INTRODUCTION TO GEOMETRIC STABILITY

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1 INTRODUCTION TO GEOMETRIC STABILITY ARTEM CHERNIKOV Lecture notes for the IMS Graduate Summer School in Logic, National University of Singapore, Jun 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 2 2. Morley s categoricity theorem Algebraic closure Strongly minimal formulas and theories Algebraic closure in strongly minimal sets and pregeometries Dimension and independence in pregeometries Uncountable categoricity of s.m. theories ω-stability Ehrenfeucht Mostowski Models Two-cardinal models and elimination of Morley s theorem, proof by Baldwin-Lachlan Further work on counting models Morley rank and forking independence in ω-stable theories M eq and canonical parameters Morley rank Definability of types Forking independence Canonical bases A Closer look at pregeometries and Zilber s trichotomy principle Characterization of local modularity Families of plane curves Group configuration theorem Germs of definable functions Hrushovski-Weil The group configuration 28 References 29 1

2 INTRODUCTION TO GEOMETRIC STABILITY 2 1. Preliminaries We recall briefly some basic facts and definitions in order to set up the notation for our discussion. M = (M, R 1,..., f 1,..., c 1,...) denotes a first-order structure in a language L. T denotes a first-order theory, M = T if it satisfies all sentences in T. Th (M) denotes the complete theory of M. Given A M, φ ( x) is an L (A)-formula (formula over a set of parameters A) if it is of the form φ ( x) = ψ ( x, b ) where ψ ( x, ȳ) 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 M n is an A-definable set if A = ψ (M) for some ψ (x) L (A). Def n (A) is the boolean algebra of all A-definable subsets of M n. Given L-structures M, N, write M 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 Ψ 0 Ψ is consistent (i.e. there is some L- structure M =Ψ 0 ), then Ψ is consistent. Fact 1.2. (Löwenheim Skolem theorem) Let M = T be given, with M ℵ 0. 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 in M, 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 in M. (2) M is κ-homogenous if any partial elementary map from M to itself with a domain of size < κ can be extended to an automorphism of M. Fact 1.4. For any T and κ, there is a κ-saturated and κ-homogeneous model M of T. Definition 1.5. For A M, we let S n (A) denote the space of all complete n-types over A. This is the Stone dual of Def n (A), hence compact (=compactness theorem), Hausdorff space with a basis of clopens given by the sets φ (x) = {p S n (A) : φ (x) p} for φ (x) L (A).

3 INTRODUCTION TO GEOMETRIC STABILITY 3 A type p (x) S n (A) is isolated if it is an isolated point of the top. space S n (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 sets and types explicitly using quantifier elimination (i.e., every formula is equivalent to one 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, Def 1 (M) consists of the finite and cofinite subsets. S 1 (M) = {tp (a/m) : a M} {p }, where p = {x a : a M} is the type of a new element. (2) We consider a vector space M = (M, 0, +,, (r (x) : r K)) where K is a field (finite or infinite), and r (x) is a function x rx. Let VS K := Th (M). It has QE, hence definable sets are given by the Boolean combinations of equations of the form r 1 x r n x n = 0. In particular, every set in Def 1 (M) is either finite or cofinite. (Note that the field R is incorporated into the language, and is not a definable as a part of the structure.) (3) ACF p - the theory of an algebraically closed field of char p, where p is prime or 0 (e.g. Th (C, +,, 0, 1) = ACF 0 ). By Tarski s QE, definable sets are precisely the Boolean combinations of algebraic sets of the form { x : p ( x) = 0}, where p is a polynomial. Again, every set in Def 1 (M) is either finite or cofinite. (4) DLO=Th (Q, <) the theory of dense linear orders without end points. By QE, Def 1 (M) consists of finite unions of (bounded or unbounded) intervals (hence can be infinite-coinfinite). S 1 (M) Dedekind cuts in (M, <) (For a cut C = (A, B), let p C := {a < x < b : a A, b B}. This set of formulas is consistent by the density 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 ℵ 0 -saturated. (*) (C, +,, 0, 1) is 2 ℵ0 -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 opportunity to 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 cardinality κ, up to isomorphism). Then T is κ-categorical for every uncountable cardinal κ. Example 2.2. (1) T is uncountably categorical indeed, every model of T is determined up to isomorphism by its size. (2) Let K be a countable field, then VS K is uncountably categorical by the standard linear algebra, any two vector spaces of the same dimension

4 INTRODUCTION TO GEOMETRIC STABILITY 4 are isomorphic, and the dimension of any uncountable vector space over a countable field is equal to its size. (3) Same holds for ACF p with the transcendence degree over the prime field in the place of dimension. Exercise 2.3. Show: (1) Assume K is a countable field. How many countable models does VS K have? (2) DLO is not κ-categorical for any uncountable κ(but ℵ 0 -categorical). (3) RG, the theory of the countable random graph, has 2 κ models of size κ, κ uncountable (but ℵ 0 -categorical). (4) (*) In fact, DLO also has 2 κ models of size κ, κ uncountable 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. Let M 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 the same set in N by elementarity). In particular, elementary substructures are algebraically closed. (2) acl (A) max (A, T ). Example 2.6. (1) If M = T, the acl (A) = A for any set A. (2) If M (Z, s), then acl (A) is the set of points reachable from A (Exercise). (3) If M = VS K, then a vector a acl (A), if and only if a is in the span of A. (4) If M = ACF p, 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 finitely many realizations in any elementary extension of M. Exercise 2.9. (1) If A M and M is A + -saturated, then p S (A) is algebraic p (M) is finite. (2) Let A, B be subsets of M and (c 0,..., c n ) a sequence of elements non algebraic over A. If M is A B + -saturated, the type tp (c 0,..., c n /A) has a realization which is disjoint from B. (3) Determine acl in DLO and RG. (4) Given A N such that N is A + -saturated, show that acl (A) = {M N : A M}.

5 INTRODUCTION TO GEOMETRIC STABILITY Strongly minimal formulas and theories. Let T be a complete theory with infinite models. Definition Let M = T, and D (x) L (M) a non-algebraic formula. (1) The set D (M) is minimal in M 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 T, VS K, ACF p are strongly minimal (by the discussion in Example 1.6). Example Minimal strongly minimal Let L = {E} and let M 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 a minimal formula. However, by compactness there is N M and a N such that the equivalence class of a is infinite. Then E (x, a) defines an infinite-coinfinite subset, hence x = x is not strongly minimal. Remark 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 Σ ψ (z, a) = { >k x (φ (x, a) ψ (x, z)) >k x (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 k ψ such that M = z ( k ψ x (φ (x, a) ψ (x, z)) k ψ x (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 Assume M = T is ω-saturated. Then any formula φ (x) L (M) such that φ (M) is minimal, is strongly minimal. Proof. If M is ω-saturated and φ (x, a) is not s.m., then for some ψ (x, z) L the set Σ ψ (z, a) is realized in M, so φ (M) is not minimal. Exercise If M is strongly minimal, A M and satisfies N := acl (A) is infinite, then N M Algebraic closure in strongly minimal sets and pregeometries. Definition Let M be an L-structure and D (x) L (M) a strongly minimal formula. We consider the closure operator defined by cl : 2 D(M) 2 D(M) cl (A) := acl (A) D (M). Proposition cl satisfies the following properties for all A D (M) and a, b D (M): (1) (Reflexivity) A cl (A), (2) (Finite character) cl (A) = {cl (A ) : A A finite},

6 INTRODUCTION TO GEOMETRIC STABILITY 6 (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) are straightforward, and we check (3). Assume that c cl ({b 1,..., b n }) and b i cl (A). Let φ (x, b 1,..., b n ) L be an algebraic formula satisfied by c. Let φ i (y) L (A) be algebraic formulas satisfied by the b i s. Then ψ (x) := ( y 1... y n φ1 (y 1 )... φ n (y n ) k zφ (z, y 1,..., y n ) φ (x, y 1,..., y n ) ) 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 is s.m. Suppose a cl (Ab) \ cl (A). Then M = φ (a, b), where φ (x, y) L (A) and φ (D, b) = n. Let ψ (y) := =n x (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) := =l y (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 a 1,..., a n+1 such that = χ (a i ). The set B i := {y D : φ (a i, y) ψ (y)} is cofinite for all i = 1,..., n + 1. Choose b B i. Then = φ (a i, b ) for each i, so {x D : = φ (x, b )} n + 1 contradicting the fact that ψ (b ) holds. Exercise Give an example of a structure M in which acl doesn t satisfy the exchange Dimension and independence in pregeometries. Definition Any abstract closure operator cl : 2 X 2 X on subsets of an infinite set X is satisfying properties (1)-(4) in Proposition 2.17 is called a pregeometry. If the underlying set is finite, it is called a matroid. There is a rather rich theory of matroids, we will discuss this more. What is important for us here is that one can develop notions of independence, dimension, basis, etc. in any pregeometry, generalizing linear independence in vector spaces and algebraic independence in algebraically closed fields. Definition Let (D, cl) be a pregeometry. A subset A D is called:

7 INTRODUCTION TO GEOMETRIC STABILITY 7 (1) independent (over C) if a / cl (A \ {a}) (resp., a / cl (C (A \ {a}))) for all 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 (Replacement lemma) Let A, B D be independent with A cl (B). (1) Suppose A 0 A, B 0 B and A 0 B 0 is a basis for acl (B) and a A \ A 0. Then b B 0 s.t. A 0 {a} (B 0 \ {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 B 0 be of min. cardinality s.t. a cl (A 0 C). As A is independent, C 1. Let b C. By exchange, b cl (A 0 {a} (C \ {b})), thus cl (A 0 {a} (B \ b)) = cl (B). If a cl (A 0 (B 0 \ {b})), then b cl (A 0 (B 0 \ {b})), contradicting A 0 B 0 being a basis. Thus A 0 B 0 is a basis. Thus A 0 {a} (B 0 \ {b}) is independent. (2) Supp. B is finite: say B = n and a 1,..., a n+1 A are distinct. Let A 0 = and B 0 = B. Using (1) inductively, find b 1,..., b n B distinct, s.t. {a 1,..., a i } (B \ {b 1,..., b i }) is a basis for cl (B) for i n. But then cl (a 1,..., a n ) = cl (B). As a n+1 cl (B), this contradicts the independence of A. Supp. B is infinite: then for any finite B 0 B, A cl (B 0 ) is finite and A B 0 B finite cl (B 0). Thus A B. (3) is immediate from (2). Hence we can define: Definition If Y D, then dim (Y ), the dimension of Y, is the cardinality of a basis for Y Uncountable categoricity of s.m. theories. We go back to a matroid given by acl on a definable strongly minimal set D. Lemma Supp. M, N = T, φ (x) L (A) is a s.m. formula, either A = or A M 0 where M 0 M, N. If a 1,..., a n φ (M) are independent over A and b 1,..., b n φ (N) are independent over A, then tp M (ā/a) = tp N ( b/a ). Proof. We assume φ (x) L (A), A M 0, M 0 M, N (case A = is a straightforward modification of the argument).

8 INTRODUCTION TO GEOMETRIC STABILITY 8 By induction on n. Assume n = 1, a φ (M) \ cl (A) and b φ (N) \ cl (A). Let ψ (x) L (A), and suppose M = 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. Because M 0 M, N and b / cl (A), also N = ψ (b). As ψ L (A) was arbitrary, tp M (a/a) = tp N (b/a). Assume the claim is true for n and a 1,..., a n+1 φ (M), b 1,..., b n+1 φ (N) are independent sequence over A. Let ā = (a 1,..., a n ), b = (b 1,..., b n ), by induction tp M (ā/a) = tp N ( b/b ). Assume M = ψ (ā, a n+1 ) for some ψ ( w, v) L (A). As a n+1 / cl (Aā), φ (M) \ ψ (ā, M) is finite. Hence n s.t. M = {v : φ (v) ψ (ā, v)} = n. Because M 0 M, N and tp M (ā/a) = tp N ( b/a ), also N = { v : φ (v) ψ ( b, v )} = n. As b n+1 / cl ( A b ), N = ψ ( b, bn+1 ), and we conclude. Lemma If M, N and φ are as above and dim (φ (M)) = dim (φ (N)), then there 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 tp M (b/b ). As g is elementary, can find c φ (N) such that N = ψ ( c, g ( d)). Then tp M (b/b ) = tp N (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 If T is a countable strongly minimal theory, then T is κ-categorical for all uncountable κ. Moreover, it has at most countably many models of cardinality ℵ 0. Proof. If T is countable, then acl (A) A + ℵ 0, hence any basis for M with M = κ > ℵ 0 has cardinality κ. If M ℵ 0, then dim (M) ℵ ω-stability. We aim to show that if T is κ-categorical for some uncountable κ, then it is a fibration over a definable strongly minimal set. For this, we isolate some consequences of κ-categoricity.

9 INTRODUCTION TO GEOMETRIC STABILITY 9 Definition T is ω-stable if for every M = T and every A M, A ℵ 0 = S 1 (A) ℵ 0. Exercise (1) Show that under ω-stability, also S n (A) ℵ 0 for all countable A and n ω. (2) Show that a countable T is ω-stable S 1 (M) = M for all infinite M = T. By Stone duality, the space of types is small definable sets are not so complicated. Example 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 + ℵ 0 ) and the unique non-algebraic type. Theorem Let T be ω-stable. If M = T, then there is a minimal formula in M. Proof. Suppose no formula is minimal. Then by induction we can build a tree of L (M)-formulas (φ η (x) : η 2 <ω ) such that: φ is x = x, η ν 2 <ω = φ ν φ η, φ η i (x) φ η (1 i) (x), φ η (M) is infinite for all η 2 <ω. Let A be the set of all parameters of all formulas φ η, A ℵ 0. For each η 2 ω, the set {φ η n (x) : n ω} is consistent, hence extends to some p η S 1 (A). And if η, ν are incomparable, then p η p ν. Hence S 1 (A) = 2 ℵ0, contradicting ω-stability. Definition Given A M = T, we say that M is prime over A if N = T and f : A N partial elementary, there is an elementary f : M N extending f. Lemma Let a countable T be ω-stable, M = T, A M. The isolated types in S n (M) are dense. Proof. If not, can build a binary tree of formulas as in Theorem Theorem Let a countable T be ω-stable, M = T and A M. There is M 0 M, a prime model over A. Proof. We find an ordinal δ and subsets (A α : α δ) of M s.t. (1) A 0 = A. (2) α limit = A α = β<α A β. (3) if no element of M \ A α realizes an isolated type over A α, stop and let δ = α; (4) otherwise, pick a α with tp (a α /A α ) isolated and let A α+1 = A α {a α }. Let M 0 be the substructure of M with M 0 := A δ. Claim 1. M 0 M. By Tarski-Vaught test: suppose M = φ (v) with φ (x) L (A δ ). By the lemma, b M s.t. tp (b/a δ ) is isolated and M = φ (b). By choice of δ, b A δ.

10 INTRODUCTION TO GEOMETRIC STABILITY 10 Claim 2. M 0 is a prime model extension of A. Let N = T and f : A N partial elementary. We find by induction f = f 0... f α f δ with f α : A α N elementary. α limit: let f α = β<α f β. Given f α : A α N partial elementary, ( let φ (v, a) isolate tp M0 (a α /A α ). Then φ (v, f α (a)) isolates f α tp M 0 (a α /A α ) ) in S1 N (f α (A)). As f α is elementary, b N s.t. N = φ (b, f α (a)). Hence f α+1 = f α {(a α, b)} is elementary. Hence f δ : M 0 N is elementary, as wanted. Theorem If T is κ-categorical for some uncountable κ, then T is ω-stable. Before we can prove it, we need an auxilliary construction Ehrenfeucht Mostowski Models. Definition Let I be a linear ordering. A sequence (a i : i I) of tuples in an L-structure M is indiscernible over B M if tp (a i1,..., a in /B) = tp (a j1,..., a jn /B) n ω, ī, j strictly increasing tuples from I. Example We saw that if T is strongly minimal, and (a i : i I) is a sequence of acl-independent elements, then it is indiscernible. Fact For any T in language L and I there is some M = T and (a i : i I) a non-constant indiscernible sequence in M. Proof. Recall Ramsey theorem: for any n, k ω and f : ω m k, there is an infinite subset S ω s.t. f is constant on all increasing m-tuples from S. Consider the partial type Σ (x i : i I) containing formulas (1) T, (2) x i x j, i j, (3) φ (x i1,..., x in ) φ (x j1,..., x jn ), φ L and ī, j increasing tuples from I. Any finite Σ 0 Σ is consistent: let (a i : i ω) be any infinite sequence in some M = T, and a finite set of formulas. By Ramsey, (a i : i ω) contains an infinite subsequence satisfying condition (3) (and the other conditions hold). Conclude by compactness. Theorem Let L be countable, T an L-theory. For all κ ℵ 0 there is M = T with M = κ such that if A M, then M realizes at most A + ℵ 0 types over A. Proof. Let T be an expansion of T in a language L L with Skolem functions (i.e. for each formula φ (x, y) L there is a function symbol f φ (y) such that T y ( xφ (x, y) φ (f φ (y), y)). Exists by taking a model of T and expanding until catching the tail.) Let M = T and contain I, an L -indiscernible sequence of order type (κ, <).

11 INTRODUCTION TO GEOMETRIC STABILITY 11 Let M, an L -substructure generated by I. In particular, the L-reduct of M is a model of T, and M = κ. Let A M. For each a A, a = t a ( x a ) for some term t a and x a from I. Let X := {x I : x occurs in x a for some a A}. Then X A + ℵ 0. By L -indiscernibility: if ȳ, z are tuples from κ with the same order type over X (denoted by ȳ X z) and t is an L -term, then t (ȳ), t ( z) have the same type over A. Hence enough to show: for any n ω, I n / X A + ℵ 0. For y I \ X, let C y := {x X : x < y}. Then z X ȳ 1 i n: (1) if y i X, then y i = z i, (2) if y i / X, then z i / X and C yi = C zi. As I is well-ordered, C y = C z C y = C z = or inf {i I : i > C y } = inf {i I : i > C z }. In particular, there are at most X + 1 possible cuts C y. Hence I n / X A + ℵ 0, 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 S 1 (A) > ℵ 0. We can find N 0 M with N 0 realizing uncountably many types from S 1 (A). By Theorem 2.37, N 1 = T, N 1 = κ, and B N 1, B = ℵ 0, N 1 realizes at most ℵ 0 types over B. Then N 0 = N1, contradicting κ-categoricity Two-cardinal models and elimination of. Definition Let κ > λ ℵ 0 be cardinals. Then M = T is a (κ, λ)-model if M = κ and for some φ (x) L, φ (M) = λ. Proposition A countable T is κ-categorical = T has no (κ, λ)-models for any λ < κ. Proof. By compactness and downwards LS, any countable T has a model M of size κ s.t. every infinite -definable subset of M also has cardinality κ. Fact (Vaught s two-cardinal theorem) If T has a (κ, λ)-model with κ > λ ℵ 0, then T has an (ℵ 1, ℵ 0 )-model. Almost any other implication of this form is false or independent from ZFC. Definition T has a Vaught pair if there are M N = T and φ (x) L (M) such that: (1) M N, (2) φ (M) is infinite, (3) φ (M) = φ (N). Lemma 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.

12 INTRODUCTION TO GEOMETRIC STABILITY 12 As φ (N) = φ (M), this is a Vaught pair. Vaught s theorem really proves the following: Lemma If T has a Vaught pair, then it has a (ℵ 1, ℵ 0 )-model. Corollary If T is ℵ 1 -categorical, then T has no Vaught pairs. Assuming ω-stability, we have the following converse. Theorem Suppose T is ω-stable and T has an (ℵ 1, ℵ 0 )-model. If κ > ℵ 1, then T has a (κ, ℵ 0 )-model. Proposition If T has no Vaught pairs, then T eliminates. That is, for each φ ( v, w) L n φ ω s.t. if M = T, ā M and φ (M, ā) > n, then φ (M, ā) is infinite. Proof. Let L := L {P (x), c 1,..., c n }. Let T be the theory of all L -structures (M, P (M), c 1,..., c n ) where: M is a model of T, P (M) L M, P (c i ) holds i, φ (M, ā) P (M). Suppose no bound n φ ω exists. Then n N = T and ā N s.t. φ (N, ā) is finite, but has more than n elements. Let M be a proper elem. ext. of N. Then φ (M, ā) = φ (N, ā) and (M, N, ā) = T. Hence T { >n xφ (x, c) : n ω} is finitely satisfiable. By compactness it has a model, which gives a Vaught pair for T. Corollary If T has no Vaught pairs, then any min. formula is s.m. Proof. If φ (x, ā) is minimal in M and T eliminates, then for each ψ (x, z) L, ( x (φ (x, ā) ψ (x, z)) x (φ (x, ā) ψ (x, z))) is an elementary property of z and holds in M, hence holds in every model of T. Exercise Show that RG has a Vaught pair Morley s theorem, proof by Baldwin-Lachlan. Now we put everything together. Theorem (Baldwin-Lachlan) Let κ be an uncountable cardinal. A countable T 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 categoricity 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 model M 0 (over ). By Theorem 2.29 and Corollary 2.47, an s.m. formula φ (v) L (M 0 ). Let M, N = T, of card. κ ℵ 1.

13 INTRODUCTION TO GEOMETRIC STABILITY 13 WLOG M 0 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) and φ (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 has at most countably many countable models. This can be refined to: Fact (Baldwin-Lachlan) If T is uncountably categorical but not ℵ 0 -categorical, then it has exactly ℵ 0 countable models Further work on counting models. Definition For a theory T, consider the function I T (κ) = the number of models of T of size κ, up to isomorphism. Note that 1 I T (κ) 2 κ for all infinite κ bigger than the cardinality of the language. Morley s theorem says: let T be a countable theory. If I T (κ) = 1 for some uncountable κ, then I T (κ) = 1 for all uncountable κ. Stability theory developed historically in Shelah s work as a chunk of machinery intended to generalize Morley s theorem to a computation of the possible spectra of complete first order theories, in particular to prove the following conjectures of Morley. Theorem [Shelah] (1) The function I T (κ) is non-decreasing on uncountable cardinals. (2) For theory T of any cardinality, there is some cardinal κ (T ) such that if I T (λ) = 1 for some λ > κ (T ), then I T (λ) = 1 for all λ > κ (T ). This project, at least for countable theories, was essentially completed by Shelah 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 to encode in a definable manner a certain finitary combinatorical configuration) on the space of first-order theories, showing that all theories on the non-structure side of the dividing line have as many models as possible, and on the structure side developing some kind of dimension theory and showing 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 case of a vector space we only need one cardinal invariant, but if we have an equivalence relation then we need to know the size of each of its classes).

14 INTRODUCTION TO GEOMETRIC STABILITY 14 A complete classification of the possible functions I T (κ) for countable theories was given by Hart, Hrushovski, Laskowski [1] (required some descriptive set-theoretic ideas in order to prove a continuum hypothesis for a certain notion 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 seen rather as a way of classifying definable sets in a structure and describing the interaction between definable sets. Eventually this theory started to be seen as having a geometric meaning. It turned out that the study of the definability of certain algebraic objects such as groups and fields is essential even for purely logical questions of categoricity. Moreover, more recently it was realized that a lot of techniques from stability can still be developed in larger contexts, and the so-called generalized stability is currently an active area of research. Exercise Show that if T is ω-stable, then for any formula φ (x, y) L and any model M = T, we cannot find a sequence (a i, b i : i ω) in M such that M = φ (a i, b j ) i j. Any such formula is called stable, and large part of the theory developed in this course 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 M eq and canonical parameters. All the notions above generalize in an obvious way to multi-sorted structures, i.e. the underlying set of our model is now partitioned into several sorts, and for each of the variables for each relation and function symbols we specify which sort they live in. Then formulas and other notions are defined and evaluated accordingly. We recall Shelah s construction of imaginaries, which allows us to view quotients of definable sets by definable equivalence relations as definable objects themselves. Given M = T, we construct a structure M eq in a language L eq extending L as follows: add a new sort M n /E, n ω and E (x, y) L defining an equivalence relation on M n. L eq contains an n-ary function symbol π E for the projection map π E : M n M n /E, π E (a) = [a] E, the E-equivalence class of a M n. 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 := Th L eq (M eq ). Exercise 3.1. T eq doesn t depend on the choice of the initial model M = T.

15 INTRODUCTION TO GEOMETRIC STABILITY 15 Every automorphism of M extends uniquely to an automorphism of M eq. If M = T is κ-saturated (-homogeneous), then M eq is also κ-saturated (resp., κ-homogeneous). For every ψ (x, x 1,..., x n ) L eq with x in the main sort and x i is of sort E i, φ (x, y 1,..., y n ) L with y i = arity (E i ) s.t. for all tuples a, a 1,..., a n M: M eq = ψ (a, π E1 (a 1 ),..., π En (a n )) M = ψ (a, a 1,..., a n ). If T is t.t., then T eq is also t.t. In particular, M eq is a monster model for T eq and T eq doesn t define any new subsets of the main sort. Definition 3.2. Given ψ (x, a) L (M), consider the equiv. rel. E ψ (y, z) x (ψ (x, y) ψ (x, z)), we define the canonical parameter ψ (x, a) of ψ (x, a) as the imaginary [a] Eψ. Note: E ψ (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 acl eq, dcl eq to denote acl and dcl in the structure M eq. Exercise 3.4. Assume that φ (M, a) = ψ (M, b) for any φ (x, a), ψ (x, b) L (M). Then φ (x, a) and ψ (x, b) are interdefinable, i.e. φ (x, a) dcl eq ( ψ (x, b) ) and ψ (x, b) dcl eq ( φ (x, a) ). Hence given X M n, we can talk about the canonical parameter X of X (well-defined up to interdefinability). Canonical parameters help dealing with automorphisms and give a Galoistheoretic characterization of definability: Lemma 3.5. Let X M n 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 (M eq /A) (i.e. σ is an automorphism of M fixing A pointwise). (3) X dcl eq (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.

16 INTRODUCTION TO GEOMETRIC STABILITY 16 (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 = φ (x, c). By (2) (1) applied to the definable set {[c] E }, we see that [c] E dcl eq (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 dcl eq (A) it is fixed by all automorphisms. Similarly for acl eq (Exercise, use the proof of the previous lemma and compactness): Lemma 3.6. Let X M n 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) X acl eq (A) 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 (by compactness of X). We then say that α is the CB-rank if X, 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 (S n (M)) < (i.e. bounded by some ordinal). (2) Let Φ (x) be a set of formulas over a small set. Let Y = {p S n (M) : Φ p}, a closed subspace of S n (M). By the Morley rank of Φ, RM (Φ), we mean CB (Y ). If CB (Y ) <, then the Morley degree of Φ, dm (Φ), is defined to be the CB mult (Y ). Exercise 3.9. Show that if CB (S 1 (M)) <, then T is t.t. We write RM (ā/a) for RM (tp (ā/a)). Lemma (1) RM (φ (x)) α + 1 if and only if there is an infinite set {φ i (x) : i ω} of pairwise contradictory formulas such that φ i (x) φ (x) and RM (φ i ) α for all i ω.

17 INTRODUCTION TO GEOMETRIC STABILITY 17 (2) RM (Φ (x)) = max {CB (p) : p S x (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 formulas in 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 φ 1,..., φ k with φ i (x) φ (x) and RM (φ i ) α. (8) Suppose A M, ā, b M and b acl (Aā). Then RM (āb/a) = RM (ā/a). (9) If X M n, Y M m 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 Prove this lemma. Lemma It T is countable, then T is ω-stable T is t.t. Proof. ( = ): ω-stability = isolated types are dense, hence every type has an ordinal-valued CB-rank. ( ): Exercise. Fact (1) A formula φ (x) is s.m. RM (φ) = dm (φ) = 1. (2) Supp. T is s.m. If A M and ā M, then RM (ā/a) = dim (ā/a) Definability of types. Definition p S n (A) is definable over B if φ (x, y) L there is d p φ (y) L (B) s.t. φ (x, a) p = d p φ (a) for all a A. Theorem If T is ω-stable and M M and p S n (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 model M = T we can find (a i, b i : i n + 1) such that = δ (a i, b j ) i j. Exercise Assume that δ (x, y 1 ), γ (x, y 2 ) are stable. Then: j δ (y 1, x) := δ (x, y 1 ) is stable, ψ (x, y 1 y 2 ) := δ (x, y 1 ) δ (x, y 2 ) and ψ (x, y 1 ) := δ (x, y 1 ) are stable (hence any Boolean combination of stable formulas is stable). Theorem 3.15 follows from the following fundamental lemma using Exercise 2.53 and compactness. Namely, given p S n (M), let a M s.t. a = p (x). Then for any δ (x, y) L, we can take d p φ (y) := ( )) j i (a δ j i, y L (M) from the lemma. Lemma Let δ (x, } y) be stable, M M and a M. Then there is a finite set {a j i : i, j < N of tuples in M s.t. for any b M, = δ (a, b) = ( )) i (a δ j i, b. Proof. Let N 1 be as given by ( ) for δ (x, y).

18 INTRODUCTION TO GEOMETRIC STABILITY 18 Claim. For any ψ (x) tp (a/m), a 1,..., a n M for some n N 1 s.t. b M, n i=1 δ (a i, b) δ (a, b). We choose inductively a i M, a i = ψ (x) and b i M s.t. = δ (a, b i ) and = δ (a i, b j ) i j. Suppose we have already found such a 1,..., a n, b 1,..., b n. Then a = ψ (x) n i=1 δ (x, b i), and this is an L (M)-formula. As a M and M M, there is also some a n+1 M satisfying it. If (a 1,..., a n+1 ) satisfies the claim, we stop. Otherwise there is some b n+1 M such that = n+1 i=1 δ (a i, b), but = δ (a, b). This process must stop after N 1 steps by the choice of N 1. Let χ (x 1,..., x N1 ; y) := δ (x 1, y)... δ (x N1, y). By Exercise 3.16, χ and χ are stable. Let N 2 be as given by ( ) for χ. We choose inductively ā 1, ā 2,... in M satisfying the claim above, and b i M s.t. (1) = χ ( ) ā i, b j i > j, (2) = δ (a, b i ) for all i. Suppose we have already chosen such ā i, b i for i = 1,..., n. By the claim, ā n+1 M s.t. = χ ( ) ā n+1, b i for i = 1,..., n and s.t. for any b M, χ ( ā n+1, b ) δ (a, b). So if n+1 i=1 χ ( ā i, y ) doesn t satisfy the requirements of the lemma, then b n+1 M s.t. = δ (a, b n+1 ) but = χ ( ) ā i, b n+1 for i = 1,..., n + 1 and the construction can be continued. But it must stop at some n N 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 Supp. A B, p S n (A), q S n (B) and p q. We say that q is a non-forking extension of p if RM (q) = RM (p). Theorem (Properties of non-forking extensions) Suppose p S n (A) and A B. (1) q S n (B), a non-forking extension of p. (2) There are at most deg M (p) n.f. extension of p in S n (B), and exactly deg M (p) in S n (M). (3) There is at most one q S n (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 S n (B). (4) If M M,p S n (M), M B M and q S n (B) extends p, then q S n (B) is a non-forking extension q is definable over M. Namely, φ (x, y) L, b B we have φ (x, b) q = d p φ (b). Using it, we can define a notion of independence between subsets of M.

19 INTRODUCTION TO GEOMETRIC STABILITY 19 Definition Given A, B, C small subsets of M, we write A B if tp (A/BC) C doesn t fork over C. Fact (Forking calculus) If T is ω-stable (or just stable), satisfies the folowing conditions. (1) Invariance: A C B and σ Aut (M) = σ (A) σ(c) σ (B). (2) Finite character: A C B A 0 C B 0 for any finite A 0 A and B 0 B. (3) Extension: If A C B, then D M σ Aut (M /CB) s.t. σ (A) C BD. (4) Local character: if A is finite and B any set, then there is some C B with C T and s.t. A C B. (5) Transitivity: A C B A BC D A C BD. (6) Symmetry: A C B B C A. (7) Stationarity over models: if M M, tp (A/M) = tp (B/M) and A M C, B M C, then tp (A/CM) = tp (B/CM). Moreover, we have: Fact Let T be any theory, M = T. Assume that there is some relation on small subsets of M satisfying (1)-(7) in Fact Then T is stable, and =. Example If T is a strongly minimal theory, -independence coincides with acl-independence Canonical bases. Definition A type p S (A) is stationary if it has a unique non-forking extension 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 Let T be stable, A = acl eq (A). Then any p S (A) is stationary. We call the tp (a/ acl eq (A)) the strong type of a over A, denoted stp (A). Definition Let p S (M) be a definable global type (i.e. definable over some small A M) in a stable theory. The canonical base of p, Cb (p), is := dcl eq ({ d p φ (y) : φ (x, y) L}). For any σ Aut (M), σ fixes the canonical parameter d p φ (y) it permutes the set p φ(x,y). Hence using Lemma 3.5 we have: Remark Let p S (M), and A a small set. Then p is A-invariant Cb (p) dcl eq (A). In particular, Cb (p) is the smallest definably closed set in M eq over which p is invariant. Proposition Let T be stable and p S (M). (1) p doesn t fork over A Cb (p) acl eq (A). (2) p doesn t fork over A and p A is stationary Cb (p) dcl eq (A). Definition Let T be stable and A M be small. Let p S (A) be a stationary type. We define Cb (p) := Cb (p ), where p is the unique global nonforking extension of p.

20 INTRODUCTION TO GEOMETRIC STABILITY 20 Proposition Let T be stable. (1) TFAE: (a) a A B, (b) Cb (a/b) acl eq (A), (c) Cb (a/a) = Cb (a/b). (2) Let p S (A) be stationary type and let (a i : i < ω) be a sequence in M such that a i = p Aa<i for all i ω (exists by saturation of M). Then Cb (p) dcl eq ((a i : i ω)). Fact Moreover, if T is s.m. then for any stationary type p, Cb (p) is a finite set, hence an element in M eq. See my notes on stability at 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, hence understanding 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 2 X 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 closed extension C of C containing a, i.e. C C s.t. C Ca and C C with C C C. (Finite charactrer) If B C, then B = B 0 B finite ( B 0 ), 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. a subset 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 immediate extensions of closed sets cl (B) = C 0 C 1... C d = cl (AB). The localization of X at A is X A := (X, cl A ) where cl A (B) := cl (AB). Equivalently, the closed sets of X A are the closed sets of X which contain A. Example 4.3. In the case of a s.m. set D with the acl closure operator, localizing at 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} for any a X. If (X, cl) is any pregeometry, there is a natural associated geometry: Let X 0 := X \cl ( ), and consider the relation on X given by a b cl ({a}) = cl ({b}). By exchange, is an equivalence relation.

21 INTRODUCTION TO GEOMETRIC STABILITY 21 Let ˆX (:= X 0 /, and define ĉl on ˆX by ĉl (A/ ) = {b/ : b cl (A)}. Then ˆX, ĉl) is a geometry. ( ) Exercise 4.4. Equivalently, ˆX, ĉl is the set of dimension 1 closed sets of X with ĉl defined by A ĉl (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 trivial pregeometry as all subsets are closed, acl (B) = B. (2) An action of a group G) on an infinite set D without fixed points in the language (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) + dim (B) dim (A B). dim (A B) Definition 4.8. A pregeometry (X, cl) is modular if for any finite-dimensional closed 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 sets the vector subspaces. For V finite dimensional, this is the acl-pregeometry of a s.m. structure M = VS K. By standard linear algebra, it is modular. This is not a geometry 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 quotiening by the action of scalar multiplication the projective space of dim (V ) 1. A form of converse holds: Fact Any non-disintegrated modular geometry of dim 4 is isomorphic to a projective geometry over a division ring. Exercise (1) Prove that disintegrated pregeometries are modular. (2) Prove that for M = VS K, (M, acl) is not disintegrated. Localization of a modular pregeometry is also modular (Exercise). However, the converse is false. Example Let V be an infinite dimensional vector space over a division ring K, 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 have trivial intersection. Localizing at 0 yields the projective geometry of the previous example. Definition (X, cl) is locally modular if the localization at any e X \ cl ( ) is modular.

22 INTRODUCTION TO GEOMETRIC STABILITY 22 Example Let K = ACF be a saturated model (hence, of infinite transcendence 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 k 1 = acl (k (d)). Then p (X, Y ) k 1 [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 impossible as then α k 1 and a, b k 1. 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, VS K or ACF. Zilber s conjecture: Every non locally modular s.m. set is essentially an ACF (e.g., interprets an ACF. Conversely, a theorem of Macintyre says that every infinite ω-stable field is algebraically closed). It was refuted by Hrushovski via a combinatorial construction: there are non locally modular s.m. sets which do not even interpret any infinite groups! [Hrushovski, Zilber] Holds for Zariski geometries Characterization of local modularity. Example If V is a vector space, a 1,..., a n is a basis for a subspace A V, b 1,..., b m is a basis for B subspace of V. Then any x Span (A B) is the sum of some a A and b B. An analogous property characterizes modularity: Lemma 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 dim A is finite. If x cl (b), we are done. So WMA x cl (b). By modularity, dim (Abx) = dim A + 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).

23 INTRODUCTION TO GEOMETRIC STABILITY 23 By exchange, x cl (ba). (2) (3) WMA A, B are finite dim. Proof by induction on dim A. If dim A = 0, then (3) holds. Supp. A = cl (A 0 a), where dim A 0 = dim A 1. Then x cl (A 0 Ba). By (2) c cl (A 0 B) s.t. x cl (c, a). By induction, a 0 A 0 and b B s.t. c cl (a 0, b). By (2) again, a cl (a 0 a) A s.t. x cl (a b). (3) (1). Supp. A, B X are fin. dim. and closed. We prove (1) by induction on dim A. If dim A = 0, done. Supp. A = cl (A 0 a), where dim A 0 = dim A 1 and assume inductively dim (A 0, B) = dim A 0 + dim B dim (A 0 B). Assume a cl (A 0 B). Then dim (A 0 B) = dim (AB) and, as a / A 0, dim A = dim A As a cl (A 0 B), by (3) a 0 A 0 and b B s.t. a cl (a 0 b). As a / cl (a 0 ), by exchange b cl (aa 0 ). Thus b A. But b / A 0, as otherwise a A 0. Hence dim (A B) = dim (A 0 B) + 1 as desired. Next, supp. a / cl (A 0 B). Then need to show A B = A 0 B. Supp. b B and b cl (A 0 a) \ cl (A 0 ). Then by exchange a cl (A 0 b), a contradiction Families of plane curves. The following definition aims to capture the idea that any parametrized family of curves on the plane D 2 is at most 1-dimensional. Definition Supp. D = φ (M) M n is s.m. Then D is linear if p S 2 (D), if φ (v 1 ) φ (v 2 ) p and RM (p) = 1, then RM (Cb (p)) 1. Theorem Let D M n be a s.m. set. TFAE: (1) for some small B D, the pregeometry D B is modular, (2) D is linear, (3) for any b D \ acl ( ), D b is modular, (4) D is locally modular. Proof. We will assume that D = M (can always be achieved without loss of generality using definability of types and working inside the induced structure on D). (1) (2) Claim. D is non-linear = D B is non-linear. Supp. p S 2 (D), RM (p) = 1, α = Cb (p) M eq, 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 D B is non-linear. Thus, adding B to the language, WMA B = and D is modular.