-rank and meager types by

146 (1995)

M -rank and meager types

by

Ludomir N e w e l s k i (Wrocław)

Abstract. Assume T is superstable and small. Using the multiplicity rank M we find locally modular types in the same manner as U -rank considerations yield regular types.

We define local versions of M-rank, which also yield meager types.

0. Introduction. Throughout, T is superstable, small, and we work in C = C ^eq . In [Ne1] we defined the multiplicity rank M and proved that M has additivity properties similar to those of U -rank. In [Ne2] we defined the notion of meager regular type and proved that every such type is locally modular. It turns out that using M-rank we can produce locally modular regular types of prescribed M-rank. These types are either trivial or meager (the second case always holds when T has < 2 ^ℵ

countable models).

We use the following notation. If s(x) is a (partial) type over C, then [s]

denotes the class of (partial) types over C containing s. For p ∈ S(A), St(p) is the set of stationarizations of p over C, St A (p) = {r|acl(A) : r ∈ St(p)}.

For B ⊇ A, S _p (B) = S(B) ∩ [p] and S _p,nf (B) = {q ∈ S(B) ∩ [p] : q does not fork over A}. We regard strong types over A as types over acl(A). We define Tr A (s) (the trace of s on A) as the set of types r(x) ∈ S(acl(A)) consistent with s(x). Thus if p ∈ S(A) then Tr _A (p) = St _A (p). In general, Tr _A (s) is closed. Tr _A (a/B) abbreviates Tr _A (tp(a/B)). If a ∪ B (A) then sometimes ^| we use St A (a/B) to denote Tr A (a/B). Also, x A denotes the tuple of variables x indexed by elements of A. We will often tacitly use the following easy fact.

Fact 0.1. Assume A ⊆ B ⊆ C. Then either Tr A (a/C) is open in Tr _A (a/B) or Tr _A (a/C) is nowhere dense in Tr _A (a/B).

P r o o f. Suppose Tr _A (a/C) is not nowhere dense in Tr _A (a/B). Since Tr A (a/C) is closed, this means that for some ϕ = ϕ(x, c ⁰ ) with c ⁰ ∈ acl(A),

∅ 6= [ϕ] ∩ Tr _A (a/B) ⊆ Tr _A (a/C). Thus we can choose a ⁰ realizing ϕ(x, c ⁰ )

1991 Mathematics Subject Classification: Primary 03C45.

Research supported by KBN grant 2-1103-91-01.

[121]

with a ⁰ ≡ a (C). Now let r ∈ Tr A (a/C). So r = stp(a ⁰⁰ /A) for some a ⁰⁰ ≡ a (C). Choose c ⁰⁰ with a ⁰⁰ c ⁰⁰ ≡ a ⁰ c ⁰ (C). Clearly, r ∈ [ϕ(x, c ⁰⁰ )] ∩ Tr _A (a/B) ⊆ Tr _A (a/C). This shows that Tr _A (a/C) is open in Tr _A (a/B).

We define the multiplicity rank M on complete types p over finite sets A, with values in Ord ∪ {∞}, by the following conditions.

(1) M(p) ≥ 0.

(2) M(p) ≥ α + 1 iff for some finite B ⊇ A and q ∈ S p,nf (B), M(q) ≥ α and St(q) is nowhere dense in St(p).

(3) For limit δ, M(p) ≥ δ iff M(p) ≥ α for every α < δ.

M(a/A) abbreviates M(tp(a/A)).

Notice that we have proved in [Ne2] that if I(T, ℵ ₀ ) < 2 ^ℵ

then M(p) <

∞ for every p.

Suppose P is a closed subset of S(acl(A)) for some finite set A. We say that forking is meager on P if for every formula ψ forking over A, the set of types r ∈ P consistent with ψ is nowhere dense in P , that is, P ∩ Tr _A (ψ) is nowhere dense in P .

Suppose p is a stationary non-trivial regular type. Let ϕ be a p-simple formula over a finite set A such that the p-weight of ϕ is 1 and for each a ∈ ϕ(C), if w _p (a/A) > 0 then stp(a/A) is regular non-orthogonal to p. Let P ϕ = {r ∈ S(acl(A)) ∩ [ϕ] : w p (r) > 0}. Assume P ϕ is closed in S(acl(A)), and for each a ∈ ϕ(C) and b ⊆ C, if w _p (a/b) = 0 then for some formula ψ(x, y) over acl(A) true of a, b, whenever ψ(a ⁰ , b ⁰ ) holds then w _p (a ⁰ /b ⁰ ) = 0.

We call any ϕ with the above properties a p-formula (over A). Given p, a p-formula ϕ over some finite set A exists by [Hr-Sh]. We say that p is meager if forking is meager on P ϕ . In [Ne2] we show that this definition does not depend on the choice of ϕ, and also that if p is meager then p is locally modular. A complete regular type p is meager iff every stationarization of p is meager.

In the next lemma we collect the properties of M we shall use.

Lemma 0.2. (1) M(ab/A) ≤ M(a/Ab) ⊕ M(b/A).

(2) If p ∈ S(A) and A ⊆ B are finite, then M(p ⁰ ) = M(p) for some p ⁰ ∈ S _p,nf (B).

(3) If a ∪ b (A) then M(a/Ab) + M(b/A) ≤ M (ab/A). ^|

(4) Assume A ⊆ B and a ∪ B (A). If St(a/B) is open in St(a/A) then ^| M(a/B) = M(a/A). If M(a/B) = M(a/A) < ∞ then St(a/B) is open in St(a/A).

(5) (Symmetry) If a ∪ b (A) and St(a/Ab) is open in St(a/A) then ^| St(b/Aa) is open in St(b/A) and M(ab/A) = M(a/A) ⊕ M(b/A).

P r o o f. (1)–(3) are proved in [Ne1]. (4) and (5) are easy.

The next theorem is a kind of open mapping theorem. It is an easy conse- quence of Shelah’s finite equivalence relation theorem ([Sh]). However, it has non-trivial applications. In fact, the open mapping theorem of Lascar–Poizat ([Ba]) follows from the finite equivalence relation theorem in a similar way.

Theorem 0.3. Assume A ⊆ B ⊆ C and f : Tr _B (ab/C) → Tr _A (a/C) is restriction to acl(A) and to formulas with free variable x a . Then f is an open surjection.

P r o o f. Clearly f is a surjection. To show that f is open, let E(x ab , x ⁰ _ab ) ∈ FE(B). Let X = {[E(x _ab , a ⁰ b ⁰ )] ∩ Tr _B (ab/C) : a ⁰ b ⁰ ≡ ab (C)}. We see that X is finite, elements of X are closed, C-conjugate, and cover Tr _B (ab/C).

Hence Tr A (a/C) = S

{f (S) : S ∈ X} and f (S) is closed for each S ∈ X.

It follows that for some (hence every) S ∈ X, f (S) has non-empty interior in Tr A (a/C). It is easy to see that in fact f (S) is open in Tr A (a/C). This shows that f is open.

In [Ne1] I pointed out that M-rank may be defined in a way similar to the definition of Morley rank. It was puzzling whether there are local versions of M-rank, just as there are local versions of Morley rank. Local Morley ranks measure (type-)definable sets by means of instances of formulas from some (finite) set ∆. A local M-rank should measure St(p) with respect to some fixed (finite) set C ⊆ Dom(p). We define some local versions of M-rank in

§2. This must be done carefully, so that the resulting rank has the additivity properties of M. We show that these local versions of M-rank can also be used to produce meager types.

1. M-rank and meager types. In this section we show how to produce meager types with the help of M-rank. The next lemma improves Lemma 0.2(2).

Lemma 1.1. Assume A ⊆ B are finite.

(1) If Tr A (a/B) is open in St A (a/A) then M(a/B) ≥ M(a/A).

(2) If p(x) ∈ S(A) and s(x) is a (partial) type over B and Tr A (s) ∩ St _A (p) is open in St _A (p) then there is a p ⁰ ∈ S _p (B)∩[s] with M(p ⁰ ) ≥ M(p).

P r o o f. (1) First we prove the following.

(a) For every finite B ⁰ ⊇ A with B ⁰ ∪ a (A) there is a B ^{| 00} with B ⁰⁰ ≡ B ⁰ (Aa), B ⁰⁰ ∪ B (A), a ^| ∪ B ^{| 00} (B) and with Tr B

(a/BB ⁰⁰ ) open in St _B

(a/B ⁰⁰ ).

Given B ⁰ as in (a) choose B ^∗ ≡ B ⁰ (Aa) with B ^{∗ |} ∪ B (A). Let p(x) =

tp(a/A), p ⁰ (x) = tp(a/B) and q = tp(a/B ^∗ ). Since Tr _A (p ⁰ ) is open in St _A (p)

and stp(a/A) ∈ Tr A (q) ∩ Tr A (p ⁰ ) ⊆ St A (p), we see that Tr A (q) ∩ Tr A (p ⁰ )

is non-empty and open in Tr _A (q). Let s(x) be the type over BB ^∗ saying

that x ∪ B ^{| ∗} (B). Let f : St B

(q) → Tr A (q) be restriction and let U = f ⁻¹ (Tr _A (q) ∩ Tr _A (p ⁰ )). Then U is open in St _B

(q). Also we have

(b) U ⊆ Tr _B

(p ⁰ (x) ∪ s(x)).

Indeed, let r ∈ U . Then r|acl(A) ∈ Tr _A (p ⁰ ), hence we can choose c realizing p ⁰ ∪ r|acl(A), and without loss of generality, c ∪ B ^{| ∗} (B). Now, B ^{∗ |} ∪ B (A) implies c ∪ B ^{| ∗} (A), hence c realizes r ∪ p ⁰ (x) ∪ s(x).

Since T is small, there is a p ⁰⁰ ∈ S _q (BB ^∗ )∩[p ⁰ ∪s] such that Tr _B

(p ⁰⁰ )∩U is open in St B

(q). By Fact 0.1 we see that Tr B

(p ⁰⁰ ) is open in St B

(q). Let a ⁰⁰ realize p ⁰⁰ . We see that a ⁰⁰ ≡ a (B), a ^{00 |} ∪ B ^∗ (B), Tr _B

(a ⁰⁰ /BB ^∗ ) is open in St _B

(a ⁰⁰ /B ^∗ ) and a ⁰⁰ B ^∗ ≡ aB ⁰ . Choose B ⁰⁰ with a ⁰⁰ B ^∗ ≡ aB ⁰⁰ (B). Clearly, B ⁰⁰ satisfies our demands in (a).

Now we prove by induction on α that M(a/A) ≥ α implies M(a/B) ≥ α.

We check only the successor step. Suppose M(a/A) ≥ α + 1. So there is a finite B ⁰ ⊇ A with a ∪ B ^{| 0} (A), M(a/B ⁰ ) ≥ α and St(a/B ⁰ ) nowhere dense in St(a/A). By (a), without loss of generality, a ∪ B ^{| 0} (B) and Tr _B

(a/BB ⁰ ) is open in St _B

(a/B ⁰ ). Thus by the inductive hypothesis, M(a/BB ⁰ ) ≥ α.

To finish, we must show

(c) St(a/BB ⁰ ) is nowhere dense in St(a/B).

In the following diagram all the maps α _i are restrictions.

St BB

(a/BB ⁰ ) St B (a/B)

St _B

(a/B ⁰ ) St _A (a/A)

α

//

α

²²

α

²² α

//

This diagram commutes. Also, Tr _A (a/B) = Rng α ₂ , hence by Theo- rem 0.3 and our assumptions, α 2 is open. By (a) and Theorem 0.3 also α 1

is open. St _BB

(a/BB ⁰ ) and St _B (a/B) are naturally homeomorphic to St(a/BB ⁰ ) and St(a/B) respectively. So to prove (c) it suffices to show that α 0 (St BB

(a/BB ⁰ )) is nowhere dense in St B (a/B).

If not, then α ₀ (St _BB

(a/BB ⁰ )) is open in St _B (a/B), hence also α ₂ α ₀ (St _BB

(a/BB ⁰ )) is open in St _A (a/A). We have

α 2 α 0 (St BB

(a/BB ⁰ )) = α 3 α 1 (St BB

(a/BB ⁰ )) ⊆ α 3 (St B

(a/B ⁰ )).

Since St(a/B ⁰ ) is nowhere dense in St(a/A), we see that α ₃ (St _B

(a/B ⁰ )) is nowhere dense in St _A (a/A). Hence the more so α ₂ α ₀ (St _BB

(a/BB ⁰ )) is no- where dense in St A (a/A), a contradiction.

(2) By the smallness of T , S p (B) ∩ [s] is countable. The sets Tr A (p ⁰ ),

p ⁰ ∈ S _p (B) ∩ [s], cover Tr _A (s) ∩ St _A (p). Hence for some p ⁰ ∈ S _p (B) ∩ [s],

Tr A (p ⁰ ) is not nowhere dense in St A (p). By Fact 0.1, Tr A (p ⁰ ) is open in

St _A (p). Hence (2) follows from (1).

The range of M is of the form I ∪ {∞} or I, for some proper initial segment I of ω ₁ . Let α = ω ^β for some β ∈ Ord, or α = β = ∞. Assume α is in the range of M. Let γ _β be the minimal γ such that M(p) ≥ α for some type p of ∞-rank γ. Notice that if β ⁰ < β and α ⁰ = ω ^β

then γ β

≤ γ β .

Theorem 1.2. Assume A is finite, q ∈ S(A), R _∞ (q) = γ _β and M(q) ≥ α. Then q is regular and locally modular. If q is non-trivial then q is meager and for some isolated q ⁰ ∈ S(A), R _∞ (q ⁰ ) = γ _β , M(q ⁰ ) ≥ α and q ⁰ is non- orthogonal to q.

We begin the proof of Theorem 1.2 with two lemmas.

Lemma 1.3. q is regular and orthogonal to every type with ∞-rank < γ _β . P r o o f. Suppose not. Then for some finite B ⊇ A and r ∈ S(B) with R _∞ (r) < γ _β , r is not almost orthogonal to some nf extension q ⁰ of q over B. Choose ψ ∈ r with R _∞ (ψ) < γ _β . Since r ^a 6⊥ q ⁰ , for some a, b realizing r, q ⁰ respectively, a ∪ b (B). This is witnessed by a formula δ(x, y) over B, ^- true of a, b. Since forking means decreasing some local rank, without loss of generality we have

(a) if δ(a ⁰ , b ⁰ ) holds and b ⁰ satisfies a nf extension of q over B, then ψ(a ⁰ ) and a ^{0 -} ∪ b ⁰ (B).

Let δ ⁰ (y) = ∃x δ(x, y). So δ ⁰ (y) does not fork over A. Let s(y) be the type over B saying:

“δ ⁰ (y) and q(y) and tp(y/B) does not fork over A.”

We see that Tr _A (s) is open in St _A (q). By Lemma 1.1(2) for some q ⁰⁰ ∈ S q,nf (B) ∩ [δ ⁰ ], we have M(q ⁰⁰ ) ≥ α. Let b ⁰⁰ realize q ⁰⁰ , and choose a ⁰⁰ with δ(a ⁰⁰ , b ⁰⁰ ). By (a), R _∞ (b ⁰⁰ /Ba ⁰⁰ ) < γ _β and R _∞ (a ⁰⁰ /B) < γ _β (as R _∞ (ψ) <

γ _β ). By the choice of γ _β , we have M(a ⁰⁰ /B), M(b ⁰⁰ /Ba ⁰⁰ ) < α. By Lemma 0.2 we have M(b ⁰⁰ /B) ≤ M(a ⁰⁰ b ⁰⁰ /B) ≤ M(a ⁰⁰ /B) ⊕ M(b ⁰⁰ /Ba ⁰⁰ ) < α, a contradiction.

Choose p ⁰ ∈ S(A) non-orthogonal to q, with R _∞ (p ⁰ ) = γ _β , M(p ⁰ ) ≥ α and Cantor–Bendixson rank CB(p ⁰ ) minimal possible. Choose ϕ ∈ p ⁰ with R ∞ (ϕ) = γ β , CB(p ⁰ ) = CB(ϕ) and CB-multiplicity of ϕ equal to 1. By Lemma 1.3, p ⁰ is regular.

Lemma 1.4. p ⁰ is orthogonal to every type in S(A)∩[ϕ]\{p ⁰ } and forking is meager on St _A (p ⁰ ).

P r o o f. Suppose r ∈ S(A) ∩ [ϕ] \ {p ⁰ } is non-orthogonal to p ⁰ . Then

for some finite B ⊇ A and a, b realizing over B nf extensions of r, p ⁰

respectively, we have a ∪ b (B). As in Lemma 1.3, for some b ^{- 0} realizing p ⁰ ,

we have b ^{0 |} ∪ B (A), M(b ⁰ /B) ≥ α and for some a ⁰ ∈ ϕ(C)\p ⁰ (C), a ^{0 -} ∪ b ⁰ (B).

If M(a ⁰ /A) ≥ α and a ^{0 |} ∪ B (A), then by the choice of p ⁰ and ϕ, tp(a ⁰ /A) is orthogonal to q, hence to p ⁰ . Thus also tp(a ⁰ /B) is orthogonal to p ⁰ , a contradiction. If a ^{0 -} ∪ B (A) then R _∞ (a ⁰ /B) < γ _β , so again (by Lemma 1.3 applied to q := p ⁰ ), tp(a ⁰ /B) is orthogonal to p ⁰ .

Hence we get M(a ⁰ /A) < α and a ^{0 |} ∪ B (A). Thus M(a ⁰ /B) < α and M(b ⁰ /Ba ⁰ ) < α, because R _∞ (b ⁰ /Ba ⁰ ) < γ _β . Hence M(b ⁰ /B) ≤ M(a ⁰ b ⁰ /B)

≤ M(a ⁰ /B) ⊕ M(b ⁰ /Ba ⁰ ) < α, a contradiction.

To prove the second clause, suppose forking is not meager on St _A (p ⁰ ).

Then for some forking formula δ(x) over a finite set B ⊇ A, Tr _A (δ) ∩ St _A (p ⁰ ) is open in St A (p ⁰ ). By Lemma 1.1, for some p ⁰⁰ ∈ S p

(B) ∩ [δ], we have M(p ⁰⁰ ) ≥ M(p ⁰ ) ≥ α. Since p ⁰⁰ forks over B, R _∞ (p ⁰⁰ ) < γ _β . Hence M(p ⁰⁰ ) <

α, a contradiction.

Now we can conclude the proof of Theorem 1.2. Suppose q is non-trivial.

So p ⁰ is non-trivial. Let p be any stationarization of p ⁰ . We shall prove that ϕ satisfies the conditions from the definition of a meager type, and forking is meager on P _ϕ , hence that p is meager. By Lemmas 1.3 and 1.4, every type in S(A) ∩ [ϕ] \ {p ⁰ } is hereditarily orthogonal to p. Also, p is orthogonal to any forking extension of p ⁰ , and every type in St A (p ⁰ ) is regular. Hence ϕ is p-simple of p-weight 1. By the claim in the proof of Lemma 1.6(2) in [Ne2], the set of r ∈ St _A (p ⁰ ) such that r is non-orthogonal to p is clopen in St _A (p ⁰ ).

Thus P ϕ is clopen in St A (p ⁰ ). Since all stationarizations of p ⁰ have the same local ranks, as in Lemma 1.3 we see that the p-weight 0 is definable on ϕ(C).

Thus ϕ is a p-formula. By Lemma 1.4, forking is meager on P ϕ , hence p is meager.

Next notice that if some type in P _ϕ is modular, then every type in P _ϕ is, since they are all stationarizations of the complete type p ⁰ . But then if c realizes one of them, then they all have forking extensions over Ac, contradicting meagerness. It follows that every type in P _ϕ is not modular.

By [Ne2, Corollary 1.8], P ϕ is open in S(acl(A)). Hence also St A (p ⁰ ) is open in S(acl(A)). This means that p ⁰ is isolated, proving the theorem.

Notice that by Lemma 1.4 and [Ne2, Lemma 1.6(1)], if I(T, ℵ ₀ ) < 2 ^ℵ

then in the proof of Theorem 1.2 the type p ⁰ is non-trivial. It follows that if I(T, ℵ ₀ ) < 2 ^ℵ

then the type q in Theorem 1.2 is non-trivial and meager.

Lemma 1.5. Let q ∈ S(A) be as in Theorem 1.2. Assume A ⁰ is finite, q ⁰ ∈ S(A ⁰ ) is isolated, R _∞ (q ⁰ ) = γ _β and q ⁰ is non-orthogonal to q. Then M(q ⁰ ) ≥ α and M(q) ≤ M(q ⁰ ) ⊕ α ⁰ for some α ⁰ < α.

P r o o f. First we prove that M(q) ≤ M(q ⁰ ) ⊕ α ⁰ for some α ⁰ < α. For

some finite B ⊇ A ∪ A ⁰ there are a, b realizing over B nf extensions of q, q ⁰

respectively, with a ∪ b (B). As in Lemma 1.3, by Lemma 0.2, changing a ^-

somewhat we can assume that M(a/B) ≥ M(q). This change affects b, but

since q ⁰ is isolated, b still realizes q ⁰ , and since q is orthogonal to any type with ∞-rank < γ _β , we have b ∪ B (A ^{| 0} ). Hence M(b/B) ≤ M(q ⁰ ). Also, a ∪ b (B) yields M(a/Bb) < α. Let α ^{- 0} = M(a/Ba). Hence we get

M(q) ≤ M(a/B) ≤ M(ab/B) ≤ M(q ⁰ ) ⊕ M(a/Bb) = M(q ⁰ ) ⊕ α ⁰ . If M(q ⁰ ) < α, then M(q) ≤ M(q ⁰ ) ⊕ α ⁰ < α, a contradiction.

Suppose M(p) = n 1 ω ^β

⊕ . . . ⊕ n k ω ^β

, where n i are finite, n 1 6= 0, and β ₁ > . . . > β _t ≥ β > β _t+1 > . . . > β _k = 0. We define M _β (p) as n ₁ ω ^β

⊕ . . . ⊕ n _t ω ^β

. If M(p) = ∞ then M _β (p) = ∞. If β = ∞ then M β (p) = 0 if M(p) < ∞ and M β (p) = ∞ otherwise.

Given β and α = ω ^β , for isolated q over a finite set such that R ∞ (q) = γ β

and M(q) ≥ α, M _β (q) is determined by the non-orthogonality class of q.

The next corollary clarifies the orthogonality relations between the locally modular types q obtained in Theorem 1.2.

Corollary 1.6. Assume α = ω ^β , α ⁰ = ω ^β

are in the range of M, A, A ⁰ are finite and q ∈ S(A), q ⁰ ∈ S(A ⁰ ) are isolated with R _∞ (q) = γ _β , R _∞ (q ⁰ ) = γ _β

, M(q) ≥ α and M(q ⁰ ) ≥ α ⁰ . If q, q ⁰ are non-orthogonal then γ β = γ β

and M β (q) = M β

(q) = M β (q ⁰ ) = M β

(q ⁰ ).

In the case when T has < 2 ^ℵ

countable models, we can strengthen Theorem 1.2.

Theorem 1.7. Assume I(T, ℵ 0 ) < 2 ^ℵ

. For q and A as in Theorem 1.2, we have M _β (q) = α.

P r o o f. By Theorem 1.2 there is an isolated p over A non-orthogonal to q, with R _∞ (p) = γ _β and M(p) ≥ α. By the claim in [Ne2, Lemma 1.6], extending A by an element of acl(A), we can assume that all types in St A (p) are non-orthogonal. Let ϕ isolate p over A. By the proof of Theorem 1.2, ϕ and P _ϕ = St _A (p) witness that p is meager. By Lemma 1.5 it suffices to prove that M(p) < α ⊕ α.

We shall use Corollary 2.15 from [Ne2], which says that in our case, for every finite B extending A, for every a realizing p with a ∪ B (A), exactly ^| one of the following holds.

(a) For some b realizing p with b ∪ B (A), tp(b/B) is isolated and ^| a ∪ b (A). ^-

(b) There are finitely many a 0 , . . . , a n realizing r = tp(a/B) such that for every b realizing r, for some i ≤ n and b ⁰ ≡

s b (A), we have b ^{0 -} ∪ a _i (A).

Suppose for a contradiction that M(p) ≥ α ⊕ α. Let a realize p. We can

find a finite set B ⊇ A with a ∪ B (A) and M(a/B) = α. The proof splits ^|

into two cases, depending on which of conditions (a), (b) holds.

C a s e 1: (a) holds. So choose b realizing p with tp(b/B) isolated, b ∪ ^| B (A) and a ∪ b (A). It is easy to see that M(b/B) = M(p), hence M(b/B) ^-

≥ α ⊕ α. Since a ∪ b (B), by the choice of γ ^- _β we have M(b/Ba) < α.

Hence α ⊕ α ≤ M(b/B) ≤ M(ab/B) ≤ M(a/B) ⊕ M(b/Ba) < α ⊕ α, a contradiction.

C a s e 2: (b) holds. Extending B by an element from acl(B), we can assume that n = 0, that is, for every b realizing r = tp(a/B), for some b ⁰ ≡

s b (A), we have b ^{0 -} ∪ a (A). In other words, every s ∈ St B (r) has a forking extension over Ba. Since T is countable, there is a single formula ψ(x) over Ba, forking over B, such that the set of s ∈ St B (r) consistent with ψ has non-empty interior in St _B (r). Without loss of generality, R _∞ (ψ) < γ _β . Since M(r) = α, by Lemma 1.1 there is a type r ⁰ ∈ S _r (Ba) ∩ [ψ] with M(r ⁰ ) ≥ α.

Since R ∞ (r ⁰ ) < γ β , this contradicts the choice of γ β . Corollary 1.8. Assume I(T, ℵ ₀ ) < 2 ^ℵ

.

(1) Assume β < β ⁰ , α = ω ^β , α ⁰ = ω ^β

and α ⁰ is in the range of M.

Then γ β

> γ β + 2. Also, γ 0 ≥ 1 and γ 1 ≥ ω.

(2) Assume A is finite, p ∈ S(A) and M(p) ≥ ω ^β . Then R _∞ (p) ≥ 1+2β.

P r o o f. (1) Without loss of generality, β ⁰ = β + 1. Choose a finite set A and an isolated p ∈ S(A) with R _∞ (p) = γ _β

and M(p) ≥ α ⁰ . So p is non- trivial, regular and meager. Without loss of generality, all stationarizations of p are non-orthogonal, and the formula ϕ isolating p witnesses that p is meager. Again we apply Corollary 2.15 from [Ne2] (see the proof of Theorem 1.7 above). Let a realize p. Since β < β ⁰ , for some finite B ⊇ A with a ∪ B (A) we have M(a/B) = α ⊕ α. Since α ⊕ α < α ^{| 0} , as in Theorem 1.7 we see that in Corollary 2.15, case (b) holds, and (without loss of generality) n = 0 there. As in Theorem 1.7, we find a forking extension r ⁰ of tp(a/B) over Ba with M(r ⁰ ) ≥ α ⊕ α. In particular, R _∞ (r ⁰ ) ≥ γ _β . In fact, R _∞ (r ⁰ ) ≥ γ _β + 1. If not, then R _∞ (r ⁰ ) = γ _β and M(r ⁰ ) ≥ α ⊕ α, which contradicts Theorem 1.7. Since r ⁰ forks over B, we get γ β

≥ R ∞ (r ⁰ ) + 1. Altogether we get γ _β

≥ γ _β + 2.

We have γ ₀ ≥ 1 trivially, and γ ₁ ≥ ω because by [Ne1], every type with finite U -rank has finite M-rank.

(2) Notice that R _∞ (p) ≥ γ _β , and by (1), γ _β ≥ 1 + 2β.

Corollary 1.8 shows that there is a bound on M-rank, depending on the

∞-rank. Can we improve the bound obtained there? Assuming I(T, ℵ ₀ ) <

2 ^ℵ

, is it true that M(p) ≤ U (p)? This is proved in [Ne1] for types with finite U -rank.

Corollary 1.9. Assume I(T, ℵ 0 ) < 2 ^ℵ

, β < β ⁰ , α = ω ^β , α ⁰ = ω ^β

and

α ⁰ is in the range of M. Suppose A and A ⁰ are finite, p ∈ S(A), p ⁰ ∈ S(A ⁰ ),

R ∞ (p) = γ β , R ∞ (p ⁰ ) = γ β

, M(p) ≥ α and M(p ⁰ ) ≥ α ⁰ . Then p and p ⁰ are meager , regular and orthogonal.

To illustrate our ideas we shall give an example of a superstable theory with a meager type p of M-rank ω.

Example 1.10. Let V = ^ω×ω 2 and let + be pointwise addition on V , modulo 2. We think of V as the product Q

n {n}×ω 2. We define certain subgroups of V . For n > 0 let

P n = {f ∈ V : f |n × n ≡ 0} and G n = {f ∈ P n : f |(ω \ n) × ω ≡ 0}.

Clearly, G _n and P _n are subgroups of (V, +) and G _n ∩ P _n+1 ⊆ G _n+1 . More- over, for each n > 1, G n /G n−1 is naturally isomorphic to ( {n−1}×(ω\n) 2, +), and G ₁ ∩P _n is isomorphic to ^{0}×(ω\n) 2. Thus the mapping g : {n−1}×(ω\n) 2

→ ^{0}×(ω\n) 2 given by g(f )((0, k)) = f ((n − 1, k)) induces a group isomor- phism f n : G n /G n−1 → G 1 ∩ P n .

Let M = (V ; +, P n , G n , f n+1 ) 0<n<ω and T = Th(M ). Then T is a su- perstable 2-dimensional 1-based theory. Up to non-orthogonality, there are two regular types in T : p ₀ , the principal generic type of M , and p ₁ , the principal generic type of G 1 . Both types are meager, p 1 is weakly minimal, R _∞ (p ₀ ) = U (p ₀ ) = ω. The functions f _n , n > 1, ensure that the weakly minimal groups G _n /G _n−1 , n > 1, are non-orthogonal to G ₁ .

Let p ∈ S(∅) be the type generated by ¬P ₁ (x). Thus p is regular non- orthogonal to p 0 , R ∞ (p) = U (p) = ω and St _∅ (p) = Str(∅) ∩ [¬P 1 (x)]. Let a realize p and let ϕ _n (x, a) be G _n (x − a), n > 0. We see that Tr _∅ (ϕ _n (x, a)) ⊆ St _∅ (p) and Tr _∅ (ϕ _n (x, a)) ∩ [P _n+1 (x − a)] is nowhere dense in Tr _∅ (ϕ _n+1 (x, a)) (the formula P n+1 (x − a) is almost over ∅). This is essentially because in V with the product topology, G _n ∩ P _n+1 is nowhere dense in G _n+1 .

Let p n ∈ S(a) be one of the countably many non-forking extensions of p such that St _∅ (p _n ) is an open subset of Tr _∅ (ϕ _n (x, a)). Of course every stationarization of p n is modular. We see that M(p n ) = n and M(p) = ω.

Also γ ₀ = 1 and γ ₁ = ω here.

Notice that γ 0 is the minimal γ such that some type without Morley

rank has ∞-rank γ. By Theorem 1.7, if I(T, ℵ ₀ ) < 2 ^ℵ

then for every type

p of ∞-rank ≤ γ ₀ , M(p) ≤ 1, and if M(p) = 1 then p is meager (hence

locally modular). This is related to Proposition 1.11 in [Pi]. Pillay proves

there (under the few models assumption) that if G is a locally modular

superstable group of rank γ ₀ , without Morley rank, then for every finite

A and a ∈ G, if tp(a/A) is non-isolated then it has finite multiplicity. In

general we cannot claim that much. The following example shows that it can

happen that a type p ∈ S(∅) has ∞-rank γ 0 , M(p) = 1 and p is non-isolated

(in this case necessarily γ ₀ ≥ ω).

Example 1.11. Let V = ^ω×ω 2 and + be as in Example 1.10. We define some subgroups of V . For n > 0 let

P n = {f ∈ V : f |{0} × n ≡ 0},

G _n = {f ∈ V : f |{0} × ω ≡ 0 and f |(ω \ (n + 1)) × ω ≡ 0}, H n = {f ∈ V : f |(ω \ n) × 1 ≡ 0}.

Thus G _n ⊆ G _n+1 ⊆ T

n P _n , H _n ⊆ H _n+1 , [G _n+1 : G _n ] is infinite and [H n+1 : H n ] = 2. Consider the structures M 0 = (V ; +, P n , G n ) n>0 and M ₁ = (V ; +, H _n ) _n>0 and their theories T ₀ , T ₁ respectively. They are 1- based. T 0 is superstable, with ∞-rank ω and γ 0 = ω, while T 1 has Morley rank 2 and ∞-rank 2. T 0 and T 1 have countably many countable models.

Now we do not have to include into M ₀ the functions f _n (as in Example 1.10) since G n /G n−1 are strongly minimal and ω-categorical.

We shall define a structure M with universe V ⁰ = V × V . Let E be the equivalence relation on V ⁰ defined by (a, b)E(a ⁰ , b ⁰ ) iff b = b ⁰ . We can naturally identify V ⁰ /E with V . Let Q n = {(a, b) ∈ V ⁰ : a ∈ H n }. For each a ∈ V ⁰ , a/E can be endowed with a structure isomorphic to M ₁ , uniformly in a. However, we do not want to have in M a full group structure on a/E. Let R be the 4-ary relation on V ⁰ defined by: R((a 0 , b 0 ), (a 1 , b 1 ), (a 2 , b 2 ), (a 3 , b 3 )) iff b ₀ = b ₁ , b ₂ = b ₃ and a ₁ − a ₀ = a ₃ − a ₂ .

We see that R gives an affine group structure on each a/E, and addi- tionally a group isomorphism between any two E-classes, after fixing single points in them.

Finally, let M = (V ⁰ ; E, Q _n , R; +, P _n , G _n ) _n>0 , where +, P _n , G _n are de- fined on V ⁰ /E via the identification of V ⁰ /E with V . Let T = Th(M ). Then T is a superstable 1-based theory with countably many countable models.

Here γ 0 = ω. Let p(x) ∈ S(∅) be the type generated by {¬Q n (x), n <

ω} ∪ {¬P ₁ (x/E)}. Clearly p is non-isolated, R _∞ (p) = ω and M(p) = 1.

Also, p is regular and meager. Notice, however, that since M/E is bi- interpretable with M 0 , M/E is a regular group of ∞-rank ω, non-orthogonal to p, with meager generics. If a realizes p, then a/E is generic in M/E and tp((a/E)/∅) is isolated, as promised by Proposition 1.11 in [Pi]. More- over, ϕ(x) = (x = x) is a p-formula in M , and isolated types are dense in P _ϕ = Str(∅)∩ T

n>0 [¬G _n (x/E)]. This is a good illustration of the phenomena encountered in [Ne2].

2. Localization. There are several motivations for this section. One is that unfortunately possibly not all meager types (even up to non-ortho- gonality) are obtained by the minimization process from §1. Another is that if, for example, S(acl(∅)) is uncountable (and hence M(q) ≥ 1 for some q ∈ S(∅); remember that T is small), then the minimization procedure from

§1 yields some meager (or trivial) p ∈ S(A) with M(p) ≥ 1. However, p may

have nothing in common with the rich topological structure of S(acl(∅)), that is, possibly Tr _∅ (p) is finite. My feeling was that we should be able to find a type p as above such that the topological structure of St(p) is related to S(acl(∅)). In other words, that the complicated structure of S(acl(∅)) really forces the existence of a locally modular type. In this section we will show that this is the case. To do this we need some local versions of M-rank, which we define below. These local versions have many additivity properties of M. Another reason for trying to define local M-ranks is an attempt to find the most basic impact that the structure of acl(C) (for a given finite C ⊆ C) has on the structure of the whole theory. So local M-ranks are in- tended to play the same role in the “theory of multiplicities” as local Morley ranks in the theory of forking. I leave it to the reader to judge how much I succeeded in this attempt.

In this section let C be a fixed finite subset of C. We define local rank M _C on types p ∈ S(A) for finite sets A ⊇ C, as follows. Suppose p = tp(a/A).

We define M _C (p) = M _C (a/A) by the following conditions:

(1) M _C (a/A) ≥ 0.

(2) M C (a/A) ≥ α + 1 iff for some finite B ⊇ A with a ∪ B (A), ^| M _C (a/B) ≥ α and Tr _C (a/B) is nowhere dense in Tr _C (a/A).

(3) M _C (a/A) ≥ δ for limit δ iff M _C (a/A) ≥ α for each α < δ.

From now on we usually assume that C is contained in the sets of pa- rameters A, B we consider. M C has the following properties.

Lemma 2.1. Assume C ⊆ A ⊆ B, A, B are finite.

(1) M C (a/A) ≤ M(a/A).

(2) If a ∪ A (C) then M ^| _C (a/A) = M(a/A).

(3) If Tr C (b/A) = Tr C (a/A) then M C (b/A) = M C (a/A).

(4) For each a there is a b with b ∪ A (C) such that Tr ^| _C (b/A) = Tr _C (a/A), hence M(b/A) = M _C (b/A) = M _C (a/A).

(5) If a ∪ B (A) and M ^| C (a/A) < ∞ then M C (a/B) = M C (a/A) iff Tr _C (a/B) is open in Tr _C (a/A).

(6) For every a there is a b with b ≡ a (A), b ∪ B (A) and M ^| C (b/B) = M _C (a/A).

P r o o f. (1) We prove by induction on α that M _C (a/A) ≥ α implies M(a/A) ≥ α. Let us check the successor step. Suppose M _C (a/A) ≥ α + 1.

Choose a finite B ⊇ A with a ∪ B (A), M ^| C (a/B) ≥ α and Tr C (a/B) nowhere dense in Tr _C (a/A). Let f _B : St _B (a/B) → Tr _C (a/B), f _A : St _A (a/A)

→ Tr C (a/A) and g : St B (a/B) → Tr A (a/B) ⊆ St A (a/A) be restrictions. By

Theorem 0.3, f _A , f _B and g are open. Also, f _B = g ◦ f _A . It follows that

Tr A (a/B) is nowhere dense in St A (a/A). Otherwise Tr A (a/B) is open in St _A (a/A), hence Tr _C (a/B) = g(Tr _A (a/B)) is open in Tr _C (a/A), a contra- diction.

As Tr A (a/B) is nowhere dense in St A (a/A), and St A (a/A) and St B (a/B) are canonically homeomorphic to St(a/A) and St(a/B) respectively, we see that St(a/B) is nowhere dense in St(a/A). By the inductive hypothesis, M(a/B) ≥ α, hence by the definition of M, M(a/A) ≥ α + 1. The proofs of the other parts of the lemma are equally easy, so we omit them.

Unfortunately, M _C does not have the additivity properties of M from Lemma 0.2. One of the reasons for that is as follows. Suppose b ⊆ A. Then St(ab/A) is homeomorphic to St(a/A), hence M(ab/A) = M(a/A). Let p(x ab ) = tp(ab/A). Consider Tr C (a/A) = Tr C (p|x a ) and Tr C (ab/A) = Tr _C (p). Let f : Tr _C (ab/A) → Tr _C (a/A) be restriction to formulas with free variable x _a . Then f is continuous and open; however, f need not to be injective. In particular, it could happen that Tr C (a/A) is finite and Tr _C (ab/A) is infinite. Then M _C (a/A) = 0 and M _C (ab/A) ≥ 1, showing that M _C (ab/A) ≤ M _C (a/Ab) ⊕ M _C (b/A) fails.

It turns out that the situation described above is the only obstacle pre- venting M _C from having the additivity properties of M. We correct this by defining an eventual version of M _C , denoted by M ^e _C . It is defined just like M C , but with (2) replaced by the following condition:

(2 ⁰ ) M ^e _C (a/A) ≥ α + 1 iff for some finite B, D extending A with B ⊆ D and a ∪ D (A), we have M ^{| e} _C (a/D) ≥ α and Tr C (aB/D) is nowhere dense in Tr _C (aB/B).

Notice that M ^e _C is an eventual version of M C just as the Morley rank is an “eventual” version of the Cantor–Bendixson rank. Also, in (2 ⁰ ) we can additionally assume that Tr C (aA/B) is open in Tr C (aA/A) (by Lemma 2.2(3) below and Theorem 0.3). We can give another, equivalent definition of M ^e _C . For finite B ⊆ C define an equivalence relation ^∼ B on S(C) by r ^∼ B r ⁰ iff r|acl(C)B = r ⁰ |acl(C)B. For X ⊆ S(C) let X/ ^∼ B be {r/ ^∼ B : r ∈ X}. We can equivalently replace (2 ⁰ ) by the following condition:

(2 ⁰⁰ ) M ^e _C (a/A) ≥ α + 1 iff for some finite B, D extending A, with B ⊆ D and a ∪ D (A), we have M ^{| e} _C (a/D) ≥ α and S

St(a/D)/ ^∼ B is nowhere dense in St(a/B).

We shall not use (2 ⁰⁰ ), however. If in (2 ⁰⁰ ) we required only that

S St(a/D)/ ^∼ B is nowhere dense in St(a/A), we would end up with M ^e _C =

M. The role of B in (2 ⁰ ) and (2 ⁰⁰ ) is puzzling. It would be natural to re-

quire in (2 ⁰ ), (2 ⁰⁰ ) additionally that St(a/B) is open in St(a/A). If we did

so, however, we might lose the property of M ^e _C that M ^e _C (p) ≥ M ^e _C (p ⁰ ) for

any nf extension p ⁰ of p (see Lemma 2.2(3) below).

Lemma 2.2. (1) M C (a/A) ≤ M ^e _C (a/A) ≤ M(a/A).

(2) If a ∪ A (C) then M ^| C (a/A) = M ^e _C (a/A) = M(a/A).

(3) If A ⊆ B and a ∪ B (A) then M ^{| e} _C (a/B) ≤ M ^e _C (a/A).

(4) If A ⊆ B, a ∪ B (A) and St(a/B) is open in St(a/A) (which implies ^| M(a/B) = M(a/A)), then M ^e _C (a/B) = M ^e _C (a/A).

(5) If A ⊆ B, then for every a there is an a ⁰ ≡ a (A) with a ^{0 |} ∪ B (A) and M ^e _C (a ⁰ /B) = M ^e _C (a ⁰ /A). Also, if additionally B ⊆ acl(A) then M ^e _C (a/A) = M ^e _C (a/B).

(6) In (2 ⁰ ) we can additionally assume that D \ A ⊆ C ₌ (C ₌ is the “real”

sort of C ^eq ).

P r o o f. (1) To show M C (a/A) ≤ M ^e _C (a/A), we prove by induction on α that M _C (a/A) ≥ α implies M ^e _C (a/A) ≥ α. Let us check the successor step.

Assume M C (a/A) ≥ α + 1. Choose a finite B ⊇ A with a ∪ B (A) such that ^| M _C (a/B) ≥ α and Tr _C (a/B) is nowhere dense in Tr _C (a/A). It follows that Tr _C (aA/B) is nowhere dense in Tr _C (aA/A). By the inductive hypothesis, M ^e _C (a/B) ≥ α, and by the definition of M ^e _C , M ^e _C (a/A) ≥ α + 1. Similarly we prove M ^e _C (a/A) ≤ M(a/A).

(2) follows from (1) and Lemma 2.1(2). (3) is easy.

(4) By (3), M ^e _C (a/B) ≤ M ^e _C (a/A). By Lemma 2.5(1) below, also M ^e _C (a/A) ≤ M ^e _C (a/B) (and no vicious circle arises).

(5) Since T is small, we can find an a ⁰ ≡ a (A) with a ⁰ ∪ B (A) and ^| St(a ⁰ /B) open in St(a/A). Hence (5) follows from (4).

(6) Suppose M ^e _C (a/D) ≥ α and Tr _C (aB/D) is nowhere dense in Tr C (aB/B) (for some B, D as in (2 ⁰ )). Choose finite B ⁰ , D ⁰ with B ⁰ ⊆ D ⁰ ⊆ C ₌ , B \ A ⊆ acl(B ⁰ ), D \ A ⊆ acl(D ⁰ ) and a ∪ D ^{| 0} (A). As in (4) we can choose B ⁰ , D ⁰ so that M ^e _C (a/DD ⁰ ) ≥ α and Tr _C (aBB ⁰ /DD ⁰ ) is nowhere dense in Tr C (aBB ⁰ /BB ⁰ ). As in (4) and (5), we see that M ^e _C (a/AD ⁰ ) ≥ α and Tr _C (aAB ⁰ /AD ⁰ ) is nowhere dense in Tr _C (aAB ⁰ /AB ⁰ ), so we are done.

Notice that Lemma 2.2(4) naturally corresponds to the following prop- erty of forking: If A ⊆ B and U (a/B) = U (a/A) then R(a/B) = R(a/A) for any (local) rank R. This shows again a similarity between our treatment of multiplicities and the theory of forking.

To evaluate M ^e _C (ab/A) in terms of M ^e _C (a/Ab) and M ^e _C (b/A) we need the following technical lemma.

Lemma 2.3. Assume C ⊆ A ⊆ B and b ∪ B (A). ^|

(1) If Tr C (a/Bb) is open in Tr C (a/Ab) and Tr C (bA/B) is open in Tr _C (bA/A) then Tr _C (a/B) is open in Tr _C (a/A).

(2) If Tr C (ab/B) is open in Tr C (ab/A) then Tr C (a/B) is open in

Tr _C (a/A).

P r o o f. (1) Let ϕ(x a ; B, y b ) be a formula over B such that for some E(x _a , x ⁰ _a ) ∈ FE(C), for each b ⁰ , ϕ(C; B, b ⁰ ) is a union of some E-classes, and (a) Tr _C (a/Ab) ∩ [ϕ(x _a ; B, b)] = Tr _C (a/Bb).

Let ϕ ⁰ (x _a , x _b ) be a formula over some c ∈ acl(A), saying: for B ⁰ ≡

s B (A) with B ⁰ ∪ x ^| a x b (A), ϕ(x a ; B ⁰ , y b ) holds. We can also assume that c ∈ dcl(B).

Now, ϕ ⁰ (x _a , b) is equivalent to ϕ(x _a ; B, b).

Indeed, ϕ ⁰ (a ⁰ , b) implies that ϕ(a ⁰ ; B ⁰ , b) holds for some B ⁰ ∪ a ^{| 0} b (A) with B ⁰ ≡

s B (A). This gives B ⁰ b ≡

s Bb (A), hence ϕ(C; B, b) = ϕ(C; B ⁰ , b), and ϕ(a ⁰ ; B, b) holds. The other direction is similar.

So we can assume ϕ = ϕ ⁰ , that is, ϕ(x _a ; B, y _b ) is both over B and over c ∈ dcl(B) ∩ acl(A). We need the following claim.

Claim. Suppose C ⊆ A ⊆ B and c ∈ acl(b) ∩ A. If Tr _C (b/B) is open in Tr _C (b/A) then Tr _C (bc/B) is open in Tr _C (bc/A).

P r o o f. For b ⁰ ≡ b (A) let X _b

= {stp(b ⁰⁰ c/C) : b ⁰⁰ ≡

s b ⁰ (C) and b ⁰⁰ ≡ b (A)}. Since c ∈ acl(b) ∩ A, X b

is finite. Indeed, let C b

= {c ⁰ : c ⁰ ≡ c (b ⁰ ) and for some b ⁰⁰ ≡ b (A), b ⁰⁰ c ≡

s b ⁰ c ⁰ (C)}. Since c ∈ acl(b)∩A, C _b

is finite and Ab ⁰ -definable. Moreover, if stp(b ⁰⁰ c/C) ∈ X b

then for some c ⁰ , stp(b ⁰⁰ c/C) = stp(b ⁰ c ⁰ /C), and c ⁰ ∈ C _b

, hence X _b

= {stp(b ⁰ c ⁰ /C) : c ⁰ ∈ C _b

}. Also, if b ⁰⁰ ≡ b ⁰ ≡ b (A) then X _b

is the b ⁰⁰ -copy of X _b

over A, hence |X _b

| = |X _b

|.

Similarly, for b ⁰ ≡ b (B) let Y _b

= {stp(b ⁰⁰ c/C) : b ⁰⁰ ≡

s b ⁰ (C) and b ⁰⁰ ≡ b (B)}

= {stp(b ⁰ c ⁰ /C) : c ⁰ ≡ c (b ⁰ ) and for some b ⁰⁰ ≡ b (B), b ⁰⁰ c ≡

s b ⁰ c ⁰ (C)}.

We see that stp(b ⁰ c/C) ∈ Y b

⊆ X b

. Notice that Tr _C (bc/A) = [

{X _b

: b ⁰ ≡ b (A)} and Tr _C (bc/B) = [

{Y _b

: b ⁰ ≡ b (B)}.

Choose a formula χ(y _bc ) over acl(C) with X _b ∩ [χ(y _bc )] = {stp(bc/C)}.

Clearly,

if b ⁰ ≡

s b (C) and b ⁰ ≡ b (A) then X b

= X b , and if b ⁰ ≡

s b (C) and b ⁰ ≡ b (B) then Y b

= Y b .

So by compactness there is a formula α(y _b ) over acl(C), true of b, such that:

if b ⁰ ≡ b (A) and α(b ⁰ ) then X _b

∩ [χ(y _bc )] has size 1, and if b ⁰ ≡ b (B) and α(b ⁰ ) then Y _b

∩ [χ(y _bc )] has size 1.

Thus if b ⁰ ≡ b (B) and α(b ⁰ ) then moreover X _b

∩ [χ(y _bc )] = Y _b

∩ [χ(y _bc )].

Since Tr _C (b/B) is open in Tr _C (b/A), for some β(y _b ) over acl(C),

Tr _C (b/B) = Tr _C (b/A) ∩ [β(y _b )].

Let γ(y bc ) be α(y b )&β(y b )&χ(y bc ). So γ(bc) holds. To finish, it suffices to show that

Tr C (bc/A) ∩ [γ(y bc )] ⊆ Tr C (bc/B).

So suppose b ⁰ c ⁰ ≡ bc (A) and γ(b ⁰ c ⁰ ) holds (and necessarily c ⁰ = c).

We want to find b ⁰⁰ with b ⁰⁰ c ≡

s b ⁰ c (C) and b ⁰⁰ ≡ b (B). Now, β (b ⁰ ) yields a b ⁰⁰ ≡

s b ⁰ (C) with b ⁰⁰ ≡ b (B). So α(b ⁰⁰ ) holds, which gives that X _b

∩ [χ(y _bc )]

equals Y _b

∩ [χ(y _bc )] and has size 1. By the definition of Y _b

, there is a b ^∗ ≡

s b ⁰⁰ (C) with b ^∗ ≡ b (B) and {stp(b ^∗ c/C)} = Y _b

∩ [χ(y _bc )]. Without loss of generality, b ^∗ = b ⁰⁰ . Since b ⁰⁰ ≡

s b ⁰ (C) and b ⁰⁰ ≡ b ⁰ (A), we get X _b

= X _b

. So stp(b ⁰ c/C) ∈ X b

∩ [χ(y bc )] = X b

∩ [χ(y bc )] = Y b

∩ [χ(y bc )] = {stp(b ⁰⁰ c/C)}.

It follows that b ⁰⁰ c ≡

s b ⁰ c (C) and b ⁰⁰ ≡ b (B), hence stp(b ⁰ c/C) ∈ Tr _C (bc/B).

This proves the claim.

Returning to the proof of the lemma, since Tr _C (bA/B) is open in Tr _C (bA/A), and c ∈ dcl(B) gives Tr _C (bA/B) ⊆ Tr _C (bA/Ac), we deduce that Tr C (bA/B) is open in Tr C (bA/Ac). Applying the claim to b := bA, A := Ac and B := Bc, we see that Tr _C (bAc/B) is open in Tr _C (bAc/Ac).

Moreover, c ∈ acl(A) gives that Tr _C (bAc/Ac) is open in Tr _C (bAc/A). Alto- gether we conclude that Tr C (bAc/B) is open in Tr C (bAc/A). Hence there is a formula δ(y _bAc ; B) almost over C such that

(b) Tr C (bAc/A) ∩ [δ(y bAc ; B)] = Tr C (bAc/B).

Let S = S _B = {stp(a ⁰ /C) : for some b ⁰ , a ⁰ b ⁰ ≡ ab (Ac) and δ(b ⁰ Ac; B) and ϕ(a ⁰ ; B, b ⁰ ) hold}.

Then S B is a closed subset of Tr C (a/A). Since c ∈ dcl(B), S B is definable over B (that is, Aut _B (C)-invariant), and for B ⁰ ≡ B (A) we can define S _B

as the B ⁰ -copy of S _B over A. Since δ and ϕ are almost over A, the set X = {S B

: B ⁰ ≡ B (A)} is finite and Tr C (a/A) = S

X. It follows that S is open in Tr _C (a/A) and stp(a/C) ∈ S. So to show that Tr _C (a/B) is open in Tr _C (a/A) it suffices to prove

(c) S ⊆ Tr _C (a/B).

So suppose stp(a ⁰ /C) ∈ S. Thus for some b ⁰ we have a ⁰ b ⁰ ≡ ab (Ac) and δ(b ⁰ Ac; B) and ϕ(a ⁰ ; B, b ⁰ ) hold. We must find a ⁰⁰ with a ⁰⁰ ≡

s a ⁰ (C) and a ⁰⁰ ≡ a (B). By (b), for some b ⁰⁰ with b ⁰⁰ Ac ≡

s b ⁰ Ac (C), we have b ⁰⁰ ≡ b (B).

Hence we can find a ₀ and a ₁ with (d) a 0 b ⁰⁰ Ac ≡

s a ⁰ b ⁰ Ac (C), ab ≡ a 1 b ⁰⁰ (B).

In particular, we have

(e) a ₀ b ⁰⁰ ≡ a ⁰ b ⁰ ≡ ab ≡ a ₁ b ⁰⁰ (Ac).