NOTES ON “STABLE DOMINATION AND INDEPENDENCE IN ACVF”

1. Classical stability theory

In this part we try to summarize the notions and results for stable theories just enough to have a feeling, which will help in the rest of the notes. We shall not attempt to answer the question“why stability theory?”, and we shall let the reader read the ‘Introduction’ chapter in the book, which is quite nice.

We assume some familiarity with basic model theory. For instance, formulas, types and of course the compactness theorem. In this section we tend to just sketch the proofs or skip them completely. In that case we direct the reader to one of the -now- classical books: Buechler, Pillay, Poizat.

Conventions and notations. From now on let T be a theory in a (multisorted) language L. Models of T are denoted by M, N. For a sort s of L, let s(M) denote the underlying set of M of sort s. Also by A ⊆ M we mean that A ⊆ ∐_{s} s(M), where s varies through all the sorts of L. We let U denote a sufficiently saturated model of T. This is to say that it is κ-saturated and κ-homogenous where κ is a cardinal bigger than the cardinality of everyting we are interested in. Sets of cardinality less than κ are called small. We denote tuples of variables by letters x, y, z. For a tuple x of variables of sort (s_{1}, …, s_{n}) and a model M of T, let M_{x} denote the set s_{1}(M) × ⋯ × s_{n}(M). Below ‘type’ means ‘complete type’, and ‘definable’ means ‘definable with parameters’. If ϕ(x) is an L-formula, then ϕ(M_{x}) denotes the set of realizations of ϕ in M. For a small set C, L(C) denotes the language obtained by augmenting L with a constant symbol for each element of C.

For a, b ∈ U_{x} and C ⊆ U, a ≡_{C} b means tp(a/C) = tp(b/C).

1.1. Type spaces. Let M be a model of T, C ⊆ M, and x a tuple of variables. We denote the set of all types over C by S_{x}(C). Note that it becomes a topological space with the open basis consisting of sets of the form

[ϕ] := {p ∈ S_{x}(C) : ϕ ∈ p},

where ϕ is an L(C)-formula, whose free variables are among x. Note that [ϕ] = S_{x} \ [¬ϕ] is also closed. Hence S_{x}(C) is a Stone space: it is compact, Hausdorff, totally disconnected space. If p ∈ S_{x}(C) is the only element of [ϕ], for some ϕ, then it is called an isolated type.

To Stone spaces, we attach an ordinal valued rank, called the Cantor-Bendixson rank defined as follows:

(1) CB(p) ≥ 0 for every p ∈ S_{x}(C),

(2) CB(p) = α if it is isolated in the (closed) subspace

{q ∈ S_{x}(C) : CB(q) ≥ α}.

We shall see later in this section that this rank becomes something more comprehensible if we go to saturated models.

Sometimes it is convenient to consider ‘local’ types: Let Δ be a finite set of L-formulas of the form δ(x, y). A Δ-type over C is a maximal set of formulas of the form δ(x, c) with δ(x, y) ∈ Δ and c ∈ C. Let S_{Δ}(C) be the set of Δ-types over C. Once again it becomes a Stone space with the Cantor-Bendixson rank, CB_{Δ}. Many of the things in the rest of this section could be handled in this setting as well.

From now on S(C) denotes ⋃ S_{x}(C), where the union is over all the tuples of variables x.

1.2. A few words on Imaginaries. Remember that T is said to have elimination of imaginaries if for every model M of T and ∅-definable equivalence relation E on s_{1}(M) × ⋯ × s_{n}(M), there is a ∅-definable function f with domain s_{1}(M)×⋯×s_{n}(M) such that for every a, b ∈ s_{1}(M)×⋯×s_{n}(M)

E(a, b) ⟺ f(a) = f(b).

We can achieve this in an artificial way by adding a sort for every such s_{1}(M) × ⋯ × s_{n}(M)/E with the projection map a ↦ a/E. The structure obtained this way is denoted by M^{eq} and its theory by T^{eq}.

Let ϕ(x, a) be an L(a)-formula, D = ϕ(M_{x}, a). Consider the ∅-definable equivalence relation E_{ϕ} defined by

E_{ϕ}(y_{1}, y_{2}) ⟺ ∀x(ϕ(x, y_{1}) ↔ ϕ(x, y_{2})).

We call ⌜D⌝ := a/E_{ϕ} the code for D; it lives in M^{eq}.

1.3. Stable theories. The adjective ‘stable’ applies to both theories and formulas as follows:

Definition 1.1. (1) Let κ be an infinite cardinal. The theory T is κ-stable if for every C ⊆ U, with |C| < κ, we have that |S_{x}(C)| < κ.

(2) T is stable if it is κ-stable for some κ.

(3) A formula ϕ(x, y) is unstable if there are a_{i} ∈ U_{x}, b_{i} ∈ U_{y} for i ∈ ℕ such that for every i, j ∈ ℕ

U ⊧ ϕ(a_{i}, b_{j}) ⟺ i < j.

Otherwise ϕ is said to be stable.

One very important property of stable theories is that the types are ‘definable’ in the following sense:

Definition 1.2. Let p ∈ S_{x}(C) and ϕ(x, y) an L-formula. An L(C)-formula ψ(y) is a ϕ-definition for p if for every a ∈ U_{y}

ϕ(x, a) ∈ p ⟺ U ⊧ ψ(a).

We shall write such ψ as d_{p}(ϕ). The type p is said to be definable over B ⊆ C, if for every ϕ(x, y), d_{p}(ϕ) is an L(B)-formula.

Remark/warning. Just because p is definable over B, we cannot conclude that there is a type over B′ ⊃ B defined by the same defining scheme. This is true if C is a model.

The following is the theorem that connects all the definitions above. (See Pillay or Buechler for a proof.)

Theorem 1.3. The following conditions are equivalent:

(1) T is stable.

(2) Every formula is stable.

(3) For every M ⊧ T, each 1-type over M is definable.

(4) For every M ⊧ T, each type over M is definable.

(5) For every M ⊧ T and every finite Δ, |S_{Δ}(M)| ≤ |M| + |L|.

(6) For every M ⊧ T and every cardinal κ, we have |S(M)| ≤ |M|^{κ}.

1.4. Stable embeddedness. A definable set D = ϕ(U_{x}, c) is stably embedded if for every definable E ⊆ U_{x}^{n} the set E ∩ D^{n} is definable over Dc. The induced structure on D, D_{ind} is the structure with underlying set D whose relations are of the form E ∩ D^{n} where E is c-definable. We say that D is stable whenever D_{ind} is stable.

Another definable set E is said to be D-internal if E ⊆ dcl(D ∪ A), where A is a finite set. Stable and stable embeddedness is the closed under internality.

Lemma 1.4. Let D = ϕ(U_{x}, c) be definable. Then the following are equivalent:

(1) D is stable and stably embedded.

(2) Every formula ψ(x_{1}, …, x_{n}, y) that implies ϕ(x_{1}, c) ∧ ⋯ ∧ ϕ(x_{n}, c) is stable.

(3) If κ > |L| and c ∈ B with κ = κ^{ℵ_{0}} and |B| = κ, then there are at most κ many 1-types over B that are realized in D.

As a result stable, stably embedded sets are closed under positive operations (union and intersection), but not under complements unless T is stable.

1.5. Invariant types. There is a class of types related to the definable ones called invariant.

Let Aut(U/C) denote the group of automorphisms of U fixing C pointwise. Note that Aut(U/C) acts on S(U) as follows: let g ∈ Aut(U/C) and p ∈ S_{x}(U), then g · p := {ϕ(x, g(a)) : ϕ(x, a) ∈ p}. A type p ∈ S(U) is said to be Aut(U/C)-invariant if it is in the stabilizer of this action. It is easy to see that this is the same as saying: for every formula ϕ(x, y) and a, b ∈ U_{y} with a ≡_{C} b, we have

ϕ(x, a) ∈ p ⟺ ϕ(x, b) ∈ p.

(This last observation uses, of course, homogeneity of U.) We call p invariant if it is Aut(U/C)-invariant for some C, and such C is called a base of p.

Note that if p is C-definable then it it is Aut(U/C)-invariant. However the converse does not hold and invariant and non-definable types will play an important role in the sequel.

Let p ∈ S_{x}(U) and q ∈ S_{y}(U) be invariant, and let C be a base for p. Let d be a realization of q|C and c a realization of p|Cd, and consider tp((c, d)/C) ∈ S_{x,y}(C). This type does not depend on the choice of c, d, and CLEARLY there is a type p ⊗ q ∈ S_{x,y}(U) such that p ⊗ q|C = tp((c, d)/C) for every such C, c, d.

Definition 1.5. We say that invariant types p ∈ S_{x}(U) and q ∈ S_{y}(U),say with a common base C, are domination equivalent if there is r ∈ S_{x,y}(C) extending both p|C and q|C such that p ∪ r ≡ q ∪ r.

The set of domination classes of invariant types from S(U) becomes a semigroup under ⊗ denoted by \overline{Inv}.

1.6. Forking an independence. We want to define precisely the notion of extending types to larger parameter sets without loss of information. There are several steps to do this.

Definition 1.6. (1) Let ϕ(x, y) be an L-formula, b ∈ U_{y} and C ⊆ U. We say that ϕ(x, b) divides over C if there are elements b_{1}, b_{2}, b_{3}, … of U_{y} such that tp(b_{i}/C) = tp(b/C) for every i ∈ ℕ and the set {ϕ(x, b_{i}) : i ∈ ℕ} of formulas is k-inconsistent for some k ∈ ℕ.

(2) A partial type π(x) over B ⊇ C forks over C if it implies a formula of the form ⋁_{j=0}^{n} ϕ_{j}(x), where each ϕ_{i}(x) divides over C.

(3) Let p = tp(a/B). Then we say a is independent from B over C if p does not fork over C. In this case we write a | ⌣_{C} B.

(4) For a set A we write A | ⌣_{C} B if for every finite tuple a from A we have a | ⌣_{C} B. If C ⊈ B, then a | ⌣_{C} B means A | ⌣_{C} B ∪ C.

A type p ∈ S(C) is called stationary if there is a unique non-forking extension of p in S(U). If C is algebraically closed, then each type in S(C) is stationary (this uses EI). If p is stationary and B ⊆ U, then we write p|B for the nonforking extension of p in S(B).

We say a, b ∈ U have the same strong type over C if they are in the same class of any C-definable finite equivalence relation. In this case we write stp(a/C) = stp(b/C). Note that this is the same as saying that tp(a/ acl(C)) = tp(b/ acl(C)) (Once again we use EI for this).

Let p ∈ S_{x}(B) be stationary. For every ϕ(x), let c_{ϕ} be the code of d_{p}(ϕ). Then the canonical base, Cb(p) of p is the definable closure of (c_{ϕ})_{ϕ}.

Properties. Let T be stable, C ⊆ B.

(1) q ∈ S(B) does not fork over C if and only if q is almost definable over C.

(2) If p = tp(a/B) has an Aut(U/ acl(C))-invariant extension in S(U), then a | ⌣_{C} B. (Stability of T is not needed for this.) If a | ⌣_{C} B, then p has a unique Aut(U/ acl(C))-invariant extension in S(U).

(3) (Existence) Let p ∈ S(C) and M contain C. Then p has a non-forking extension in S(M). If C is algebraically closed, then this extension is unique.

(4) (Finite character) p = tp(a/B) does not fork over C if and only if there is a formula ϕ in p such that ϕ does not fork over C. Otherwise said: a | ⌣_{C} B if and only if a | ⌣_{C} B_{0} for some finite B_{0} ⊆ B.

(5) (Transitivity) Let C ⊆ D ⊆ B. Then a | ⌣_{C} B iff a | ⌣_{C} D and a | ⌣_{D} B.

(6) (Symmetry) For every a, b ∈ U, a | ⌣_{C} b iff b | ⌣_{C} a.

(7) (Invariance) For every a, b ∈ U and every f ∈ Aut(U) we have a | ⌣_{C} b iff f(a) | ⌣_{f(C)} f(b).

(8) If p ∈ S(B) does not fork over C and has only finitely many realizations, then p|C has finitely many realizations.

(9) Let p ∈ S(B) be stationary. Then Cb(p) ⊆ dcl(B). Also Cb(p) ⊆ acl(C) iff p does not fork over C. Moreover Cb(p) ⊆ dcl(C) iff p does not fork over C and p|C is stationary.

1.7. An easier case: Totally transcendental theories. We say that T is totally transcendental if the CB-rank of each type is ordinal valued. This implies that T is ω-stable. If the language is countable, then the converse holds as well. The Morley rank, MR(p) of a type p is the CB-rank of p in U. We define the Morley rank of a formula ϕ to be the maximum of the Morley ranks of the types containing ϕ. If T is totally transcendental, then a | ⌣_{C} B iff MR(a/C) = MR(a/B).

For a definable set D, we can characterize the Morley rank as follows:

(1) MR(D) ≥ 0 iff D ≠ ∅.

(2) For an ordinal λ, MR(D) ≥ λ + 1 iff for every i ∈ ℕ, there is D_{i} ⊆ D such that MR(D_{i}) ≥ λ and D_{i} ∩ D_{j} = ∅ for i ≠ j.

(3) If λ is a limit ordinal, then MR(D) ≥ λ iff MR(D) ≥ γ for every ordinal γ < λ.

Let D be C-definable and p ∈ S_{x}(C). Then

MR(D) = max{MR(q) : q ∈ S_{x}(C), D(x) ∈ q}

and

MR(p) = min{MR(ϕ) : ϕ is an L(C)-formula contained in q}.

2. Stable part

Definition 2.1. For C ⊆ U, let L_{C} be the language with a sort s_{D} for every C-definable stable, stably embedded set D and a relation symbol of sort s_{D} for each C-definable relation on each D. The obvious structure attached to U in this language is denoted by St_{C}.

It is easy to see that St_{C} is stable.

For a subset A of U we write St_{C}(A) or A^{St} for St_{C} ∩ dcl(AC).

Condition (∗_{C}): For b ∈ St_{C}, D_{b} stable stably embedded Cb-definable, there is ψ(x) ∈ tp(b/C) such that for every b′, D_{b′} is stable stably embedded.

Note. This holds for every C if St_{B} is St_{∅}-internal for every B, and this latter property holds in ACVF.

Lemma 2.2. If (∗_{C}) holds then for every C ⊆ C′ ⊆ St_{C} every element of St_{C′} is interdefinable over C with an element of St_{C}.

Definition 2.3. Let A, B be arbitrary sets.

(1) A | ⌣_{C}^{d} B holds if A^{St} | ⌣_{C} B^{St} and tp(B/CA^{St}) ⊢ tp(B/CA).

(2) tp(A/C) is stably dominated if for every B ⊆ U if A^{St} | ⌣_{C} B^{St} then A | ⌣_{C}^{d} B.

Lemma 2.4. tp(B/CA^{St}) ⊢ tp(B/CA) ⇔ tp(A/CB^{St}) ⊢ tp(A/CB) ⇔ tp(B^{St}/CA^{St}) ∪ tp(B/C) ⊢ tp(B/CA).

It follows from this lemma that | ⌣_{C}^{d} is of finite character and is symmetric.

Proposition 2.5. For A, p = tp(A/C) is stably dominated if and only if for every definable set D the set {X ⊧ p : X^{St} | ⌣_{C}^{d} d^{St}} is either disjoint from D or is included in D where d := ⌜D⌝.

Proposition 2.6. Let M ⊧ T and C ⊆ M.

(1) If tp(A/C) is stably dominated, then tp(A/C) is extendable to an acl(C)-definable type over M such that whenever A′ realizes that type we have A′ | ⌣_{C}^{d} M.

(2) If M is sufficiently saturated |A|, |C| < |M| and tp(A/C) is stably dominated, then tp(A/ acl(C)) has a unique Aut(M/ acl(C))- invariant extension (over M).

From now on we say that an Aut(U/C)-invariant type p is stably dominated if p|C is.

Corollary 2.7. If tp(a/M) is stably dominated for every a and M ⊧ T, then T is stable.

Proposition 2.8. Let p be an Aut(U/C)-invariant stably dominated type. Then for every formula ϕ over U, ϕ ∈ p if and only if for some θ ∈ p|C, θ ∧ ¬ϕ divides over C.

We have yet another notion of independence that will supposedly help us to handle stationary types. From now on we write b instead of C.

Definition 2.9. Suppose that at least one of X, Y is a subset of St_{b}. Then we write X | ⌣_{b}^{s} Y if X^{St} | ⌣_{b} Y^{St} and (s1) for every a ∈ acl(b) ∩ dcl(Xb) we have tp(a/b) ⊢ tp(a/Y b).

We have the following characterizations of (s1).

Proposition 2.10. The following are equivalent to (s1): (s2) for every a ∈ dcl(Xb), tp(a/b) ⊢ tp(a/ dcl(Y b)) ∩ acl(b). (s3) for every a ∈ acl(b) ∩ dcl(Xb), tp(a/b) ⊢ tp(a/ dcl(Y b) ∩ acl(b)). If we also assume that X ⊆ St_{b} and X | ⌣_{b} Y^{St}, then (s1) is equivalent to the following (s4) for every a ∈ X, tp(a/b) is stationary as far as St_{b}(Y), i.e. if a′ ≡_{b} a and a′ | ⌣_{b} Y^{St}, then a′ ≡_{St_{b}(Y)} a.

This ‘stationary’ independence is symmetric and transitive.

Lemma 2.11. Let a ∈ St_{b}. Then

(1) a | ⌣_{acl(b)} Y^{St} (in St_{b}) if and only if tp(a/Y b) extends to a type over U that is Aut(U/ acl(b))-invariant.

(2) a | ⌣_{b}^{s} Y if and only if tp(a/Y b) is the unique extension of tp(a/b) which extends to an Aut(U/ acl(b))-invariant type.

(3) If p is Aut(U/b)-invariant and a ⊧ p|Y b, then a | ⌣_{b}^{s} Y.

Now we introduce a notion of canonical base suitable for this new independence.

Definition 2.12. Let Y ⊆ St_{b}. Then Cb(X/Y; b) is the smallest Z such that Z = dcl(Zb) ⊆ dcl(Y b) and X | ⌣_{Z}^{s} Y.

This canonical base always exists.

Definition 2.13. (1) A ∗-function is a tuple (f_{i})_{i∈I} is a tuple of definable functions. Such a ∗-function is said to be C-definable if each f_{i} is.

(2) A type p over C is stably dominated via a C-definable A ∗-function (f_{i})_{i} if for every realization a of p|C we have that f(a) ⊆ St_{b} and for every e if f(a) | ⌣_{acl(C)}^{s} e, then tp(e/Cf(a)) ⊧ tp(e/Ca).

Proposition 2.14. Let c be an enumeration of acl(C). Let f be a Cdefinable ∗-function from p = tp(a/C) to St_{C}. Then p is stably dominated via f if the following hold

(1) tp(a/Cf(a)) ⊧ tp(a/cf(a)),

(2) tp(a/ acl(C)) is stably dominated via f.

Remark. If tp(a/C) is stably dominated via a C-definable ∗-function f from tp(a/C) to St_{C}, then for any C′ with C ⊆ C′ ⊆ acl(C), tp(a/C′) is stably dominated via f.

Corollary 2.15. (1) If tp(a/ acl(C)) is stably dominated via a C-definable ∗-function f, then tp(a/C) is stably dominated via f′ where f′(a) is an enumeration of f(a) ∪ (acl(C) ∩ dcl(Ca)).

(2) tp(a/ acl(C)) is stably dominated via an acl(C)-definable ∗-function F, then tp(a/ acl(C)) is stably dominated via a C-definable ∗-function.

(3) tp(a/C) is stably dominated if and only if tp(a/ acl(C)) is stably dominated.

Definition 2.16. We say that A is stably dominated by B over C if for every D with B^{St} | ⌣_{C} D^{St} we have tp(D/CB) ⊢ tp(D/CA).

Proposition 2.17. (1) If tp(a/C) is stably dominated and St_{C}(a) = C,

then a ∈ dcl(C).

(2) Let B ⊆ St_{C} and let A be dominated by B over C. Then tp(A/C) is stably dominated.

(3) Suppose tp(A/C) is stably dominated and a′ ⊆ dcl(AC). Then tp(a′/C) is stably dominated.

Next we want to understand which extensions of stably dominated types are stably dominated.

Suppose that every extension of tp(a/C) is stably dominated and let b = St_{C}(a). Then b ⊇ C and hence tp(a/b) is stably dominated. Also St_{b}(a) = b. Thus by the first part of the previous proposition we get that a ∈ dcl(b) and that a ∈ St_{C}. So we cannot achieve increasing/decreasing in general. Increasing has the following answer.

Proposition 2.18. Let p be an Aut(U/ acl(C))-invariant type over U, and let B ⊇ C. If p|C is stably dominated, then so is p|B.

Proof. Let a be a realization of p|B. Let St_{B}(a) | ⌣_{B} St_{B}(D) and D ≡_{St_{B}(a)} D′. We need to show that D ≡_{Ba} D′.

By Lemma 2.11, we have that St_{C}(a) | ⌣_{C} St_{C}(B). Also St_{C}(a) | ⌣_{B} St_{C}(BD) in St_{B} and so St_{C}(a) | ⌣_{St_{C}(B)} St_{C}(BD) in St_{C}. Hence by transitivity

St_{C}(a) | ⌣_{C} St_{C}(BD) (in St_{C}).

Now by the stable domination of tp(a/C) we get that BD ≡_{Ca} BD′. So D ≡_{Ba} D′. □

The decreasing o parameters is quite harder and we shall not give a proof here. We first state an easier case where a stable domination is uniform in extensions, then we state a theorem generalizing it by removing the uniformity condition.

Proposition 2.19. Let g be a C-definable ∗-function and p, q Aut(U/C)- invariant types. Assume that for a realization b of q|C, p|Cb is stably dominated via g. Then p|C is stably dominated via g.

Theorem 2.20. Let p, q be Aut(U/C)-invariant. Suppose that p|Cb is stably dominated for every realization b of q|C. Then p|C is stably dominated.

We have the following nice consequence of the proof which seems to apply to the case of ACVF.

Corollary 2.21. Let Γ be a ∅ -definable stably embedded set with a ∅- definable linear ordering on it such that for every γ ∈ Γ and E ⊆ Γ, tp(γ/E) extends to an Aut(U/E)-invariant type. Also let f be a ∅-definable function. If tp(a/B∪(Γ∩Ba)) is stably dominated, then so is tp(f(a)/B∪(Γ∩Bf(a))).