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INTRODUCTION TO GEOMETRIC STABILITY

ARTEM CHERNIKOV

Lecture notes for the IMS Graduate Summer School in Logic, National Universityof Singapore, Jun 2017. The material is based on a number of sources including:

• Lecture notes on “Elements of Geometric Stability Theory” by Boris Zilber• “Model Theory: An Introduction” by David Marker• "A Course in Model Theory" by Katrin Tent and Martin Ziegler• “Geometric stability theory” by Anand Pillay• Lecture notes “Geometric Stability Theory” by Martin Bays

All comments and corrections are very welcome.Last update: June 24, 2017

Contents

1. Preliminaries 22. Morley’s categoricity theorem 32.1. Algebraic closure 42.2. Strongly minimal formulas and theories 52.3. Algebraic closure in strongly minimal sets and pregeometries 52.4. Dimension and independence in pregeometries 62.5. Uncountable categoricity of s.m. theories 72.6. ω-stability 82.7. Ehrenfeucht–Mostowski Models 102.8. Two-cardinal models and elimination of ∃∞. 112.9. Morley’s theorem, proof by Baldwin-Lachlan 122.10. Further work on counting models 133. Morley rank and forking independence in ω-stable theories 143.1. M eq and canonical parameters 143.2. Morley rank 163.3. Definability of types 173.4. Forking independence 183.5. Canonical bases 194. A Closer look at pregeometries and Zilber’s trichotomy principle 204.1. Characterization of local modularity 224.2. Families of plane curves 235. Group configuration theorem 255.1. Germs of definable functions 255.2. Hrushovski-Weil 265.3. The group configuration 28References 29

1

https://people.maths.ox.ac.uk/zilber/est.pdf

INTRODUCTION TO GEOMETRIC STABILITY 2

1. Preliminaries

We recall briefly some basic facts and definitions in order to set up the notationfor our discussion.

• M = (M,R1, . . . , f1, . . . , c1, . . .) denotes a first-order structure in a lan-guage L.

• T denotes a first-order theory,M |= T if it satisfies all sentences in T .• Th (M) denotes the complete theory ofM.• Given A ⊆M , φ (x) is an L (A)-formula (formula over a set of parametersA) if it is of the form φ (x) = ψ

(x, b)where ψ (x, y) is an L-formula and b

is tuple from A.• We will abuse notation and write x, y, z, etc. to denote tuples of variables

when there is no confusion.• If Ψ (x) ⊆ L (A), a set of L (A) formulas and a ∈ M , we write a |= Ψ (x)⇐⇒ a satisfies all formulas in Ψ.

• Given B ⊆M |x|, Ψ (B) := {b ∈ B :M |= Ψ (a)}.• X ⊆Mn is an A-definable set if A = ψ (M) for some ψ (x) ∈ L (A).• Defn (A) is the boolean algebra of all A-definable subsets of Mn.• Given L-structuresM,N , writeM≡ N to denote elementary equivalence

(Th (M) = Th (N )).• Given a (partial) map f :M→ N , f is elementary if for all a ∈ Dom (f)

and φ ∈ L,M |= φ (a) ⇐⇒ N |= φ (f (a)).• M � N is an elementary substructure if the embedding map is elementary.

Fact 1.1. (Compactness theorem) Let L be any language and Ψ a set of L-sentences(of any cardinality). If every finite Ψ0 ⊆ Ψ is consistent (i.e. there is some L-structureM |=Ψ0), then Ψ is consistent.

Fact 1.2. (Löwenheim–Skolem theorem) Let M |= T be given, with |M| ≥ ℵ0.Then for any cardinal κ ≥ |L| there is some N with |N | = κ and such that:

• M � N if κ > |M|,• N �M if κ < |M|.• For A ⊆ M , a partial type Φ (x) over A is consistent collection of L (A)-

formulas.• Φ (x) is a (complete) type if φ (x) ∈ Φ or ¬φ (x) ∈ Φ for all φ ∈ Φ.• For a tuple b inM, tp (b/A) := {φ (x) : b |= φ (x) , φ (x) ∈ L (A)}.

Definition 1.3. Let κ be an infinite cardinal.(1) M is κ-saturated if ∀A ⊆ M with |A| < κ, every 1-type over A can be

realized inM.(2) M is κ-homogenous if any partial elementary map fromM to itself with a

domain of size < κ can be extended to an automorphism ofM.

Fact 1.4. For any T and κ, there is a κ-saturated and κ-homogeneous model Mof T .

Definition 1.5. For A ⊆M , we let Sn (A) denote the space of all complete n-typesover A.

• This is the Stone dual of Defn (A), hence compact (=compactness theo-rem), Hausdorff space with a basis of clopens given by the sets 〈φ (x)〉 ={p ∈ Sn (A) : φ (x) ∈ p} for φ (x) ∈ L (A).


• A type p (x) ∈ Sn (A) is isolated if it is an isolated point of the top. spaceSn (A), i.e. ∃φ (x) ∈ p s.t. φ (x) ` p (x).

Example 1.6. In many natural structures it is possible to describe definable setsand types explicitly using quantifier elimination (i.e., every formula is equivalent toone not involving quantifiers, see Marker’s book for the details).

(1) T∞ — the theory of an infinite set, in the language of equality.By QE, Def1 (M) consists of the finite and cofinite subsets.S1 (M) = {tp (a/M) : a ∈M} ∪ {p∗}, where p∗ = {x 6= a : a ∈M} is thetype of a “new element”.

(2) We consider a vector spaceM = (M, 0,+,−, (r (x) : r ∈ K)) where K is afield (finite or infinite), and r (x) is a function x 7→ rx.Let VSK := Th (M).It has QE, hence definable sets are given by the Boolean combinations ofequations of the form r1x1 + . . .+ rnxn = 0.In particular, every set in Def1 (M) is either finite or cofinite.(Note that the field R is incorporated into the language, and is not a de-finable as a part of the structure.)

(3) ACFp - the theory of an algebraically closed field of char p, where p is primeor 0 (e.g. Th (C,+,×, 0, 1) = ACF0).By Tarski’s QE, definable sets are precisely the Boolean combinations ofalgebraic sets of the form {x : p (x) = 0}, where p is a polynomial.Again, every set in Def1 (M) is either finite or cofinite.

(4) DLO=Th (Q, <) — the theory of dense linear orders without end points.By QE, Def1 (M) consists of finite unions of (bounded or unbounded) in-tervals (hence can be infinite-coinfinite).S1 (M) ↔ Dedekind cuts in (M,<) (For a cut C = (A,B), let pC :={a < x < b : a ∈ A, b ∈ B}. This set of formulas is consistent by the den-sity of the order, and defines a complete type by QE)

Exercise 1.7. Show:• If M is κ-saturated and A ⊆ M , |A| < κ then every n-type over A is

realized in M .• (R,+,×, 0, 1) is not ℵ0-saturated.• (*) (C,+,×, 0, 1) is 2ℵ0-saturated (we will see later how it follows from the

general theory).

2. Morley’s categoricity theorem

In this section we prove Morley’s categoricity theorem, using it as an opportunityto introduce some of the fundamental ideas of stability theory.

Fact 2.1. (Morley, 1965) Let T be a countable complete first-order theory κ-categorical for some uncountable cardinal κ (i.e. admitting a unique model of car-dinality κ, up to isomorphism). Then T is κ-categorical for every uncountablecardinal κ.

Example 2.2. (1) T∞ is uncountably categorical — indeed, every model ofT∞ is determined up to isomorphism by its size.

(2) Let K be a countable field, then VSK is uncountably categorical — bythe standard linear algebra, any two vector spaces of the same dimension


are isomorphic, and the dimension of any uncountable vector space over acountable field is equal to its size.

(3) Same holds for ACFp with the transcendence degree over the prime field inthe place of dimension.

Exercise 2.3. Show:(1) Assume K is a countable field. How many countable models does VSK

have?(2) DLO is not κ-categorical for any uncountable κ(but ℵ0-categorical).(3) RG, the theory of the countable random graph, has 2κ models of size κ, ∀κ

uncountable (but ℵ0-categorical).(4) (*) In fact, DLO also has 2κ models of size κ, ∀κ uncountable.

2.1. Algebraic closure.• If M is an L-structure and φ (x) is an L (M)-formula, then φ (M) :={a ∈M : a |= φ (a)}.

Definition 2.4. LetM be a structure and A a subset of M .(1) A formula φ (x) ∈ L (A) is algebraic if φ (M) is finite.(2) An element a ∈ M is algebraic over A if it satisfies an algebraic L (A)-

formula. We say “a is algebraic” meaning “a is algebraic over ∅”.(3) The algebraic closure of A is the set acl (A) = {a ∈M : a is algebraic over A}.(4) A is algebraically closed if acl (A) = A.

Remark 2.5. (1) Given A ⊆M �N , acl (A) calculated in N is equal to acl (A)calculated in M (an L (A)-formula defining a finite set in M defines thesame set in N by elementarity). In particular, elementary substructuresare algebraically closed.

(2) |acl (A)| ≤ max (A, |T |).

Example 2.6. (1) IfM |= T∞, the acl (A) = A for any set A.(2) IfM≡ (Z, s), then acl (A) is the set of points reachable from A (Exercise).(3) IfM |= VSK , then a vector a ∈ acl (A), if and only if a is in the span of A.(4) If M |= ACFp, then a ∈ acl (A) ⇐⇒ a there is a polynomial with

coefficients in the field generated by A such that a is its root).

Definition 2.7. A type p (x) ∈ S (A) is algebraic if p contains an algebraic formula.

Remark 2.8. (1) Any such type is isolated by an algebraic formula φ (x) ∈L (A), namely any φ (x) ∈ p with the minimal number of solutions in M(Check!).

(2) Using it and compactness, p ∈ S (A) is algebraic ⇐⇒ p has only finitelymany realizations in any elementary extension ofM.

Exercise 2.9. (1) If A ⊆ M and M is |A|+-saturated, then p ∈ S (A) isalgebraic ⇐⇒ p (M) is finite.

(2) Let A,B be subsets of M and (c0, . . . , cn) a sequence of elements nonalgebraic over A. If M is |A ∪B|+-saturated, the type tp (c0, . . . , cn/A)has a realization which is disjoint from B.

(3) Determine acl in DLO and RG.(4) GivenA ⊆ N such thatN is |A|+-saturated, show that acl (A) =

⋂{M � N : A ⊆M}.


2.2. Strongly minimal formulas and theories. Let T be a complete theorywith infinite models.

Definition 2.10. LetM |= T , and D (x) ∈ L (M) a non-algebraic formula.(1) The set D (M) is minimal inM if for all ψ (x) ∈ L (M), D (M)∧ψ (M) is

either finite or cofinite in φ (M).(2) The formula D (x) is strongly minimal if D (x) defines a minimal set in any

elementary extensions N <M.(3) T is strongly minimal if the formula “x = x” is strongly minimal.

Example 2.11. T∞,VSK ,ACFp are strongly minimal (by the discussion in Ex-ample 1.6).

Example 2.12. Minimal 6= strongly minimalLet L = {E} and letM be an L structure in which E is an equivalence relation

with one class of size n for all n ∈ ω and no infinite classes. Then x = x is aminimal formula. However, by compactness there is N �M and a ∈ N such thatthe equivalence class of a is infinite. Then E (x, a) defines an infinite-coinfinitesubset, hence x = x is not strongly minimal.

Remark 2.13. The property “φ (x, a) is strongly minimal” depends only on tp (a).Namely, φ (x, a) is s.m. ⇐⇒ for all ψ (x, z) ∈ L the set

Σψ (z, a) ={∃>kx (φ (x, a) ∧ ψ (x, z)) ∧ ∃>kx (φ (x, a) ∧ ¬ψ (x, z)) : k ∈ N

}cannot be realized in any elementary extension.

By compactness, this means that for all ψ (x, z), there is a bound kψ such that

M |= ∀z(∃≤kψx (φ (x, a) ∧ ψ (x, z)) ∧ ∃≤kψx (φ (x, a) ∧ ¬ψ (x, z))

).

— an elementary property of a, i.e. expressible by a first-order formula.Hence, it makes sense to say “φ (x, b) is a s.m. formula” without specifying a

model.

Corollary 2.14. AssumeM |= T is ω-saturated. Then any formula φ (x) ∈ L (M)such that φ (M) is minimal, is strongly minimal.

Proof. IfM is ω-saturated and φ (x, a) is not s.m., then for some ψ (x, z) ∈ L theset Σψ (z, a) is realized inM, so φ (M) is not minimal. �

Exercise 2.15. If M is strongly minimal, A ⊆ M and satisfies N := acl (A) isinfinite, then N �M.

2.3. Algebraic closure in strongly minimal sets and pregeometries.

Definition 2.16. LetM be an L-structure and D (x) ∈ L (M) a strongly minimalformula. We consider the closure operator

cl : 2D(M) → 2D(M)

defined bycl (A) := acl (A) ∩D (M) .

Proposition 2.17. cl satisfies the following properties for all A ⊆ D (M) anda, b ∈ D (M):

(1) (Reflexivity) A ⊆ cl (A),(2) (Finite character) cl (A) =

⋃{cl (A′) : A′ ⊆ A finite},


(3) (Transitivity) cl (cl (A)) = cl (A),(4) (Exchange) a ∈ cl (Ab) \ cl (A) =⇒ b ∈ cl (Aa) (abusing notation slightly,

we write Ab to denote A ∪ {b}).

Proof. Properties (1)-(3) hold without any assumption on D. (1) and (2) arestraightforward, and we check (3).

• Assume that c ∈ cl ({b1, . . . , bn}) and bi ∈ cl (A).• Let φ (x, b1, . . . , bn) ∈ L be an algebraic formula satisfied by c.• Let φi (y) ∈ L (A) be algebraic formulas satisfied by the bi’s.• Then ψ (x) :=

∃y1 . . . yn(φ1 (y1) ∧ . . . φn (yn) ∧ ∃≤kzφ (z, y1, . . . , yn) ∧ φ (x, y1, . . . , yn)

)is an algebraic L (A)-formula satisfied by c.

On the other hand, (4) is more subtle, and we show that it holds assuming D iss.m.

• Suppose a ∈ cl (Ab) \ cl (A).• ThenM |= φ (a, b), where φ (x, y) ∈ L (A) and |φ (D, b)| = n.• Let ψ (y) := ∃=nx (D (x) ∧ φ (x, y)).• If |ψ (D)| <∞, then b ∈ cl (A) , hence a ∈ cl (A) (as in the argument above)

— a contradiction.• Thus, ψ (y) defines a cofinite subset of D.• If {y ∈ D : φ (a, y) ∧ ψ (y)} is finite, we are done because then b ∈ cl (Aa).• Towards contradiction, assume |D \ {y : φ (a, y) ∧ ψ (y)}| = l for some l ∈ω.

• Let χ (x) := ∃=ly (D (y) ∧ ¬ (φ (x, y) ∧ ψ (y))).• If χ (x) defines a finite subset of D, then a ∈ cl (A), a contradiction.• Thus χ (x) defines a cofinite subset of D.• Choose a1, . . . , an+1 such that |= χ (ai).• The set Bi := {y ∈ D : φ (ai, y) ∧ ψ (y)} is cofinite for all i = 1, . . . , n+ 1.• Choose b′ ∈

⋂Bi.

• Then |= φ (ai, b′) for each i,

• so |{x ∈ D :|= φ (x, b′)}| ≥ n+ 1 — contradicting the fact that ψ (b′) holds.�

Exercise 2.18. Give an example of a structureM in which acl doesn’t satisfy theexchange.

2.4. Dimension and independence in pregeometries.

Definition 2.19. Any abstract closure operator cl : 2X → 2X on subsets of aninfinite set X is satisfying properties (1)-(4) in Proposition 2.17 is called a prege-ometry.

• If the underlying set is finite, it is called a matroid. There is a rather richtheory of matroids, we will discuss this more.

• What is important for us here is that one can develop notions of inde-pendence, dimension, basis, etc. in any pregeometry, generalizing linearindependence in vector spaces and algebraic independence in algebraicallyclosed fields.

Definition 2.20. Let (D, cl) be a pregeometry. A subset A ⊆ D is called:


(1) independent (over C) if a /∈ cl (A \ {a}) (resp., a /∈ cl (C ∪ (A \ {a}))) forall a ∈ A;

(2) a generating set if D = cl (A);(3) a basis for Y ⊆ D if A ⊆ Y is an independent generating set.

• Note: any maximal independent subset of Y is a basis for Y .• Just as in vector spaces, we have:

Lemma 2.21. (Replacement lemma) Let A,B ⊆ D be independent with A ⊆ cl (B).(1) Suppose A0 ⊆ A,B0 ⊆ B and A0∪B0 is a basis for acl (B) and a ∈ A\A0.

Then ∃b ∈ B0 s.t. A0 ∪ {a} ∪ (B0 \ {b}) is a basis for acl (B).(2) |A| ≤ |B|.(3) If A,B are bases for Y ⊆ D, then |A| = |B|.

Proof. (1)• Let C ⊆ B0 be of min. cardinality s.t. a ∈ cl (A0 ∪ C).• As A is independent, |C| ≥ 1. Let b ∈ C.• By exchange, b ∈ cl (A0 ∪ {a} ∪ (C \ {b})), thus• cl (A0 ∪ {a} ∪ (B \ b)) = cl (B).• If a ∈ cl (A0 ∪ (B0 \ {b})), then b ∈ cl (A0 ∪ (B0 \ {b})), contradicting A0 ∪B0 being a basis.• Thus A0 ∪B0 is a basis.• Thus A0 ∪ {a} ∪ (B0 \ {b}) is independent.

(2)• Supp. B is finite:• say |B| = n and a1, . . . , an+1 ∈ A are distinct.• Let A0 = ∅ and B0 = B.• Using (1) inductively, find b1, . . . , bn ∈ B distinct, s.t.• {a1, . . . , ai} ∪ (B \ {b1, . . . , bi}) is a basis for cl (B) for i ≤ n.• But then cl (a1, . . . , an) = cl (B).• As an+1 ∈ cl (B), this contradicts the independence of A.• Supp. B is infinite:• then for any finite B0 ⊂ B, A ∩ cl (B0) is finite and• A ⊆

⋃B0⊆B finite cl (B0).

• Thus |A| ≤ |B|.(3) is immediate from (2). �

Hence we can define:

Definition 2.22. If Y ⊆ D, then dim (Y ), the dimension of Y , is the cardinalityof a basis for Y .

2.5. Uncountable categoricity of s.m. theories. We go back to a matroidgiven by acl on a definable strongly minimal set D.

Lemma 2.23. Supp. M,N |= T , φ (x) ∈ L (A) is a s.m. formula, either A = ∅or A ⊆M0 whereM0 �M,N .

If a1, . . . , an ∈ φ (M) are independent over A and b1, . . . , bn ∈ φ (N) are inde-pendent over A, then tpM (a/A) = tpN

(b/A

).

Proof. We assume φ (x) ∈ L (A), A ⊆ M0,M0 � M,N (case A = ∅ is a straight-forward modification of the argument).


• By induction on n.• Assume n = 1, a ∈ φ (M) \ cl (A) and b ∈ φ (N) \ cl (A).• Let ψ (x) ∈ L (A), and supposeM |= ψ (a).• As a /∈ cl (A), φ (M) ∩ ψ (M) is cofinite in φ (M).• Thus ∃n s.t. M |= |{x : φ (x) ∧ ¬ψ (x)}| = n .• BecauseM0 �M,N and b /∈ cl (A) , also N |= ψ (b).• As ψ ∈ L (A) was arbitrary, tpM (a/A) = tpN (b/A).• Assume the claim is true for n and a1, . . . , an+1 ∈ φ (M) , b1, . . . , bn+1 ∈φ (N) are independent sequence over A.

• Let a = (a1, . . . , an) , b = (b1, . . . , bn), by induction tpM (a/A) = tpN(b/B

).

• AssumeM |= ψ (a, an+1) for some ψ (w, v) ∈ L (A).• As an+1 /∈ cl (Aa), φ (M) \ ψ (a,M) is finite.• Hence ∃n s.t. M |= |{v : φ (v) ∧ ¬ψ (a, v)}| = n.• BecauseM0 �M,N and tpM (a/A) = tpN

(b/A

), alsoN |=

∣∣{v : φ (v) ∧ ¬ψ(b, v)}∣∣ =

n.• As bn+1 /∈ cl

(Ab), N |= ψ

(b, bn+1

), and we conclude.

�

Lemma 2.24. If M,N and φ are as above and dim (φ (M)) = dim (φ (N)), thenthere is a bijective partial elementary map f : φ (M)→ φ (N).

In particular, if T is s.m. and M,N |= T , then M ∼= N ⇐⇒ dim (M) =dim (N).

Proof. • Let B be a basis for φ (M) and C a basis for φ (N ).• As |B| = |C| be assumption, let f : B → C be any bijection.• By Lemma 2.23, f is elementary.• Let I := {g : B′ → C ′ : B ⊆ B′ ⊆ φ (M) , C ⊆ C ′ ⊆ φ (N) , f ⊆ g partial elementary}.• By Zorn’s lemma, there is a maximal g : B′ → C ′ in I.• Supp. b ∈ φ (M) \B′.• As b ∈ cl (B′), there is a formula ψ

(v, d)isolating tpM (b/B′).

• As g is elementary, can find c ∈ φ (N) such that N |= ψ(c, g(d)).

• Then tpM (b/B′) = tpN (c/C ′), and we can extend g by sending b to c.• This contradicts the maximality of g!• Thus φ (M) = B′.• An analogous argument shows that C ′ = φ (N).

�

Theorem 2.25. If T is a countable strongly minimal theory, then T is κ-categoricalfor all uncountable κ.

Moreover, it has at most countably many models of cardinality ℵ0.

Proof. If T is countable, then |acl (A)| ≤ |A| + ℵ0, hence any basis for M with|M| = κ > ℵ0 has cardinality κ.

If |M | ≤ ℵ0, then dim (M) ≤ ℵ0. �

2.6. ω-stability.

• We aim to show that if T is κ-categorical for some uncountable κ, then itis a “fibration” over a definable strongly minimal set.

• For this, we isolate some consequences of κ-categoricity.


Definition 2.26. T is ω-stable if for every M |= T and every A ⊆ M , |A| ≤ ℵ0=⇒ |S1 (A)| ≤ ℵ0.Exercise 2.27. (1) Show that under ω-stability, also |Sn (A)| ≤ ℵ0 for all

countable A and n ∈ ω.(2) Show that a countable T is ω-stable ⇐⇒ |S1 (M)| = |M | for all infinite

M |= T .

• By Stone duality, the space of types is small ⇐⇒ definable sets are not socomplicated.

Example 2.28. It T is strongly minimal, then T is ω-stable.Indeed, ∀ A ⊆M |= T , the types over A are the algebraic types (all realized in

acl (A), |acl (A)| ≤ |A|+ ℵ0) and the unique non-algebraic type.

Theorem 2.29. Let T be ω-stable. IfM |= T , then there is a minimal formula inM.

Proof. Suppose no formula is minimal. Then by induction we can build a tree ofL (M)-formulas (φη (x) : η ∈ 2<ω) such that:

• φ∅ is “x = x”,• η ⊆ ν ∈ 2<ω =⇒ φν ` φη,• φη_i (x) ` φη_(1−i) (x),• φη (M) is infinite for all η ∈ 2<ω.

Let A be the set of all parameters of all formulas φη, |A| ≤ ℵ0.For each η ∈ 2ω, the set {φη�n (x) : n ∈ ω} is consistent, hence extends to some

pη ∈ S1 (A).And if η, ν are incomparable, then pη 6= pν . Hence |S1 (A)| = 2ℵ0 , contradicting

ω-stability. �

Definition 2.30. Given A ⊆M |= T , we say thatM is prime over A if ∀N |= Tand f : A→ N partial elementary, there is an elementary f∗ :M→ N extendingf .

Lemma 2.31. Let a countable T be ω-stable,M |= T , A ⊆M . The isolated typesin Sn (M) are dense.

Proof. If not, can build a binary tree of formulas as in Theorem 2.29. �

Theorem 2.32. Let a countable T be ω-stable, M |= T and A ⊆ M . There isM0 �M, a prime model over A.

Proof. • We find an ordinal δ and subsets (Aα : α ≤ δ) of M s.t.(1) A0 = A.(2) α limit =⇒ Aα =

⋃β<αAβ .

(3) if no element of M \Aα realizes an isolated type over Aα, stop and letδ = α;

(4) otherwise, pick aα with tp (aα/Aα) isolated and let Aα+1 = Aα∪{aα}.• LetM0 be the substructure ofM with M0 := Aδ.• Claim 1. M0 �M.• By Tarski-Vaught test: supposeM |= φ (v) with φ (x) ∈ L (Aδ).• By the lemma, ∃b ∈M s.t. tp (b/Aδ) is isolated andM |= φ (b).• By choice of δ, b ∈ Aδ.


• Claim 2. M0 is a prime model extension of A.• Let N |= T and f : A→ N partial elementary.• We find by induction f = f0 ⊆ . . . ⊆ fα ⊆ fδ with fα : Aα → N elementary.• α limit: let fα =

⋃β<α fβ .

• Given fα : Aα → N partial elementary, let φ (v, a) isolate tpM0 (aα/Aα).• Then φ (v, fα (a)) isolates fα

(tpM0 (aα/Aα)

)in SN1 (fα (A)).

• As fα is elementary, ∃b ∈ N s.t. N |= φ (b, fα (a)).• Hence fα+1 = fα ∪ {(aα, b)} is elementary.

• Hence fδ :M0 → N is elementary, as wanted.�

Theorem 2.33. If T is κ-categorical for some uncountable κ, then T is ω-stable.

• Before we can prove it, we need an auxilliary construction.

2.7. Ehrenfeucht–Mostowski Models.

Definition 2.34. Let I be a linear ordering. A sequence (ai : i ∈ I) of tuples in anL-structureM is indiscernible over B ⊆M if tp (ai1 , . . . , ain/B) = tp (aj1 , . . . , ajn/B)∀n ∈ ω, i, j strictly increasing tuples from I.

Example 2.35. We saw that if T is strongly minimal, and (ai : i ∈ I) is a sequenceof acl-independent elements, then it is indiscernible.

Fact 2.36. For any T in language L and I there is some M |= T and (ai : i ∈ I)a non-constant indiscernible sequence inM.

Proof. • Recall Ramsey theorem: for any n, k ∈ ω and f : ωm → k, there isan infinite subset S ⊆ ω s.t. f is constant on all increasing m-tuples fromS.• Consider the partial type Σ (xi : i ∈ I) containing formulas

(1) T ,(2) xi 6= xj , ∀i 6= j,(3) φ (xi1 , . . . , xin) ↔ φ (xj1 , . . . , xjn), ∀φ ∈ L and i, j increasing tuples

from I.• Any finite Σ0 ⊆ Σ is consistent:

– let (ai : i ∈ ω) be any infinite sequence in someM |= T , and ∆ a finiteset of formulas.

– By Ramsey, (ai : i ∈ ω) contains an infinite subsequence satisfying con-dition (3) (and the other conditions hold).

• Conclude by compactness.�

Theorem 2.37. Let L be countable, T an L-theory. For all κ ≥ ℵ0 there isM |= Twith |M | = κ such that if A ⊆M , thenM realizes at most |A|+ ℵ0 types over A.

Proof. • Let T ∗ be an expansion of T in a language L∗ ⊇ L with Skolemfunctions (i.e. for each formula φ (x, y) ∈ L∗ there is a function symbolfφ (y) such that T ∗ ` ∀y (∃xφ (x, y)→ φ (fφ (y) , y)). Exists by taking amodel of T and expanding until catching the tail.)

• Let M∗ |= T ∗ and contain I, an L∗-indiscernible sequence of order type(κ,<).


• LetM, an L∗-substructure generated by I.• In particular, the L-reduct ofM is a model of T , and |M| = κ.• Let A ⊆M . For each a ∈ A, a = ta (xa) for some term ta and xa from I.• Let X := {x ∈ I : x occurs in xa for some a ∈ A}.• Then |X| ≤ |A|+ ℵ0.• By L∗-indiscernibility: if y, z are tuples from κ with the same order type

over X (denoted by y ∼X z) and t is an L∗-term, then t (y) , t (z) have thesame type over A.

• Hence enough to show: for any n ∈ ω, |In/ ∼X | ≤ |A|+ ℵ0.• For y ∈ I \X, let Cy := {x ∈ X : x < y}.• Then z ∼X y ⇐⇒ ∀1 ≤ i ≤ n:

(1) if yi ∈ X, then yi = zi,(2) if yi /∈ X, then zi /∈ X and Cyi = Czi .

• As I is well-ordered, Cy = Cz ⇐⇒ Cy = Cz = ∅ or inf {i ∈ I : i > Cy} =inf {i ∈ I : i > Cz}.

• In particular, there are at most |X|+ 1 possible cuts Cy.• Hence |In/ ∼X | ≤ |A|+ ℵ0, as wanted.

�

Proof. (of Theorem 2.33).Let κ be any uncountable cardinal.Assume T is not ω-stable, then ∃M |= T and A ⊆M countable with |S1 (A)| >

ℵ0. We can find N0 �M with |N0| realizing uncountably many types from S1 (A).By Theorem 2.37, ∃N1 |= T , |N1| = κ, and ∀B ⊆ N1, |B| = ℵ0, N1 realizes at mostℵ0 types over B.

Then N0 6∼= N1, contradicting κ-categoricity. �

2.8. Two-cardinal models and elimination of ∃∞.

Definition 2.38. Let κ > λ ≥ ℵ0 be cardinals. ThenM |= T is a (κ, λ)-model if|M | = κ and for some φ (x) ∈ L, |φ (M)| = λ.

Proposition 2.39. A countable T is κ-categorical =⇒ T has no (κ, λ)-modelsfor any λ < κ.

Proof. By compactness and downwards LS, any countable T has a modelM of sizeκ s.t. every infinite ∅-definable subset ofM also has cardinality κ. �

Fact 2.40. (Vaught’s two-cardinal theorem) If T has a (κ, λ)-model with κ > λ ≥ℵ0, then T has an (ℵ1,ℵ0)-model.

• Almost any other implication of this form is false or independent from ZFC.

Definition 2.41. T has a Vaught pair if there areM� N |= T and φ (x) ∈ L (M)such that:

(1) M 6= N ,(2) φ (M) is infinite,(3) φ (M) = φ (N).

Lemma 2.42. If T has a (κ, λ)-model, then it has a Vaught pair.

Proof. • Let N be a (κ, λ)-model.• Supp. |φ (N)| = λ.• By the LS theorem, ∃M ≺ N s.t. φ (N) ⊆M and |M | = λ.


• As φ (N) = φ (M), this is a Vaught pair.�

• Vaught’s theorem really proves the following:

Lemma 2.43. If T has a Vaught pair, then it has a (ℵ1,ℵ0)-model.

Corollary 2.44. If T is ℵ1-categorical, then T has no Vaught pairs.

• Assuming ω-stability, we have the following converse.

Theorem 2.45. Suppose T is ω-stable and T has an (ℵ1,ℵ0)-model. If κ > ℵ1,then T has a (κ,ℵ0)-model.

Proposition 2.46. If T has no Vaught pairs, then T eliminates ∃∞. That is, foreach φ (v, w) ∈ L ∃nφ ∈ ω s.t. ifM |= T , a ∈M and |φ (M, a)| > n, then φ (M, a)is infinite.

Proof. • Let L∗ := L ∪ {P (x) , c1, . . . , cn}.• Let T ∗ be the theory of all L∗-structures (M,P (M) , c1, . . . , cn) where:

– M is a model of T ,– P (M) ≺L M ,– P (ci) holds ∀i,– φ (M, a) ⊆ P (M).

• Suppose no bound nφ ∈ ω exists.• Then ∀n∃N |= T and a ∈ N s.t. φ (N, a) is finite, but has more than n

elements.• LetM be a proper elem. ext. of N .• Then φ (M, a) = φ (N, a) and (M,N, a) |= T ∗.• Hence T ∗ ∪ {∃>nxφ (x, c) : n ∈ ω} is finitely satisfiable.• By compactness it has a model, which gives a Vaught pair for T .

�

Corollary 2.47. If T has no Vaught pairs, then any min. formula is s.m.

Proof. If φ (x, a) is minimal inM and T eliminates ∃∞, then for each ψ (x, z) ∈ L,¬ (∃∞x (φ (x, a) ∧ ψ (x, z)) ∧ ∃∞x (φ (x, a) ∧ ¬ψ (x, z)))

is an elementary property of z and holds inM, hence holds in every model of T . �

Exercise 2.48. Show that RG has a Vaught pair.

2.9. Morley’s theorem, proof by Baldwin-Lachlan.• Now we put everything together.

Theorem 2.49. (Baldwin-Lachlan) Let κ be an uncountable cardinal. A countableT is κ-categorical ⇐⇒ T is ω-stable and has no Vaughtian pairs.

• As the RHS doesn’t depend on the specific uncountable κ, Morley’s cate-goricity theorem follows.

Proof. (⇒) We saw that T is κ-cat =⇒ T is ω-stable and has no Vaught pairs.(⇐)

• As T is ω-stable, it has a prime modelM0 (over ∅).• By Theorem 2.29 and Corollary 2.47, ∃ an s.m. formula φ (v) ∈ L (M0).• LetM,N |= T , of card. κ ≥ ℵ1.


• WLOGM0 �M,N .• Then |φ (M)| = |φ (M)| = κ (as T has no (κ, λ)-models)• Hence dim (φ (M)) = dim (φ (N)) = κ.• By Lemma 2.24, there is f : φ (M)→ φ (N) a partial elementary bijection.• Claim. M is prime over φ (M).

– No proper K ≺ M conains φ (M), as otherwise φ (K) = φ (M) andφ (M) is infinite, hence (M,K) would be a Vaught pair.

– As T is ω-stable, by Theorem 2.32, ∃K �M a prime model over φ (M).– By the above, K =M.

• Hence, by def. of primality, f extends to an elementary f ′ : M → N .• But as in the claim, N has no proper elem. submodels containing φ (N).• Hence f ′ is surjective, so an isomorphism.

�

• It follows from the proof that if T is uncountably categorical, then it hasat most countably many countable models.

• This can be refined to:

Fact 2.50. (Baldwin-Lachlan) If T is uncountably categorical but not ℵ0-categorical,then it has exactly ℵ0 countable models.

2.10. Further work on counting models.

Definition 2.51. For a theory T , consider the function IT (κ) = the number ofmodels of T of size κ, up to isomorphism.

• Note that 1 ≤ IT (κ) ≤ 2κ for all infinite κ bigger than the cardinality ofthe language.

• Morley’s theorem says: let T be a countable theory. If IT (κ) = 1 for someuncountable κ, then IT (κ) = 1 for all uncountable κ.

Stability theory developed historically in Shelah’s work as a chunk of machineryintended to generalize Morley’s theorem to a computation of the possible “spectra”of complete first order theories, in particular to prove the following conjectures ofMorley.

Theorem 2.52. [Shelah](1) The function IT (κ) is non-decreasing on uncountable cardinals.(2) For theory T of any cardinality, there is some cardinal κ (T ) such that if

IT (λ) = 1 for some λ > κ (T ), then IT (λ) = 1 for all λ > κ (T ).

• This project, at least for countable theories, was essentially completed byShelah in the early 80’s with the “Main gap theorem” [2].

• Shelah isolated a bunch of “dividing lines” (in the form of T being able toencode in a definable manner a certain finitary combinatorical configura-tion) on the space of first-order theories, showing that all theories on thenon-structure side of the dividing line have as many models as possible,and on the structure side developing some kind of dimension theory andshowing that the isomorphism type of a model can be described by some“small” invariants implying e.g. that there are few models (so in the caseof a vector space we only need one cardinal invariant, but if we have anequivalence relation then we need to know the size of each of its classes).


• A complete classification of the possible functions IT (κ) for countable theo-ries was given by Hart, Hrushovski, Laskowski [1] (required some descriptiveset-theoretic ideas in order to prove a “continuum hypothesis” for a certainnotion of dimension).

• Later on, motivated by the aim to understand totally categorical theories,other perspectives developed, in the work of Macintyre, Zilber, Cherlin,Poizat, Hrushovski, Pillay and many others, in which stability theory is seenrather as a way of classifying definable sets in a structure and describingthe interaction between definable sets. Eventually this theory started to beseen as having a “geometric meaning”. It turned out that the study of thedefinability of certain algebraic objects such as groups and fields is essentialeven for purely logical questions of categoricity.

• Moreover, more recently it was realized that a lot of techniques from stabil-ity can still be developed in larger contexts, and the so-called “generalizedstability” is currently an active area of research.

Exercise 2.53. Show that if T is ω-stable, then for any formula φ (x, y) ∈ Land any model M |= T , we cannot find a sequence (ai, bi : i ∈ ω) in M such thatM |= φ (ai, bj) ⇐⇒ i ≤ j.

Any such formula is called stable, and large part of the theory developed in thiscourse can be generalized to theories in which every formula is stable.

3. Morley rank and forking independence in ω-stable theories

• We fix a theory T and a κ-saturated and κ-homogeneous “monster model”M |= T , with κ = κ (M) sufficiently large.

• All sets, tuples and models we consider will be contained in M.• We say that a set A ⊆M is small, or bounded, if |A| < κ.

3.1. M eq and canonical parameters.• All the notions above generalize in an obvious way to multi-sorted struc-tures, i.e. the underlying set of our model is now partitioned into severalsorts, and for each of the variables for each relation and function symbolswe specify which sort they live in. Then formulas and other notions aredefined and evaluated accordingly.

• We recall Shelah’s construction of imaginaries, which allows us to viewquotients of definable sets by definable equivalence relations as definableobjects themselves.

• GivenM |= T , we construct a structureMeq in a language Leq extendingL as follows:– add a new sort Mn/E, ∀ n ∈ ω and E (x, y) ∈ L defining an equiva-

lence relation on Mn.– Leq contains an n-ary function symbol πE for the projection map πE :Mn →Mn/E, πE (a) = [a]E , the E-equivalence class of a ∈Mn.

– Identify M with M/ =.– The sort M/ = is called the main sort, and the elements from the

other sorts are called imaginaries.• Then T eq := ThLeq (Meq).

Exercise 3.1. • T eq doesn’t depend on the choice of the initial modelM |=T .


• Every automorphism ofM extends uniquely to an automorphism ofMeq.• If M |= T is κ-saturated (-homogeneous), then Meq is also κ-saturated

(resp., κ-homogeneous).• For every ψ (x, x1, . . . , xn) ∈ Leq with x in the main sort and xi is of

sort Ei, ∃φ (x, y1, . . . , yn) ∈ L with |yi| = arity (Ei) s.t. for all tuplesa, a1, . . . , an ∈M :

Meq |= ψ (a, πE1(a1) , . . . , πEn (an)) ⇐⇒ M |= ψ (a, a1, . . . , an) .

• If T is t.t., then T eq is also t.t.

• In particular, Meq is a monster model for T eq and T eq doesn’t define anynew subsets of the main sort.

Definition 3.2. Given ψ (x, a) ∈ L (M), consider the equiv. rel. Eψ (y, z) ⇐⇒∀x (ψ (x, y)↔ ψ (x, z)), we define the canonical parameter dψ (x, a)e of ψ (x, a) asthe imaginary [a]Eψ .

• Note: Eψ (y, z) holds ⇐⇒ ψ (M, y) = ψ (M, z).

Definition 3.3. • An element a ∈ M is definable over A ⊆ M if ∃φ (x) ∈L (A) s.t. |= φ (a) and |φ (M)| = 1.

• dcl (A) := {a ∈M : a is definable over A}

• Note: dcl (A) ⊆ acl (A).• We will write acleq,dcleq to denote acl and dcl in the structure Meq.

Exercise 3.4. Assume that φ (M, a) = ψ (M, b) for any φ (x, a) , ψ (x, b) ∈ L (M).Then dφ (x, a)e and dψ (x, b)e are interdefinable, i.e. dφ (x, a)e ∈ dcleq (dψ (x, b)e)and dψ (x, b)e ∈ dcleq (dφ (x, a)e). Hence given X ⊆ Mn, we can talk about thecanonical parameter dXe of X (well-defined up to interdefinability).

• Canonical parameters help dealing with automorphisms and give a Galois-theoretic characterization of definability:

Lemma 3.5. Let X ⊆Mn be definable and A ⊆M a small set (i.e. |A| < κ (M)).TFAE:

(1) X is A-definable.(2) X is A-invariant, i.e. σ (X) = X (setwise) for all σ ∈ Aut (M /A) =

Aut (Meq /A) (i.e. σ is an automorphism of M fixing A pointwise).(3) dXe ∈ dcleq (A).

Proof. (1) =⇒ (2):• Supp. X = ψ (M, b), b ∈ A.• For any σ ∈ Aut (M /A) we have σ (b) = b, hence

a ∈ X ⇐⇒ |= φ (a, b) ⇐⇒ |= φ (σ (a) , σ (b)) ⇐⇒ |= φ (σ (a) , b) ⇐⇒ σ (a) ∈ X.

(2) =⇒ (1):• Supp. X = φ (M, b) with b ∈M and X is A-invariant.• Let p (y) := tp (b/A).• By A-invariance of X, p (x) ` ∀x (φ (x, y)↔ φ (x, b)).• By compactness θ (y) ` ∀x (φ (x, y)↔ φ (x, b)) for some θ (y) ∈ p (y).• Let χ (x) := ∃z (θ (z) ∧ φ (x, z)), then χ (x) ∈ L (A).• Check: χ (M) = X.


(2)⇐⇒ (3):• Let φ (x, y) ∈ L, c ∈M s.t. X = φ (M, c).• Let E (y, z) be the definable e.r. φ (M, y) = φ (M, z).• Thus [c]E = dφ (x, c)e.• By (2) ⇐⇒ (1) applied to the definable set {[c]E}, we see that [c]E ∈

dcleq (A) ⇐⇒ σ ([c]E) = [c]E for all σ ∈ Aut (M /A).• For any σ ∈ Aut (M /A) we have:

σ ([c]E) = [c]E ⇐⇒ E (c, σ (c)) ⇐⇒ φ (M, c) = φ (M, σ (c)) ⇐⇒ σ (X) = X,

• and [c]E ∈ dcleq (A) ⇐⇒ it is fixed by all automorphisms.�

• Similarly for acleq (Exercise, use the proof of the previous lemma and com-pactness):

Lemma 3.6. Let X ⊆Mn be definable and A small. TFAE:(1) X has a finite Aut (M /A)-orbit.(2) X has a bounded Aut (M /A)-orbit (bounded = smaller than κ (M)).(3) dXe ∈ acleq (A).

3.2. Morley rank.• In this section we develop an (ordinal-valued) notion of “dimension” for

definable sets and types in ω-stable theories.• It can be defined topologically using the Cantor-Bendixson rank on the

associated spaces of types. Recall:

Definition 3.7. Let X be a compact Hausdorff topological space.(1) For a point p ∈ X, its Cantor-Bendixson rank CB (p) is defined by induction

on an ordinal α:(a) CB (p) ≥ 0 for all p ∈ X,(b) CB (p) = α iff p is isolated in the subspace {q ∈ X : CB (q) ≥ α}.(c) CB (p) =∞ if (b) doesn’t hold for any α.

(2) If CB (p) <∞ for all p ∈ X, then {CB (p) : p ∈ X} has the greatest element,say α, and the set {p ∈ X : CB (p) = α} is finite, say of cardinality n (bycompactness ofX). We then say that α is the CB-rank ifX, or CB (X) = α,and that n is the CB-multiplicity of X,CB−mult (X) = n.

Definition 3.8. (1) T is called totally transcendental, or t.t., if for every n ∈ ω,CB (Sn (M)) <∞ (i.e. bounded by some ordinal).

(2) Let Φ (x) be a set of formulas over a small set. Let Y = {p ∈ Sn (M) : Φ ⊆ p},a closed subspace of Sn (M). By the Morley rank of Φ, RM (Φ), we meanCB (Y ). If CB (Y ) < ∞, then the Morley degree of Φ, dM (Φ), is definedto be the CB−mult (Y ).

Exercise 3.9. Show that if CB (S1 (M)) <∞, then T is t.t.

• We write RM (a/A) for RM (tp (a/A)).

Lemma 3.10. (1) RM (φ (x)) ≥ α + 1 if and only if there is an infinite set{φi (x) : i ∈ ω} of pairwise contradictory formulas such that φi (x) ` φ (x)and RM (φi) ≥ α for all i ∈ ω.


(2) RM (Φ (x)) = max {CB (p) : p ∈ Sx (M) ,Φ ⊆ p}.(3) RM (Φ (x)) = min {RM (Φ′ (x)) : Φ′ ⊆ Φ finite }. If RM (Φ) < ∞, then

dM (Φ) is the minimum of dM (φ) where φ is a finite conjunction of formulasin Φ and RM (φ) = RM (Φ).

(4) RM (φ (x)) = 0 iff φ (x) is algebraic.(5) RM (φ (x) ∨ ψ (x)) = max {RM (φ) ,RM (ψ)}.(6) For any set Φ (x) of formulas over A, there is a complete type p (x) ∈

S|x| (A) such that RM (Φ) = RM (p).(7) If RM (φ) = α then dM (φ) is the greatest k ∈ ω such that there are pairwise

contradictory φ1, . . . , φk with φi (x) ` φ (x) and RM (φi) ≥ α.(8) Suppose A ⊆M, a, b ∈M and b ∈ acl (Aa). Then RM (ab/A) = RM (a/A).(9) If X ⊆ Mn, Y ⊆ Mm are definable and f : X → Y is a definable function

s.t. RM(f−1 (a)

)= k for all a ∈ Y , then RM (X) = RM (Y ) + k.

(10) In particular, definable finite-to-one maps preserve RM.

Exercise 3.11. Prove this lemma.

Lemma 3.12. It T is countable, then T is ω-stable ⇐⇒ T is t.t.

Proof. ( =⇒ ): ω-stability =⇒ isolated types are dense, hence every type has anordinal-valued CB-rank.

(⇐): Exercise. �

Fact 3.13. (1) A formula φ (x) is s.m. ⇐⇒ RM (φ) = dM (φ) = 1.(2) Supp. T is s.m. If A ⊆M and a ∈M, then RM (a/A) = dim (a/A).

3.3. Definability of types.

Definition 3.14. p ∈ Sn (A) is definable over B if ∀φ (x, y) ∈ L there is dpφ (y) ∈L (B) s.t. φ (x, a) ∈ p ⇐⇒ |= dpφ (a) for all a ∈ A.

Theorem 3.15. If T is ω-stable andM≺M and p ∈ Sn (M), then p is definable.

• Recall that a formula δ (x, y) is stable if(∗) there is some n = n (φ) ∈ ω such that in no modelM |= T we can find(ai, bi : i ∈ n+ 1) such that |= δ (ai, bj) ⇐⇒ i ≤ j.

Exercise 3.16. Assume that δ (x, y1) , γ (x, y2) are stable. Then:• δ′ (y1, x) := δ (x, y1) is stable,• ψ (x, y1y2) := δ (x, y1) ∨ δ (x, y2) and ψ′ (x, y1) := ¬δ (x, y1) are stable

(hence any Boolean combination of stable formulas is stable).

• Theorem 3.15 follows from the following fundamental lemma using Exercise2.53 and compactness. Namely, given p ∈ Sn (M), let a ∈M s.t. a |= p (x).Then for any δ (x, y) ∈ L, we can take dpφ (y) :=

∨j

(∧i δ(aji , y

))∈

L (M) from the lemma.

Lemma 3.17. Let δ (x, y) be stable, M � M and a ∈ M. Then there is a fi-nite set

{aji : i, j < N

}of tuples in M s.t. for any b ∈ M , |= δ (a, b) ⇐⇒ |=∨

j

(∧i δ(aji , b

)).

Proof. • Let N1 be as given by (∗) for δ (x, y).


• Claim. For any ψ (x) ∈ tp (a/M), ∃a1, . . . , an ∈ M for some n ≤ N1 s.t.∀b ∈M ,

∧ni=1 δ (ai, b)→ δ (a, b).

– We choose inductively ai ∈M,ai |= ψ (x) and bi ∈M s.t. |= ¬δ (a, bi)and |= δ (ai, bj) ⇐⇒ i ≤ j.

– Suppose we have already found such a1, . . . , an, b1, . . . , bn.– Then a |= ψ (x) ∧

∧ni=1 ¬δ (x, bi), and this is an L (M)-formula.

– As a ∈M andM�M, there is also some an+1 ∈M satisfying it.– If (a1, . . . , an+1) satisfies the claim, we stop.– Otherwise there is some bn+1 ∈ M such that |=

∧n+1i=1 δ (ai, b), but

|= ¬δ (a, b).– This process must stop after ≤ N1 steps by the choice of N1.

• Let χ (x1, . . . , xN1 ; y) := δ (x1, y)∧ . . .∧ δ (xN1 , y). By Exercise 3.16, χ and¬χ are stable.

• Let N2 be as given by (∗) for ¬χ.• We choose inductively a1, a2, . . . in M satisfying the claim above, and bi ∈M s.t.(1) |= χ

(ai, bj

)⇐⇒ i > j,

(2) |= δ (a, bi) for all i.• Suppose we have already chosen such ai, bi for i = 1, . . . , n.• By the claim, ∃an+1 ∈ M s.t. |= χ

(an+1, bi

)for i = 1, . . . , n and s.t. for

any b ∈M , χ(an+1, b

)→ δ (a, b).

• So if∨n+1i=1 χ

(ai, y

)doesn’t satisfy the requirements of the lemma, then

∃bn+1 ∈ M s.t. |= δ (a, bn+1) but |= ¬χ(ai, bn+1

)for i = 1, . . . , n + 1 and

the construction can be continued.• But it must stop at some n ≤ N2.

�

3.4. Forking independence.• We assume T is ω-stable.• Using this notion of rank, we can define a “free” extension of a type to a

larger set of parameters that doesn’t add any new restrictions.

Definition 3.18. Supp. A ⊆ B, p ∈ Sn (A) , q ∈ Sn (B) and p ⊆ q. We say that qis a non-forking extension of p if RM (q) = RM (p).

Theorem 3.19. (Properties of non-forking extensions) Suppose p ∈ Sn (A) andA ⊆ B.

(1) ∃q ∈ Sn (B), a non-forking extension of p.(2) There are at most degM (p) n.f. extension of p in Sn (B), and exactly

degM (p) in Sn (M).(3) There is at most one q ∈ Sn (B), a n.f. extension of p with deg (p) =

deg (q).In part, deg (p) = 1 =⇒ p has a unique n.f. extension in Sn (B).

(4) If M � M,p ∈ Sn (M), M ⊆ B ⊆ M and q ∈ Sn (B) extends p, thenq ∈ Sn (B) is a non-forking extension ⇐⇒ q is definable over M .Namely, ∀φ (x, y) ∈ L, b ∈ B we have φ (x, b) ∈ q ⇐⇒ |= dpφ (b).

• Using it, we can define a notion of independence between subsets of M.


Definition 3.20. Given A,B,C small subsets ofM, we write A |CB if tp (A/BC)

doesn’t fork over C.

Fact 3.21. (Forking calculus) If T is ω-stable (or just stable), | satisfies thefolowing conditions.

(1) Invariance: A |CB and σ ∈ Aut (M) =⇒ σ (A) |

σ(C)σ (B).

(2) Finite character: A |CB ⇐⇒ A0 | C B0 for any finite A0 ⊆ A and

B0 ⊆ B.(3) Extension: If A |

CB, then ∀D ⊆M∃σ ∈ Aut (M /CB) s.t. σ (A) |

CBD.

(4) Local character: if A is finite and B any set, then there is some C ⊆ Bwith |C| ≤ |T | and s.t. A |

CB.

(5) Transitivity: A |CB ∧A |

BCD ⇐⇒ A |

CBD.

(6) Symmetry: A |CB ⇐⇒ B |

CA.

(7) Stationarity over models: ifM ≺M, tp (A/M) = tp (B/M) and A |MC,B |

MC,

then tp (A/CM) = tp (B/CM).

• Moreover, we have:

Fact 3.22. Let T be any theory, M |= T . Assume that there is some relation| ∗ on small subsets of M satisfying (1)-(7) in Fact 3.21. Then T is stable, and| ∗ = | .

Example 3.23. If T is a strongly minimal theory, | -independence coincides withacl-independence.

3.5. Canonical bases.

Definition 3.24. A type p ∈ S (A) is stationary if it has a unique non-forkingextension to a global type q ∈ S (M).

• We already saw that any typ p ∈ S (M) over a model is stationary.• A more careful analysis yields the following:

Lemma 3.25. Let T be stable, A = acleq (A). Then any p ∈ S (A) is stationary.

• We call the tp (a/ acleq (A)) the strong type of a over A, denoted stp (A).

Definition 3.26. Let p ∈ S (M) be a definable global type (i.e. definable oversome small A ⊆M) in a stable theory.

The canonical base of p, Cb (p), is := dcleq ({ddpφ (y)e : φ (x, y) ∈ L}).• For any σ ∈ Aut (M), σ fixes the canonical parameter dpφ (y) ⇐⇒ it

permutes the set p �φ(x,y). Hence using Lemma 3.5 we have:

Remark 3.27. Let p ∈ S (M), and A a small set. Then p is A-invariant ⇐⇒Cb (p) ⊆ dcleq (A).

In particular, Cb (p) is the smallest definably closed set in Meq over which p isinvariant.

Proposition 3.28. Let T be stable and p ∈ S (M).(1) p doesn’t fork over A ⇐⇒ Cb (p) ⊆ acleq (A).(2) p doesn’t fork over A and p �A is stationary ⇐⇒ Cb (p) ⊆ dcleq (A).

Definition 3.29. Let T be stable and A ⊆ M be small. Let p ∈ S (A) be astationary type. We define Cb (p) := Cb (p∗), where p∗ is the unique global non-forking extension of p.


Proposition 3.30. Let T be stable.(1) TFAE:

(a) a |AB,

(b) Cb (a/B) ⊆ acleq (A),(c) Cb (a/A) = Cb (a/B).

(2) Let p ∈ S (A) be stationary type and let (ai : i < ω) be a sequence in Msuch that ai |= p|Aa<i for all i ∈ ω (exists by saturation of M). ThenCb (p) ⊆ dcleq ((ai : i ∈ ω)).

Fact 3.31. Moreover, if T is s.m. then for any stationary type p, Cb (p) is a finiteset, hence an element in Meq.

• See my notes on stability at http://www.math.ucla.edu/~chernikov/teaching/StabilityTheory285D/StabilityNotes.pdf for more details.

4. A Closer look at pregeometries and Zilber’s trichotomy principle

• Recall: every uncountably categorical model is prime over a s.m. set, henceunderstanding what s.m. sets can look like is crucial.

• We return to pregeometries and have a closer look at their properties.

Remark 4.1. (1) Equivalently, a pregeometry is a set X with a collection C ⊆2X of closed subsets such that:• C is closed under intersections: B ⊆ C =⇒ (

⋂B) ∈ C.

• (Exchange) If C ∈ C and a ∈ X \ C, there is an immediate closedextension C ′ of C containing a, i.e.∃C ′ ∈ C s.t. C ′ ⊇ Ca and 6 ∃C ′′ ∈ C with C ( C ′′ ( C ′.• (Finite charactrer) If B ⊆ C, then

∨B =

⋃B0⊆B finite (

∨B0), where∨

B :=⋂{C ∈ C|

⋃B ⊆ C} is the smallest closed set containing all

B ∈ B.(2) Setting

∧B :=

⋂B and A ≤ B ⇐⇒ A ⊆ B, C forms a complete lattice.

(3) Given a closure operator cl on X, define C := {cl (A) : A ⊆ X}.(4) Conversely, given a set C of closed subsets, define cl (A) :=

⋂{C ∈ C : A ⊆ C}.

• For A,B ⊆ X, dim (A/B) is the cardinality of a basis for A over B (i.e. asubset A′ ⊆ A of minimal size such that cl (A′B) = cl (AB)).

Exercise 4.2. Equivalently, dim (A/B) is the length d of any chain of immediateextensions of closed sets cl (B) = C0 ( C1 ( . . . ( Cd = cl (AB).

• The localization of X at A is XA := (X, clA) where clA (B) := cl (AB).• Equivalently, the closed sets of XA are the closed sets of X which containA.

Example 4.3. In the case of a s.m. set D with the acl closure operator, localizingat A ⊆ D corresponds to adding the constant for all elements of A to the language.

• A geometry is a pregeometry (X, cl) s.t. cl (∅) = ∅ and cl ({a}) = {a} forany a ∈ X.

• If (X, cl) is any pregeometry, there is a natural associated geometry:Let X0 := X \cl (∅), and consider the relation ∼ on X given by a ∼ b ⇐⇒cl ({a}) = cl ({b}).By exchange, ∼ is an equivalence relation.

http://www.math.ucla.edu/~chernikov/teaching/StabilityTheory285D/StabilityNotes.pdf

http://www.math.ucla.edu/~chernikov/teaching/StabilityTheory285D/StabilityNotes.pdf


Let X := X0/ ∼, and define cl on X by cl (A/ ∼) = {b/ ∼: b ∈ cl (A)}.Then

(X, cl

)is a geometry.

Exercise 4.4. Equivalently,(X, cl

)is the set of dimension 1 closed sets of X with

cl defined by A ∈ cl (B) ⇐⇒ A ⊆ cl (⋃B).

Definition 4.5. A pregeometry is trivial, or disintegrated, if cl (A) =⋃a∈A cl (A).

• Equivalently,∨

=⋃

in the lattice of closed sets.

Example 4.6. (1) M |= T∞ is a strongly minimal set with (M, acl) a trivialpregeometry as all subsets are closed, acl (B) = B.

(2) An action of a group G on an infinite set D without fixed points in thelanguage

(D, (g (x))g∈G

)where g (x) = g ·x is a s.m. structure with trivial

pregeometry.Closed sets are unions of G-orbits, acl (B) = {g · b : g ∈ G, b ∈ B}.

Exercise 4.7. Any pregeometry satisfies the submodular law : dim (A ∪B) ≤dim (A) + dim (B)− dim (A ∩B).

Definition 4.8. A pregeometry (X, cl) is modular if for any finite-dimensionalclosed A,B ⊆ X,

dim (A ∪B) = dim (A) + dim (B)− dim (A ∩B) .

• In other words, dim (A/B) = dim (A/A ∩B).

Example 4.9. Let V be a vector space over a division ring K, with closed setsthe vector subspaces. For V finite dimensional, this is the acl-pregeometry of as.m. structureM |= VSK . By standard linear algebra, it is modular. This is not ageometry as cl (∅) = {0} and for any a ∈ V \ {0}, cl (a) is the line through a and 0.

The corresponding geometry is obtained by projectivising — deleting 0 and quo-tiening by the action of scalar multiplication — the projective space of dim (V )−1.

• A form of converse holds:

Fact 4.10. Any non-disintegrated modular geometry of dim ≥ 4 is isomorphic toa projective geometry over a division ring.

Exercise 4.11. (1) Prove that disintegrated pregeometries are modular.(2) Prove that forM |= VSK , (M, acl) is not disintegrated.

• Localization of a modular pregeometry is also modular (Exercise). However,the converse is false.

Example 4.12. Let V be an infinite dimensional vector space over a division ringK, with closed sets the affine spaces, i.e. translates of subspaces, and cl (∅) = ∅.

This is the acl-pregeometry of the s.m. structure (V, ({z = λx+ (1− λ) y} : λ ∈ K)).Here cl ({a}) = {a}, hence it is a geometry.

It is not modular: parallel lines within a common plane are dependent, but havetrivial intersection.

Localizing at 0 yields the projective geometry of the previous example.

Definition 4.13. (X, cl) is locally modular if the localization at any e ∈ X \ cl (∅)is modular.


Example 4.14. Let K |= ACF be a saturated model (hence, of infinite transcen-dence degree). Then (K, acl) is not locally modular.

• Let k ≺ K be a subfield of transcendence degree n.• We show that even localizing at k the geometry is not modular.• Let a, b, x be acl-independent over k.• Let y = ax+ b.• Then dim (k (x, y, a, b) /k) = 3 while dim (k (x, y) /k) = dim (k (a, b) /k) =

2.• We show acl (k (x, y)) ∩ acl (k (a, b)) = k contradicting modularity.• To see this, suppose d ∈ acl (k (a, b)) and y ∈ acl (k (d, x)). Let k1 =

acl (k (d)).• Then ∃p (X,Y ) ∈ k1 [X,Y ] an irreducible polynomial s.t. p (x, y) = 0.• By model completeness of ACF, p (X,Y ) is still irreducible over acl (k (a, b)).• Thus p (X,Y ) is α (Y − ax− b) for some α ∈ acl (k (a, b)) which is impos-

sible as then α ∈ k1 and a, b ∈ k1.

• ACF’s are the only known natural examples of non locally modular s.m.sets.

• Zilber’s philosophy: categorical objects are not random, and are canonical.Trichotomy principle: every s.m. set should be essentially T∞, VSK orACF.

• Zilber’s conjecture: Every non locally modular s.m. set is essentiallyan ACF (e.g., interprets an ACF. Conversely, a theorem of Macintyre saysthat every infinite ω-stable field is algebraically closed).

• It was refuted by Hrushovski via a combinatorial construction: there arenon locally modular s.m. sets which do not even interpret any infinitegroups!

• [Hrushovski, Zilber] Holds for “Zariski geometries”.

4.1. Characterization of local modularity.

Example 4.15. If V is a vector space, a1, . . . , an is a basis for a subspace A ⊆ V ,b1, . . . , bm is a basis for B subspace of V . Then any x ∈ Span (A ∪B) is the sumof some a ∈ A and b ∈ B.

• An analogous property characterizes modularity:

Lemma 4.16. Let (X, cl) be a pregeometry. TFAE:(1) (X, cl) is modular,(2) If A ⊆ X is closed, non-empty, b ∈ X and x ∈ cl (A, b), then ∃a ∈ A s.t.

x ∈ cl (a, b).(3) If A,B ⊆ X are closed, non-empty, and x ∈ cl (A,B), then ∃a ∈ A, b ∈ B

s.t. x ∈ cl (a, b).

Proof. (1)⇒ (2):• By fin. char., WMA dimA is finite.• If x ∈ cl (b), we are done.• So WMA x 6∈ cl (b).• By modularity, dim (Abx) = dimA+ dim (bx)− dim (A ∩ cl (bx)) and• dim (Abx) = dim (Ab) = dim (A) + dim b− dim (A ∩ cl (b)).• Because dim (bx) = dim (b) + 1, ∃a ∈ A s.t. a ∈ cl (bx) \ cl (b).


• By exchange, x ∈ cl (ba).(2)⇒ (3)

• WMA A,B are finite dim.• Proof by induction on dimA.• If dimA = 0, then (3) holds.• Supp. A = cl (A0a), where dimA0 = dimA− 1.• Then x ∈ cl (A0Ba).• By (2) ∃c ∈ cl (A0B) s.t. x ∈ cl (c, a).• By induction, ∃a0 ∈ A0 and b ∈ B s.t. c ∈ cl (a0, b).• By (2) again, ∃a∗ ∈ cl (a0a) ⊆ A s.t. x ∈ cl (a∗b).

(3)⇒ (1).• Supp. A,B ⊆ X are fin. dim. and closed.• We prove (1) by induction on dimA.• If dimA = 0, done.• Supp. A = cl (A0a), where dimA0 = dimA − 1 and assume inductively

dim (A0, B) = dimA0 + dimB − dim (A0 ∩B).• Assume a ∈ cl (A0B).• Then dim (A0B) = dim (AB) and, as a /∈ A0, dimA = dimA0 + 1.• As a ∈ cl (A0B), by (3) ∃a0 ∈ A0 and b ∈ B s.t. a ∈ cl (a0b).• As a /∈ cl (a0), by exchange b ∈ cl (aa0). Thus b ∈ A.• But b /∈ A0, as otherwise a ∈ A0.• Hence dim (A ∩B) = dim (A0 ∩B) + 1 as desired.• Next, supp. a /∈ cl (A0B). Then need to show A ∩B = A0 ∩B.• Supp. b ∈ B and b ∈ cl (A0a) \ cl (A0).• Then by exchange a ∈ cl (A0b), a contradiction.

�

4.2. Families of plane curves.• The following definition aims to capture the idea that any parametrized

family of “curves” on the “plane” D2 is at most 1-dimensional.

Definition 4.17. Supp. D = φ (M) ⊆Mn is s.m. Then D is linear if ∀p ∈ S2 (D),if φ (v1) ∧ φ (v2) ∈ p and RM (p) = 1, then RM (Cb (p)) ≤ 1.

Theorem 4.18. Let D ⊆Mn be a s.m. set. TFAE:(1) for some small B ⊆ D, the pregeometry DB is modular,(2) D is linear,(3) for any b ∈ D \ acl (∅), Db is modular,(4) D is locally modular.

Proof. We will assume that D = M (can always be achieved without loss of gener-ality using definability of types and working inside the induced structure on D).

(1)⇒ (2)

• Claim. D is non-linear =⇒ DB is non-linear.• Supp. p ∈ S2 (D), RM (p) = 1, α = Cb (p) ∈Meq, RM (α) ≥ 2.• If α′ realizes a n.f. ext. of tp (α) to B, then α′ is a canonical base for a

rank 1 type and DB is non-linear.

• Thus, adding B to the language, WMA B = ∅ and D is modular.


• Let p ∈ S2 (D) with RM (p) = 1.• Let φ (v1, v2, a) ne a s.m. formula in p.• Let (b1, b2) |= p (v1, v2).• Let X := acl (a) ∩ acl (b1, b2).• By modularity, dim (X) = dim (a) + dim (b1, b2)− dim (a, b1, b2).• As dim (a, b1, b2) = dim (a) + 1 and 1 ≤ dim (b1, b2) ≤ 2, dimX ≤ 1.• Thus, dim (b1b2/a) = dim (b1, b2, a) = dim (a) = dim (b1, b2) − dim (X) =

dim (b1, b2/X).• Thus dim (b1b2/X) = 1 and p d.n.f. over X.• Hence Cb (p) ⊆ acleq (X) by the above theorem, so RM (α) ≤ 1.

(2)⇒ (3)

• Let b ∈ D \ acl (∅). We will use Lemma 4.16.• Supp. B is fin.dim. closed set.• Supp. a1 ∈ acl (a2Bb).• We must find: d ∈ acl (Bb) s.t. a1 ∈ acl (a2db).• WMA: a1 /∈ acl (Bb), a2 /∈ acl (Bb) and a1 /∈ acl (a2b) (otherwise we are

done).• Thus, dim (a1a2/b) = 2 and dim (a1a2/Bb) = 1.• Let α ∈ acleq (Bb) be a canonical base for the type of a1, a2 over acleq (Bb).•• Claim. α ∈ acleq (a1a2).• As α is a canonical base, RM (a1a2α) = 1.• As RM (a1a2/α) < RM (a1a2), RM (α/a1a2) < RM (α).• By (2), RM (α) = 1. Thus α ∈ acleq (a1a2).•• As RM (a1a2/b) = 2 and RM (a1a2/α) = 1, α /∈ acl (b).• Thus b /∈ acl (α) (as b /∈ acl (∅)).• Hence a1, b have the same type over α, hence by saturation of M, ∃d ∈ D

s.t. tp (a1a2/α) = tp (bd/α).• then d ∈ acl (bα) ⊆ (acl (a1a2b) ∩ acl (Bb))and d /∈ acl b.• We have d /∈ acl (a2b) (if d ∈ acl (a2b), then as d ∈ acl b, we have a2 ∈

acl (db) ⊆ acl (Bb) — a contradiction).• Thus, as d ∈ acl (a1a2b) \ acl (a2b), a1 ∈ acl (a2bd) — as wanted.

The other implications are easy. �

• We mention one more characterization of local modularity.

Definition 4.19. Supp. T is an ω-stable theory, M |= T . T is 1-based if wheneverA,B ⊆Meq, A = acleq (A) , B = acleq (B), then A |

A∩B B.

• The following is fundamental fact:

Theorem 4.20. Suppose that M is a s.m., non-trivial, locally modular. Then thereis an infinite group definable in Meq.

• Returning to our background question, [Zilber], [Hrushovski] at a largerlevel of generality, proved the following:

Fact 4.21. If D is a s.m. set in an ω-categorical theory, then D is 1-based (solocally modular).


5. Group configuration theorem

5.1. Germs of definable functions.

Definition 5.1. Let p, q ∈ S (M).(1) If f1, f2 are definable partial functions defined at p (meaning p (x) ` x ∈

dom (fi)), then we say that they have the same germ at p if p (x) ` f1 (x) =f2 (x).

(2) The germ of f at p is the equivalence class f of f under this equivalencerelation.

(3) We write f : p→ q if p (x) ` q (f (x)) for some (any) representative f .(4) If f : p → q has a representative f defined over b and a |= p|b, let f (a) :=

f (a).(5) This is well-defined and f (a) |= q|b (Check!).(6) Given f : p → q and σ ∈ Aut (M), we have a well-defined germ σ

(f)

:

σ (p)→ σ (q).(7) b ∈Meq is a code for f if ∀σ ∈ Aut (M), b = σ (b) ⇐⇒ f = σ

(f).

(8) If b is a code for f , we define⌈f⌉

:= dcleq (b).(9) If p, q are stationary types and p, q are their global non-forking extensions,

a germ at p is defined as a germ at p, and we write f : p → q to denotef : p→ q.

(10) For a stationary p, we write p|A = p �A — the unique non-forking extensionof p to A.

Remark 5.2. • Composition of germs on global types (hence also on station-ary types) is well-defined.

• Hence we have a category with objects=stationary types and morphisms=germs.• Write f : p→q if f is invertible in this category.• Equivalently, f is injective on p (clearly necessary, and if holds then by

compactness f is already injective on some φ (x) ∈ p, hence f has a well-defined definable inverse).

• Note:⌈f ◦ g

⌉⊆ dcleq

(⌈f⌉, dge

), and

⌈f−1

⌉=⌈f⌉.

Remark 5.3. • As T is stable, p and q are definable.• Hence given an ∅-definable family fz of partial functions (i.e. some formulaF (x, y; z) s.t. for any z, fz (x, y) is a partial function), equality of germs offb, fc at p is a definable equivalence relation E (b, c):consider the formula φ (x; b, c) := x ∈ dom (fb)∩ dom (fc)∧ fb (x) = fc (x).Then:

fb = fc ⇐⇒ φ (x; b, c) ∈ p ⇐⇒ |= dpφ (b, c) ,

hence taking E (b, c) := dpφ (b, c), |= E (b, c) ⇐⇒ fb = fc.• Then, since σ

(fb

)= fσ(b), we have

⌈fb

⌉= dcleq (b/E).

Definition 5.4. • If p, q, s ∈ S (∅) are stationary, a family fs of germs p→ q

is the family fs :=(fb : b |= s

)of germs at p of an ∅-definable family fz of

partial functions, which is such that fb : p→ q whenever b |= s.• The family is


– canonical if b is a code for fb for all b |= s.– generically transitive if fb (x) | x for some (any) b and x such thatb |= s and x |= p|b.

Exercise 5.5. fs is generically transitive ⇐⇒ ∀x |= p, y |= q|x , there is b |= s

s.t. fb (x) = y.

Proposition 5.6. (Finding families of germs)Suppose p, q, s ∈ S (∅) are stationary, and fs is a family of germs p → q. Let

b |= s, x |= p|b and y = fb (x). Then

x | b, y | b, y ∈ dcleq (bx) (∗)Conversely, if any (b, x, y) satisfy (∗), let fb (x) = y be a formula witnessing y ∈

dcleq (bx) (where fb is a partial definable function), let s := tp (b) , p := tp (x) , q :=

tp (y). Then fs is a family of germs p→ q.Moreover, we have:(1) fb is invertible ⇐⇒ also x ∈ dcleq (by);(2) the family fs is generically transitive ⇐⇒ x | y;(3)

⌈fb

⌉= Cb (xy/b) (so fs is canonical ⇐⇒ Cb (xy/b) = b).

Proof. (1) , (2) — Exercise.(3)

• Let σ ∈ Aut (M).• Let p be the global non-forking extension of p, hence σ (p) = p.• Let r be the global non-forking extension of stp (xy/b).• Then r (x, y) is equivalent to p (x) ∪ {y = fb (x)}.• We have σ (Cb (xy/b)) = Cb (xy/b) ⇐⇒ σ (Cb (r)) = Cb (r) ⇐⇒ σ (r) =r ⇐⇒ p (x) ∪

{y = fσ(b) (x)

}≡ p (x) ∪ {y = fb (x)} ⇐⇒ p (x) `

fσ(b) (x) = fb (x) ⇐⇒ σ(fb

)= fb.

�

5.2. Hrushovski-Weil.• This is a slightly more general version of the “group chunk theorem” of

Hrushovski-Weil.

Definition 5.7. (1) A∧-definable homogeneous space is given by a

∧-definable

set S and a∧-definable group G (with the graph of the operation ∗ of the

group and the action G × S → S are relatively definable), with G actingtransitively on X.

(2) S is connected if ∃ a stationary type s extending S s.t. if g ∈ G and b |= s|g, then g ∗ b |= s|g (i.e. Stab (s) = G). Then s is called the generic type ofG.

(3) Generics/connectedness of a group G are defined by considering the actionof G on itself.

Exercise 5.8. If (G,S) is definable of finite Morley rank, S is connected ⇐⇒deg (S) = 1, and tp (a) for a ∈ S is generic ⇐⇒ RM (a) = RM (S).

Theorem 5.9. (Hrushovski-Weil) Let p, s ∈ S (∅) be stationary, and supp. fs is agenerically transitive canonical family of invertible germs p→ p. Supp. fs is closed


under inverse and generic composition, meaning that ∀ b |= s there exists b′ |= s

s.t. f−1b = fb′ and for b1, b2 |= s, b1 | b2, there exists b3 |= s s.t. fb1 ◦ fb2 = fb3and bi | b3 for i = 1, 2.

Then in Meq there is a∧-definable over ∅ homogeneous space (G,S), a definable

embedding of s into G as its unique generic type, and a definable embedding of pinto S as its unique generic type, s.t. the generic action of s on p agrees with thatof G on S, i.e. fb (a) = b ∗ a for b |= s and a |= p|b.

Proof.• Let G be the group of germs generated by fs.

Claim 5.10. Any element of G is a composition of two generators.

Proof. • The family is closed under inverses =⇒ the identity is the compo-sition of two generators.

• s is a complete type + generic composition and automorphism =⇒ anygenerator is the composition of two generators.

• So: suffices to show any composition of three generators fb1 ◦ fb2 ◦ fb3 is thecomposition of two.

• Let b′ |= s|b1b2b3 .• Then fb1 ◦ fb2 ◦ fb3 = fb1 ◦ fb′ ◦ f−1b′ ◦ fb2 ◦ fb3 .• Now b′ | b2, so f−1b′ ◦ fb2 = fb′′ for some b′′ |= s, b′′ | b′, b′′ | b2.• Now b′ |

b2b3, so b′′ | b2 b3 since b′′ ∈ dcleq (b′b2).

• Since b′′ | b2, we have b′′ | b3 by transitivity of | .• Also b′ | b1.• So the germs fb1 ◦ fb′ and f−1b′ ◦ fb2 ◦ fb3 appear in the family fs by generic

composition.�

• Now G is∧-definable as pairs of realizations of s, modulo equality of the

corresponding composition of germs, and the group operation is defined bycomposition of germs.

Claim. G is connected with the generic type s: If g ∈ G, then for some (hence any)b |= s|g, we have g ∗ b |= s|g.

Proof. • This holds for g |= s by assumption.• Let g ∈ G, say g = g1 ∗ g2 with g1, g2 |= s.• Let b |= s|g1g2 .• Then g2 ∗ b |= s|g1g2 , so g1 ∗ g2 ∗ b |= s|g1g2 .• Now g = g1 ∗ g2 ∈ dcleq (g1g2). so b |= s|g and g ∗ b |= s|g.

�

• G acts generically on p by application of germs.

Claim 5.11. This action is transitive: if a, a′ |= p, then a′ ∈ G ∗ a

Proof. Indeed, let c |= p|aa′ .Then by generic transitivity of fs, ∃g, g′ |= s s.t. g ∗ a = c and g′ ∗ c = a′.Then (g′ ∗ g) ∗ a = a′. �

• Define S := (G× p) /E where (g, a)E (g′, a′) ⇐⇒ (h ∗ g)∗a = (h ∗ g′)∗a′for h |= s|aa′gg′ .


• E is definable by definability of the type s.• Define the action of G by h ∗ ((g, a) /E) := (h ∗ g, a) /E.• This is well-defined:

– if (g, a)E (g′, a′) and h ∈ G, then if h′ |= s|gg′aa′h, then also h′ ∗ h |=sgg′aa′h by genericity.

– Hence h′ ∗ h ∗ g ∗ a = h′ ∗ h ∗ g′ ∗ a′, so (h ∗ g, a)E (h ∗ g′, a′).– p embeds via a 7→ (1, a) /E.– G ∗ p = S, so the action is transitive.

• Finally, p is the generic type: if g ∈ G and a |= p|g, then g ∗ a |= p|g as thisis the action of a germ.

• For the group configuration theorem, we need to generalize to families ofbijections between two types.

Lemma 5.12. (Hrushovski-Weil for bijections) Supp. acleq (∅) = dcleq (∅), letp, q, r ∈ S (∅), and supp. fr is a gen. trans. canonical family of invertible germsp→ q. Let b1, b2 |= r be independent and f−1b1 ◦ fb2 = gc with gs a canonical familyof invertible germs p→ p and s = tp (c), and suppose c | bi for i = 1, 2. Then gssatisfies the assumptions, hence the conclusions, of the Hrushovski-Weil theorem.

Proof. • Let c′ |= s|c. Let b |= r|c,c′ .• Then, as c | bi, bc ≡ b2c, so say b′1bc ≡ b1b2c.• Similarly bc′ ≡ b1c, so say bb′2c′ ≡ b1b2c.• Then gc ◦ gc′ = f−1b′1

◦ fb ◦ f−1b ◦ fb′2 = f−1b′1◦ fb′2 .

• Now b′i | b by choice of b′i, since b1 | b2.• Also b′1 | b b

′2, since c | b c

′, since c | c′ and b | cc′.• Hence b′1 | bb′2, so b′1 | b′2 and b | b′1b′2.• Hence f−1b′1 ◦ fb′2 = gc′′ for some c′′ |= s|b′i .• Then as b | b′1b′2, we have b | c′′b′1, hence c′′ | b′1b, so c′′ | c.• Similarly c′′ | c′.• Finally, must check gs is gen. trans.• Let x |= p, let b′2 |= r|x, b′1 |= r|xb′2 .• Let y := fb′2 (x) , z := f−1b′1

(y).• Then y | x by gen. trans., and b′1 | yx, so y | xb′1, i.e. y |= q|xb′1 , soz |= p|xb′1 , in particular x | z.

• Since b′1b′2 ≡ b1b2, this proves gen. trans. of gs.�

• In an ω-stable theory, we can obtain all these objects definably.

Fact 5.13. Supp. M is ω-stable and G ⊆ M is an∧-definable group. Then G is

definable.

Exercise 5.14. Give an example of a type definable group in some theory T whichis not definable.

5.3. The group configuration.

Definition 5.15. A tuple (a, b, c, x, y, z) is a group configuration if• any non-collinear triple is independent,• acleq (ab) = acleq (bc) = acleq (ac),• acleq (ax) = acleq (ay) and acleq (a) = Cb (xy/a). Similarly for bzy and czx.


Example 5.16. Supp. (G,S) is a connected∧-definable (over ∅) homogeneous

space. Let (a, b, x) be an independent triple with a, b generics of G and x a genericog S. Let c = ba, y = cx, z = bx (then also y = az). Then (a, b, c, x, y, z) is a groupconfiguration (of (G,S)).

• Remarkably, the converse is also true:

Theorem 5.17. [Hrushovski] Supp. (a, b, c, x, y, z) is a group configuration. Then,after possibly expanding the language by parameters B with B | abcxyz, thereis a connected

∧-definable homogeneous space (G,S) and a group configuration

(a′, b′, c′, x′, y′, z′) of (G,S) such that a′ is interalgebraic with a, b′ is interalgebraicwith b, etc.

Proof. Idea:(1) Reduce acleq to dcleq: replacing the elements of the original group con-

figuration with interalgebraic ones, we may assume that (b, z, y) define agenerically transitive canonical family of invertible germs via Proposition5.6.

(2) Prove that the family of germs obtained this way satisfies the independenceassumption of Lemma 5.12 — thus obtaining a

∧-definable homogeneous

space.(3) Connect the resulting homogeneous space to the original group configura-

tion.�

For the details, see Pillay, “Geometric stability theory”, Chapter 5.4.

Remark 5.18. See also https://terrytao.wordpress.com/2013/11/16/qualitative-probability-theory-types-and-the-group-chunk-and-group-configuration-theorems/for a different exposition.

Corollary 5.19. Let D be a non-trivial, locally modular, strongly minimal set.Then there is an infinite group definable in Meq.

Proof. • Expand by parameters to make D modular (possibly by definition).• By non-triviality (after possibly expanding by more parametersm ∃a, b, c ∈D s.t.

RM (ab) = RM (bc) = RM (ca) = RM (abc) = 2.

• Let b′c′ |= stp (bc/a) |a.• Then RM (bcb′c′) = 2.• Then by modularity RM (acl (bc′) ∩ acl (b′c)) = 1.• Then there is d ∈ (acl (bc′) ∩ acl (b′c)) \ acl (∅).• Then (a, b, c, b′, c′, d) is a group configuration.• By Theorem 5.17 in Meq there is a

∧-definable group, which is definable

by Fact 5.13.�

References

[1] Bradd Hart, Ehud Hrushovski, and Michael C Laskowski. The uncountable spectra of countabletheories. Annals of Mathematics-Second Series, 152(1):207–258, 2000.

[2] Saharon Shelah. Classification theory: and the number of non-isomorphic models. Elsevier,1990.

https://terrytao.wordpress.com/2013/11/16/qualitative-probability-theory-types-and-the-group-chunk-and-group-configuration-theorems/

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