B_{1} over B_{2}. By Claim 1, there are C_{0}, C_{1}, C_{2} ⊆ D containing p such that C_{i} is interalgebraic with B_{i} over p. In particular, C_{0} forks with C_{1} over C_{2}. (Note that p ∈ B_{2}, C_{2}). Now as D is modular over p, there must exist d ∈ acl(C_{0}C_{2}) ∩ acl(C_{1}C_{2}) ∩ D such that d ∉ acl(C_{2}) = acl(B_{2}). However, since acl(C_{0}C_{2}) = acl(acl(C_{0}) ∪ acl(C_{2})) = acl(B_{0}pB_{2}) = acl(B_{0}B_{2}) and similarly acl(C_{1}C_{2}) = acl(B_{1}B_{2}), d ∈ acl(C_{0}C_{2}) ∩ acl(C_{1}C_{2}) = acl(B_{0}B_{2}) ∩ acl(B_{1}B_{2}) ⊆ B_{2}, this is a contradiction. Hence Claim 2 is proved.

Now we can finish the proof of the fact. Let I = ⟨c̄_{i}|i ∈ ω⟩ be an indiscernible sequence from D^{eq}. If q ∈ D is independent from I, then after a c_{0} conjugation we may assume that I is indiscernible over q. Now by claim 2 (with a similar proof to 3.1, 1⇒4), I \ {c̄_{0}} is a Morley sequence over qc̄_{0}. As q is independent from I, by transitivity I \ {c̄_{0}} is Morley over c̄_{0}. Hence, D is 1-based by 3.1. □

The converse of 3.2 is false in general. As in [21], one can easily construct a counterexample by adding a generic predicate P (x) to a vector space over a finite field. In Remark 3.7, we will deal with this example again.

Definition 3.3. We use G(D) to denote the collection of all rank-1 elements (over dom(D)) in D^{eq}.

Clearly D ⊆ G(D), and for any small set A containing dom(D), G(D)_{A} forms a pregeometry with the usual algebraic closure relation over A. We recall the definition of linearity.

Definition 3.4. For a set A containing dom(D), we say that D_{A} (G(D)_{A}, resp.) is linear if for any two points a, b ∈ D (G(D) resp.) and parameters B with SU(ab/AB) = 1, SU(C/A) ≤ 1 where C = cb(ab/AB).

Remark 3.5. (1) When defining linearity above, it is safe to assume that the parameters B form a subset of D (or G(D)) since we can preserve the rank by taking a Morley sequence as the parameter.

(2) If a canonical base Cb(p) of p(= Lstp of some tuple from D^{eq} over parameters containing dom(D)) has rank 1, then we can find a singleton c in D^{eq} which can be considered as the canonical base up to interalgebraicity. Namely, as T has elimination of hyperimaginaries, any hyperimaginary e of ordinal SU-rank is interdefinable with a sequence (e_{i} : i < ω) of imaginaries and there must be a subsequence such that e ⊂ acl(e_{1}, … e_{n}). In this case, we can take e = Cb(p) and find an element c in the sequence such that SU(c) = 1 and Cb(p) ⊆ acl(c). From now on, for this case (SU(Cb(p)) = 1), when we refer to a canonical case of p, we actually mean any such c ∈ D^{eq}. Since we are mainly interested in the geometry of D, as we shall see, this abuse of notation will not cause any problems. (If T does not have elimination of hyperimaginaries, then G(D) is the set of rank-1 hyperimaginaries living in D^{heq}.)

(3) We now single out an easy but useful argument in the proof of Claim 1 in 3.2. Let D ⊆ F ⊆ G(D), and let (F, cl(−)∩F) form a modular subpregeometry. Then for any A ⊆ D^{eq}, there is B ⊆ F such that A and B are interalgebraic: For each c ∈ D^{eq}, there is a finite tuple d̄ ∈ D such that c ∈ dcl(d̄). Hence, there is A_{0} ⊆ D ⊆ F such that A ⊆ dcl(A_{0}). Let A_{1}(⊆ F) realizing tp(A_{0}/A) be independent from A_{0} over A. Let B = acl(A_{0}) ∩ acl(A_{1}) ∩ F. Since F is modular, A_{0} and A_{1} are independent over B. Then by the remark in the proof of 3.2, acl(B) = acl(A).

Hence, if a ∈ G(D), then there is a tuple b̄ = (b_{1}, ..., b_{n}) ∈ F such that a, b̄ are interalgebraic. In particular, b_{1} ∈ acl(a), so a forks with b_{1}. Since rk(a) = 1, a ∈ acl(b_{1}), i.e. a, b_{1} are interalgebraic. This says that the canonical geometries of F and G are identical.

Theorem 3.6. The following are all equivalent.

(1) D ( D^{eq}) is 1-based.

(2) D is linear.

(3) G(D) is linear.

(4) G(D) is modular.

(5) G(D)_{A} is linear, for any (some) small A.

(6) G(D)_{A} is modular, for any (some) small A.

Proof. The proof of 1⇒2 is the same as the stable case. For completeness, we supply a proof. Let D be 1-based. Suppose that SU(Lstp(ab/B)) = 1 (a, b, B from D). By 1-basedness, C = cb(ab/B) is in acl(ab). Now,

SU(abC) = SU(ab/C) + SU(C) = 1 + SU(C) = SU(C/ab) + SU(ab) ≤ 2.

Hence SU(C) ≤ 1. Thus D is linear.

2⇒3. Assume that D is linear. Now let x, y ∈ G(D), B ⊆ G(D) and SU(xy/B) = 1 while

SU(xy) = 2 (1). Let C = cb(xy/B). Then C ⊆ acl(B) and SU(xy/C) = 1 ( \begin{matrix}xy & ⌣ | & B \\ & C \end{matrix} )

(⋆). We will show that C ⊆ acl(xy). Now {x, y} is in the definable closure of a finite independent subset of D. In fact, we can find an independent subset F ∪ {a, b} of D such that, {x, y} and {a, b} are interalgebraic over F, and F, {x, y} are independent. Then there

is F′ ⊆ D such that F and F′ realize the same type over xy, and \begin{aligned}F′ & ⌣ | & abBF \\ & xy \end{aligned} , hence

\begin{matrix}F′ & ⌣ | & xyabBF \\ & ∅ \end{matrix} (2). Now as F and F′ realize the same type over xy, there is \{a′b′\} such

that abF and a′b′F′ realize the same type over xy. Note that acl(xyF′) = acl(a′b′F′) (3).

Claim 1) SU(a′b′/F′B) = 1: First we note that \begin{aligned}a′b′ & ↓̸ & B \\ & F′ \end{aligned} . (Otherwise, by (2) and

(3), {x, y} is independent from B, which contradicts (1).) Now if SU(a′b′/F′B) = 0, then by (3) x, y ∈ acl(F′B), and then by (2) x, y ∈ acl(B), again contradicting (1). Hence SU(a′b′/F′B) = 1. Claim 1 is proved.

Therefore by Claim 1 and the linearity of D, SU(a′b′/acl(a′b′) ∩ acl(F′B)) = 1. Now

acl(a′b′) ∩ acl(F′B) ⊆ acl(a′b′F′) ∩ acl(F′B) ⊆ acl(F′B).

Thus by transitivity, Claim 1 and (3),

SU(a′b′/W) = 1, where W = acl(a′b′F′) ∩ acl(F′B) = acl(xyF′) ∩ acl(F′B) (4).

Claim 2) SU(xy/W) = SU(xy/W C) = 1(= SU(xy/C)): Since F′ ⊆ W, by (3) and

(4), \begin{aligned}xy & ↓̸ & W \\ & F′ \end{aligned} . Hence SU(xy/W) = 0 or 1. But if SU(xy/W) = 0 then by (2), xy ∈

acl(B) which is not the case. Thus SU(xy/W) = 1. Similarly, since SU(xy/C) = 1, if SU(xy/W C) ≠ 1, then by (⋆) and (4), xy ∈ acl(W C) ⊆ acl(CF′B) ⊆ acl(F′B). Then by (2), xy ∈ acl(B). This contradicts (1) again. Hence SU(xy/W C) = 1 and Claim 2 is proved.

Now since Lstp(xy/C) is a canonical type, by Claim 2, C ⊆ acl(W). Therefore by (4),

C ⊆ acl(xyF′). Then C ⊆ acl(xy), since (2) with (⋆) says that \begin{aligned}C & ⌣ | & F′ \\ & xy \end{aligned} . We have

proved that G(D) is linear.

3⇒6 and 4. Suppose that G(D) is linear. Let a, b ∈ G(D), and SU(ab/A) = 2, whereas SU(ab/AB) = 1 for B ⊆ G(D). Now, there is a Morley sequence I of tp(ab/AB) in G(D) such that SU(ab/AB) = SU(ab/ABI) = SU(ab/I) = 1. By linearity of G(D), c ∈ acl(ab) ∩ acl(I) ∩ acl(AB) where c = cb(ab/I) = cb(ab/AB), and SU(c) = 1. Hence c ∈ G(D) by the previous remark 3.5.2. Moreover, by the rank calculation of SU(cab/A), SU(c/A) = 1, i.e c ∉ acl(A). Thus G(D)_{A} is modular by 1.3.

4⇒1. Proof of this will be similar to the proof of fact 3.2. Suppose not, say there are a, b ∈ D^{eq} which are dependent over U = acl(a) ∩ acl(b). Now since G(D) is modular, by Remark 3.5.3, there are a_{0}, b_{0}, U_{0} ⊆ G(D) which are interalgebraic with a, b, U respectively. In particular, a_{0} forks with b_{0} over U_{0}. Then again by modularity, there must exist c_{0} ∈ acl(a_{0}U_{0}) ∩ acl(b_{0}U_{0}) ∩ G(D) such that c_{0} ∉ acl(U_{0}) = acl(U). However, since acl(a_{0}U_{0}) = acl(acl(a_{0}) ∪ acl(U_{0})) = acl(aU) and similarly acl(b_{0}U_{0}) = acl(bU), c_{0} ∈ acl(aU) ∩ acl(bU) ⊆ U, a contradiction. Therefore, 4⇒1 is proved.

3⇒5. Suppose that G(D) is linear. Let us show that G(D)_{A} is linear. Let a, b ∈ G(D), and SU(ab/B) = 1 for B containing A. Now there is a Morley sequence I of tp(ab/B) in G(D) such that SU(ab/B) = SU(ab/BI) = SU(ab/I) = 1. Then by linearity of G(D), SU(c/A) ≤ SU(c) ≤ 1 where c = cb(ab/I) = cb(ab/B). Thus G(D)_{A} is linear.

5 or 6⇒3. The proof will be similar to [17, 2.2.6(i)⇒(ii)]. Let a, b ∈ G(D), B ⊆ G(D) and SU(ab/B) = 1. Let c = cb(ab/B). If SU(ab) = 1, i.e. {a, b} is independent from B, then as c ∈ acl(∅), there is nothing to prove. Thus we can assume that SU(ab) = 2. Now we can further assume that abB is independent from A. Hence, SU(ab/BA) = 1, whereas SU(ab/A) = 2. Then by modularity (or linearity) of G(D)_{A}, there is d ∈ acl(abA)∩ acl(AB) and SU(d/A) = 1. Hence d and ab are dependent over A. Thus SU(ab/Ad) = 1 = SU(ab/AB). Therefore c = cb(ab/AB) ∈ acl(Ad), and as SU(d/A) = 1, by a rank calculation SU(c/A) ≤ 1. Since c and A are independent, SU(c) ≤ 1. □

Remark 3.7. (1) Similarily, D is linear iff D_{A} is linear for any (some) A. We note that the geometry of a nonforking extension of D over A is a mere subgeometry of D_{A}. However one can still see the fact that linearity (1-basedness) is invariant via parallelism: Let q be a nonforking extension of D over A. It suffices to check that D is linear when q is linear. The proof is the same as 3.6(5)⇒(3) with the trick that when an independent pair (a_{0}, a_{1}) is chosen from D, by amalgamation there is (b_{0}, b_{1}) such that tp(a_{0}, a_{1}) = tp(b_{0}, b_{1}) and each b_{i} realizes q.

However, as is known, (local) modularity is not invariant via parallelism in general. (Frank O. Wagner was the first person to point this out.) Affine or projective space equipped with a binary random graph relation serves as an example, by a similar reason to that described in (2).

(2) Vassiliev introduced a notion of generic substructure M̄_{1} of M̄ for a supersimple SUrank 1 theory (the submodel M̄_{1} of M̄ does not have to be a small submodel). From [21], it follows that when we consider D as the universe of a model M̄, D is 1-based iff DM̄_{1} is modular. (He also proved the equivalence of 1 and 2 in the previous theorem 3.6 using a generic pair argument.)

However, in general, even if D is 1-based, D_{A} need not be modular over any (small) set A: Let V be a (saturated) affine space over a field, and let P be a generic predicate on V, as in [5]. Then there are two 1-types over ∅ isolated by P (x) and ¬P (x), both of which have SU-rank 1. By the genericity of P, both 1-types are 1-based. V is already non-modular over ∅. Now let W be a non-empty small subspace of V and consider a line l, spanned by {a, b} in V, independent from W, so dim(l/W) = 2. Now let l′, spanned by {a′b′}, be parallel to l such that dim(l/W l′) = 1. Then there is c ∈ V \ W such that acl(cW) = acl(lW) ∩ acl(l′W). However, as W is small, by genericity of P, we can find l, l′ such that {a, b, a′, b′} ⊆ P whereas P ∩ acl(cW) = P ∩ W. Then clearly P cannot be modular over W and similarily for ¬P. (We noticed later that a similar example is described in [21].)

To conclude, there are examples of non-trivial, 1-based D which are either non locally modular or locally modular but are neither affine nor projective (section 2) and examples of 1-based D which are affine/projective but not stable (an affine space with a binary random graph relation).

Since triviality of a (pre)geometry implies modularity, we have the following lemma by Remark 3.5.3.

Lemma 3.8. D is trivial iff G(D) is trivial.

Theorem 3.9. Suppose D is 1-based. Then D is trivial or the geometry of G(D) is projective geometry over a division ring.

Proof. Assume that D is non-trivial. By the above lemma and Fact 1.5, it suffices to show that any independent pair (a, b) ∈ G(D) (that is, dim(ab) = 2) is non-trivial in G(D) (i.e. there is c ∈ cl(ab) \ (cl(a) ∪ cl(b)) where cl is the closure of G(D)).

Claim 1) Any independent pair (c, d) ∈ D is non-trivial in G(D): Since G(D) is modular, if there is an independent pair from D trivial in G(D), then any independent pair from D is trivial in G(D). (Otherwise, by type amalgamation one can find a non-modular triangle which violates modularity of G(D). See the proof of 1.8.) Then any independent triple (a, b, c) ∈ D is also trivial in G(D) (otherwise, say d ∈ cl(abc) \ (cl(a) ∪ cl(b) ∪ cl(c)), then ab and cd violate modularity of G(D)). Iterating this argument shows that D is trivial, which is not the case.

Claim 2) Any independent pair (a, b) ∈ G(D) with a ∈ D is non-trivial in G(D): Suppose not, say the independent pair (a, b) ∈ G(D) with a ∈ D is trivial. Now since b ∈ G(D), there is an independent tuple c̄ = (c_{1}, ...c_{n}) ∈ D such that b ∈ cl(c_{1}, ...c_{n}). By claim 1, we can assume n > 1. We can further assume that any n − 1 subtuple of c̄ is independent from b, and {a, b, c_{1}, ..., c_{n−1}} is independent. Hence dim(a, b, c̄) = n + 1 (*). By claim 1, there is d ∈ cl(ac_{1}) \ cl(a) ∪ cl(c_{1}). Then it follows that d ∉ cl(c_{2}, ..., c_{n}), (otherwise a ∈ cl(c̄) violating (*)). Hence the dimension of the set B = {d, c_{2}, ..., c_{n}} is n, whereas by (*) and triviality of (a, b), it can easily be seen that cl(B) ∩ cl(ab) = ∅. However, by modularity of G(D), this contradicts (*). Hence claim 2 follows.

Claim 3) Any independent pair (d, e) ∈ G(D) is non-trivial: Suppose not, say the independent pair (d, e) ∈ G(D) is trivial. There is a point c ∈ D which is independent from (d, e). By claim 2, (c, d) and (c, e) are non-trivial. Again this violates modularity of G(D), as (c, d, e) forms a non-modular triangle. This finishes the proof of claim 3 and the theorem. □

The theorem shows Vassiliev’s result cheaply. Namely, the geometry of 1-based D is a subgeometry of projective geometry over a division ring.

From now on D will be 1-based.

The (pre)geometry of G(D) is not definable (it is a union of definable sets). However, with the reducibility condition we will discuss below, G(D) can be dealt with as a definable object, at least for the ω-categorical case.

Definition 3.10. We say G(D) is reducible into (or has a reduction in) D^{k} if for any point a ∈ G(D) there is a k-tuple b̄ ∈ D such that a ∈ acl(b̄).

When D is locally modular, G(D) has a reduction in D^{2}. In fact, more is true.

Lemma 3.11. (D locally modular.) For c ∈ G(D) and any d ∈ D, there is e ∈ D such that c ∈ acl(de).

Proof. By Claim 1 in the proof of 3.2, there is ā = (a_{1}, ..., a_{n}) ∈ D such that acl(ād) = acl(dc). If ā ∈ acl(d), then c ∈ acl(d) and we are done. Hence we may assume that some a_{i}, say a_{1}, is not in acl(d). Then, since rk(a_{1}) = 1, a_{1} forks with c over d. By transitivity, c forks with a_{1}d. Since rk(c) = 1, c ∈ acl(da_{1}). □

Now, as by-products, we have the following lemmas.

Lemma 3.12. (D is non-trivial locally modular, p ∈ D.) The geometry of D_{p} and the geometry of G(D) are isomorphic as projective geometries.

Proof. Since G(D)′, the geometry of G(D), is that of infinite dimensional projective space over some ring, G(D)′_{p} ≅ G(D)′. Now as D ⊆ G(D), clearly D_{p} is a subpregeometry of G(D)_{p}. Also, by the previous lemma, given x ∈ G(D) \ cl(p), there is e ∈ D such that cl(px) = cl(ex) in G(D). Hence D′_{p} ≅ G(D)′_{p} ≅ G(D)′, and the lemma holds. □

Lemma 3.13. (D locally modular.) If G(D)′ is projective over a field with Card(F) ≤ 3, then D′, the geometry of D, is affine or projective.

Proof. This can be shown by checking the possible cases, using amalgamation and 3.11, working in a projective plane. Let us first prove this when Card(F) = 3:

Claim 1) There is no independent {a, b} ⊆ D′ such that cl(ab) ∩ D′ = {a, b}: if there exist such a, b then, by amalgamation, we can find c ∈ D′ such that dim(abc) = 3 and any closed set in D′ generated by a 2-subset of {a, b, c} is itself. Let b′ ∈ (cl(bc) ∩ G(D)′) \ D′. Then by 3.11, we can fine d ∈ D′ such that b′ ∈ cl(ad). By modularity and the fact that any line in G(D)′ has size 4, there are c′ ∈ cl(ac) ∩ cl(bd) ∩ G(D) and c′′ ∈ cl(ac) \ D′ such that c′′ ≠ a, c, c′. Then it is easy to see that a, b, c ∉ cl(dc′′). Hence ∅ ≠ cl(dc′′)∩cl(bc) ∈ G(D)\D, and also, cl(dc′′) ∩ cl(ab) ∈ G(D) \ D. Therefore cl(dc′′) ∩ D′ = {d}, which contradicts 3.11. Hence claim 1 is proved.

Claim 2) There cannot exist independent pairs {a_{1}, a_{2}}, {b_{1}, b_{2}} ⊆ D′ such that Card(cl(a_{1}a_{2})∩ D′) = 3 and Card(cl(b_{1}b_{2}) ∩ D′) = 4: suppose that there are such points a_{i} and b_{i} in D′. By amalgamation, there are {u, v, w} ⊆ D′ such that Card(cl(uv)∩D′) = Card(cl(vw)∩D′) = 3, Card(cl(uw) ∩ D′) = 4 and dim(uvw) = 3. Choose v′ ∈ cl(uv) \ D′ and w′ ∈ cl(vw) \ D′v′. Then, as Card(cl(uw) ∩ D′) = 4, by modularity there is z ∈ cl(uw) ∩ cl(v′w′) ∩ D′. Then 1 ≤ Card(cl(v′w′) ∩ D′) ≤ 2, contradicting 3.11 and Claim 1. Hence claim 2 follows and the lemma is established for Card(F) = 3 by Fact 1.6.

When Card(F) = 2, by a similar argument, it can be seen that one cannot amalgamate independent (c_{1}c_{2}), (c_{1}c_{2}) and (d_{1}d_{2}) from D′ such that Card(cl(c_{1}c_{2}) ∩ D′) = 2 and Card(cl(d_{1}d_{2}) ∩ D′) = 3. Then 1.6 says the lemma holds for this case as well. □

Remark 3.14. If D is a generic subset of locally modular U, then G(D) also has a reduction in D^{2}: Let c ∈ G(D). Then by 3.11, there are d ∈ D, e ∈ U such that c ∈ acl(de). Now

there exists (a, b) realizing tp(de/c) such that \begin{aligned}ab & ⌣ | & de \\ & c \end{aligned} . Then by the remark in the proof

of 3.2, acl(g) = acl(c) where g = cb(ab/de). By the genericity of D, there exists (a′, b′) ∈ D such that tp^{−}(ab/de) = tp^{−}(a′b′/de) in the language of U without the predicate D. Then a′b′ preserves the canonical base of tp(ab/de). Hence acl(c) = acl(g) ⊆ acl(a′b′). We are done. (In this case, of course, strong reducibility in D^{2} as in 3.11 does not follow.)

In general, the following holds. Hence any G(D) is reducible into D^{3} for 1-based D.

Theorem 3.15. Let u ∈ G(D).

(1) Suppose that (x, y) is a fixed independent pair from D. Then u ∈ acl(x′y′z) where x′, y′, z ∈ D and tp(xy) = tp(x′y′).

(2) Suppose that there are independent {x, y} from D such that for any z ∈ acl(xy) ∩ G(D), z ∈ acl(z′) for some z′ ∈ (acl(xy) ∩ D), i.e. a full line is already in D. Then G(D) is reducible into D^{2}.

(3) For given d_{1} ∈ D, there is d_{2}d_{3} ∈ D such that u ∈ acl(d_{1}d_{2}d_{3}).

Proof. We first prove (1) using induction on n where u ∈ G(D) has a reduction in D^{n}. The case n = 0, that is u ∈ acl(∅), is trivial. Now suppose that u ∈ acl(a_{1}...a_{n+1}) (a_{i} ∈ D) with the induction hypothesis for n. We will verify (1) for n + 1. We can assume that {a_{1}, ..., a_{n+1}} is independent. Now let g = cb(ua_{1}/a_{2}...a_{n+1}). Thus SU(g) = 1 and g ∈ G(D). Since a_{1} ∉ acl(g) ⊆ acl(a_{2}...a_{n+1}), u is in the line generated by {g, a_{1}} (*). Then, since g ∈ acl(a_{2}...a_{n+1}), by the induction hypothesis, g ∈ acl(bcd) for some c, d ∈ D such that tp(cd) = tp(xy). If g is already in acl(cd), then from (*), u ∈ acl(ga_{1}) ⊆ acl(cda_{1}). Thus, in this case, the theorem holds. Therefore we only need to consider the case when g ∉ acl(cd). We can also clearly assume that g ∉ acl(b) (otherwise we are done). Then, since G(D) is modular, we can find v ∈ acl(cd) ∩ acl(bg) ∩ G(D) such that g ∈ acl(bv) (†), and dim(vg) = 2. Also, at least one of {v, c} or {v, d} ( say {v, d}) is independent (**). Now, as d, a_{1} ∈ D, Lstp(d) = Lstp(a_{1}). Hence, we can find v′ such that Lstp(vd) = Lstp(v′a_{1}) (⋆). In particular, Lstp(v) = Lstp(v′). Then by (*),(**),(⋆), we can amalgamate types tp(v′/a_{1}) and tp(v/g), so that we obtain v′′ ⊧ tp(v′/a_{1}) ∪ tp(v/g). Hence, there are b′, c′ such that tp(vgb) = tp(v′′gb′), tp(vdc) = tp(v′′a_{1}c′) (by (⋆)), and so by (†), tp(c′a_{1}) = tp(xy), v′′ ∈ acl(c′a_{1}) and g ∈ acl(b′v′′) (‡). Then by (*),(‡),

u ∈ acl(ga_{1}) ⊆ acl(ga_{1}v′′) ⊆ acl(a_{1}b′v′′) ⊆ acl(c′a_{1}b′).

Since tp(c′a_{1}) = tp(xy), the (n + 1)th induction hypothesis for (1) is deduced. Therefore we have proved (1). (2) comes from (1) and the modularity of G(D).

For (3), use the proof of (1), replacing g with u and a_{1} with d_{1}. □

Remark 3.16. From the above result 3.15, we can identify any element u in G(D) as e = d̄/E ∈ G(D) (i.e. acl(u) = acl(e)) where d̄ ∈ D^{6} and E is a definable equivalence relation on D^{6} (over dom(D)), as follows: As usual we take dom(D) = ∅ (by naming dom(D)) for notational simplicity. By 3.15, there are triples (a, b, c), (a′, b′, c′) ∈ D^{3} such

that u ∈ acl(abc), (a′b′c′) ⊧ tp(abc/u) and \begin{aligned}abc & ⌣ | & a′b′c′ \\ & u \end{aligned} . Then by the remark in 3.2,

acl(u) = acl(g) where g = cb(abc/a′b′c′). Now if an automorphism fixes (abca′b′c′) then as the parallelism class of Lstp(abc/a′b′c′) is setwise fixed, so is cb(abc/a′b′c′) , i.e. g ∈ dcl(abca′b′c′). Hence we can identify g = (abca′b′c′)/F̄ where F̄ is a type-definable equivalence relation on D^{6}. Therefore, since SU(g) = 1, as in 3.5.2, there is a definable equivalence relation E ∈ F̄ such that (abca′b′c′)/E in G(D) and g are interalgebraic. As u and g are interalgebraic, the result follows.

In particular, if D is ω-categorical, then as there are only finitely many types and equivalence relations over dom(D) on D^{6}, by the above remark, the geometry of G(D) is isomorphic to that of some union of finitely many definable sets in finitely many sorts in G(D). Since the union itself is ω-categorical even as a 1-sorted structure, we have the following results.

Corollary 3.17. D ω-categorical. Then, D is trivial or the geometry of G(D) is projective over a finite field.

Definition 3.18. Let G(D) have a reduction in D^{k} where k is minimal such. (Hence k ≤ 3.) By a hypersurface of D^{k}, we mean a Lstp(a_{1}...a_{k}/A) of rank k − 1 (a_{i} ∈ D). We call a hypersurface of D^{k} with rank-0 canonical base rigid.

The Projective closure of D, denoted P (D), is defined as:

P (D) = D ∪ {e|e is the canonical base of a non-rigid hypersurface of D^{k}}.

By a rank calculation, the rank of the canonical base of a hypersurface of D^{k} is ≤ 1. Hence P (D) is a collection of canonical bases e = cb(a_{1}...a_{k}/e) where SU(a_{1}...a_{k}/e) = k − 1 for independent (a_{1}, ..., a_{k}) ∈ D^{k}. Since SU(e) = 1, by Remark 3.5.2 we can assume that P (D) ⊆ G(D) by identifying e with some interalgebraic g ∈ G(D). If D is ω-categorical, we can moreover assume, as in 3.16, that P (D) is a dom(D)-definable subset of G(D).

Lemma 3.19. (Notation as in 3.16) For any point u ∈ G(D), there is g ∈ P (D) such that g and u are interalgebraic. (Hence the geometries of P (D) and G(D) coincide.)

Proof. This is immediate from Remark 3.16, using the fact that there SU(abc/a′b′c′) = 2. □

Then we get the following theorem immediately. This answers the main conjecture in [21] positively.

Theorem 3.20. Suppose D is ω-categorical. Then P (D) is dom(D)-definable (in D^{eq}), as is its geometry P (D)′. Hence, if D is non-trivial, P (D)′ has a strongly minimal stable reduct (over dom(D)) preserving the projective geometry over a finite field F. Hence, over a finite parameter in D^{eq}, D interprets a vector space (as projective and affine space do) over F.

(Recently Ben-Yaacov, Tomasic and Wagner obtained an almost hyperimaginary group configuration in [2]. Using this, they also proved the interpretability of a vector space in an ω-categorical non-trivial D [20].)

Now we study the relationship between the whole theory and rank-1 types. As in the stable case, the notion of triviality can be imposed on the underlying theory.

Definition 3.21. We say that T is trivial if for a, b, c, A ⊆ M̄^{eq}, whenever {a, b, c} is pairwise independent over A, then {a, b, c} is independent over A.

By mimicking the proofs in the stable case found in [17, 4.2.5], with some novel argument, we can show 3.22 below. The following easy fact will be used. In any simple T, if a finite tuple ā is not in acl(∅), then there is a complete extension p of tp(ā) such that SU(p) = 1: Let tp(ā) = p_{o} ⊆ p_{1}... ⊆ p_{i}.. be a maximal non-algebraic forking chain. (The length is at most |T |.) Then p = ∪_{i}p_{i} is non-algebraic as well and, by maximality, SU(p) = 1.

Lemma 3.22. Suppose that T is 1-based. Then T is trivial if and only if all rank-1 types are trivial.

Proof. We only need to prove the right to left direction, as the other direction is obvious. Suppose that there exists a non-trivial triple (a, b, c) over A, i.e. (a, b, c) is pairwise independent over A, whereas {a, b, c} is dependent over A. We will find a non-trivial rank-1 type.

Claim 1) We may assume that a ∈ acl(bcA), b ∈ acl(acA) and c ∈ acl(baA), (but {a, b, c} is still pairwise independent over A): if we choose a_{0} ∈ cb(a/bcA) \ acl(A), b_{0} ∈ cb(b/a_{0}cA) \ acl(A), c_{0} ∈ cb(c/a_{0}b_{0}A) \ acl(A) such that cb(a/bcA) ⊂ acl(a_{0}) and similarly for b_{0}, c_{0}, then, by 1-basedness, these elements will suffice.

Now, for notational convenience, let A = ∅.

Claim 2) There is F such that SU(a/F) = SU(b/F) = SU(c/F) = 1 and (a, b, c) is still a non-trivial triple over F: By the remark before 3.22, there is d such that SU(a/d) = 1.

We assume d = cb(a/d). Then by 1-basedness, d ∈ acl(a) and \begin{aligned}bd & ⌣ | & cd \\ & F \end{aligned} where F =

acl(bd) ∩ acl(cd). Then \begin{aligned}b & ⌣ | & c \\ & F \end{aligned} . Now we have SU(a/F) = 1, since otherwise a ∈ acl(bd),

so a ∈ acl(d) (as \begin{matrix}ad & ⌣ | & b \\ & ∅ \end{matrix} ) contradicting the fact that SU(a/d) = 1. Similarly \begin{aligned}a & ⌣ | & b \\ & F \end{aligned}

and \begin{aligned}a & ⌣ | & c \\ & F \end{aligned} . Now by the rank calculation of SU(ab/cF), SU(b/cF) = SU(b/F) = 1 since

SU(a/cF) = 1. Similarly by the rank calculation of SU(ac/bF), SU(c/bF) = SU(c/F) = 1. Hence, claim 2 is established.

Claim 3) tp(a/F) is non-trivial. (This finishes the proof.): First, there are a_{1}, a_{2}, b′, c′ such

that tp(a_{1}c′/F b) = tp(ac/F b), \begin{aligned}c′ & ⌣ | & abc \\ & F \end{aligned} , tp(a_{2}b′/F c) = tp(ab/F c) (*) and \begin{aligned}b′ & ⌣ | & abcc′ \\ & F \end{aligned} .

Hence \begin{matrix}b′c′ & ⌣ | & abc \\ & F \end{matrix} (**). Now we subclaim that a, a_{1}, a_{2}, all realizing the rank-1 type

tp(a/F), form a non-trivial triple over F b′c′. For pairwise independence, we only show

\begin{aligned}a_{1} & ⌣ | & a_{2} \\ & F b′c′ \end{aligned} , the other cases follow similarily. We have \begin{aligned}b & ⌣ | & c \\ & F \end{aligned} , hence \begin{aligned}b & ⌣ | & c \\ & F b′c′ \end{aligned} by

(**). Thus \begin{aligned}F bc′ & ⌣ | & F cb′ \\ & F b′c′ \end{aligned} . Then it follows \begin{aligned}a_{1} & ⌣ | & a_{2} \\ & F b′c′ \end{aligned} by Claim 2 and (*). Now as

a ∈ acl(bc), again by (*) and Claim 2, each of a, a_{1}, a_{2} is algebraic over F b′c′ together with the other two. Hence the subclaim is verified.

Secondly, by 1-basedness, there is a tuple a_{3}a_{4}a_{5} realizing tp(aa_{1}a_{2}/F b′c′) such that

\begin{aligned}aa_{1}a_{2} & ⌣ | & F b′c′ \\ & a_{3}a_{4}a_{5} \end{aligned} and \begin{aligned}aa_{1}a_{2} & ⌣ | & a_{3}a_{4}a_{5} \\ & F b′c′ \end{aligned} . Then, it can be easily seen that (a, a_{1}, a_{2})

forms a non-trivial triple over F a_{3}a_{4}a_{5}. This means that the localized geometry of tp(a/F) at {a_{3}, a_{4}, a_{5}} is non-trivial since all the a and a_{i} realize the rank-1 type tp(a/F). Hence, the original type tp(a/F) cannot be trivial either. We have proved claim 3 and the lemma. □

Recall that if T is 1-based, then for any A and tuple ā, there is ā′ with |ā| = |ā′| such that D(tp(ā/A), φ, k) = D(tp(ā/ā′), φ, k) for each φ, k (by taking an element in a Morley sequence for example). Thus, if additionally T is ω-categorical, then T is supersimple since every forking chain must have finite length. Moreover, since SU(ā/A) = SU(ā/ā′), any ntype in T has finite SU-rank ([9, 4.7]). (Evans and Wagner [8] showed that any supersimple, ω-categorical group or any CM-trivial theory also has finite SU-rank.)

By previous results in this section, we obtain the following.

Corollary 3.23. Let T be non-trivial, 1-based and ω-categorical. Then an infinite dimensional vector space over a finite field, in particular the infinite additive group, is definable in M̄^{eq} over a finite parameter.

References

[1] E. Artin, Geometric algebra, Interscience Publishers, New York (1957).

[2] I. Ben-Yaacov, I. Tomasic and F. O. Wagner, ‘Constructing an almost hyperdefinable group’, preprint 2001.

[3] S. Buechler, ‘The geometry of weakly minimal types’, J. of Symbolic Logic 50 (1985) 1044-1053.

[4] S. Buechler, A. Pillay and F. O. Wagner ‘Supersimple theories’, J. of AMS 14 (1998) 109-124.

[5] Z. Chatzidakis and A. Pillay, ‘Generic structures and simple theories’, Ann. of Pure and Applied Logic 95 (1998) 71-92.

[6] G. Cherlin, L. Harrington, and A. H. Lachlan, ‘ℵ_{0}-categorical, ℵ_{0}-stable structures’, Ann. Pure and Applied Logic 28 (1985) 103-135.

[7] J. Doyen and X. Hubaut, ‘Finite regular locally projective spaces’, Math. Z. 119 (1971) 83-88 .

[8] D. Evans and F. O. Wagner, ‘Supersimple ω-categorical groups’, J. of Symbolic Logic 65 (2000) 767-776.

[9] B. Hart, B. Kim and A. Pillay, ‘Coordinatisation and canonical bases in simple theories’, Journal of Symbolic Logic, 65 (2000) 293-309.

[10] E. Hrushovski, ‘Unimodular minimal theories’, Journal of London Math. Soc., 46 (1992) 385-396.

[11] E. Hrushovski, ‘Locally modular regular types’, in Classification theory, Lecture Notes in Math. 1292, Springer-Verlag, Berlin (1987) 132-164.

[12] E. Hrushovski, ‘The Mordell-Lang conjecture for function fields’, J. of AMS 9 (1996) 667-690.

[13] E. Hrushovski, ‘Simplicity and the Lascar group’, preprint (1998).

[14] E. Hrushovski and S. Shelah, ‘A dichotomy theorem for regular types’, Ann. of Pure and Applied Logic 45 (1989) 157-169.

[15] B. Kim, ‘Forking in simple unstable theories’, Journal of London Math. Soc. (2) 57 (1998) 257-267.

[16] B. Kim and A. Pillay, ‘Simple theories’, Ann. of Pure and Applied Logic 88 (1997) 149-164.

[17] A. Pillay, Geometric stability theory, Oxford University Press, Oxford (1996).

[18] S. Shelah, ‘Simple unstable theories’, Annals of Math. Logic 19 (1980), 177-203.

[19] S. Shelah, Classification theory, 2nd edn, North-Holland, Amsterdam (1990).

[20] I. Tomasic and F. O. Wagner, ‘Applications of the group configuration theorems in simple theories’, preprint 2001.

[21] E. Vassiliev, ‘Expansions of rank 1 structures’, Ph. D. Thesis, Univ. of Notre Dame (2001).

[22] F. O. Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht (2000).

[23] B. I. Zilber, Uncountably categorical theories, AMS Translations of Mathematical Monographs, Vol. 117 (1993).

Mathematics Department, MIT, 77 Mass. Ave., Cambridge, MA 02139

  E-mail address: tdpdp@math.mit.edu

Mathematics Department, MIT, 77 Mass. Ave., Cambridge, MA 02139

  E-mail address: bkim@math.mit.edu