GALOIS-STABILITY FOR TAME ABSTRACT ELEMENTARY CLASSES

RAMI GROSSBERG AND MONICA VANDIEREN

Abstract. We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this new context. We assume that K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:

Theorem 0.1. Suppose that K is not only tame, but LS(K)-tame. If μ ≥ Hanf(K) and K is Galois stable in μ, then κ_{μ}(K) < ℶ_{(2^{Hanf(K)})^{+}}, where κ_{μ}(K) is a relative of κ(T) from first order logic.

Hanf(K) is the Hanf number of the class K. It is known that Hanf(K) ≤ ℶ_{(2^{LS(K)})^{+}}

The theorem generalizes a result from [Sh3]. It is used to prove both the existence of Morley sequences for non-splitting (improving Claim 4.15 of [Sh 394] and a result from [GrLe1]) and the following initial step towards a stability spectrum theorem for tame classes:

Theorem 0.2. If K is Galois-stable in some μ > ℶ_{(2^{Hanf(K)})^{+}}, then K is stable in every κ with κ^{μ} = κ. E.g. under GCH we have that K Galois-stable in μ implies that K is Galois-stable in μ^{+n} for all n < ω.

Introduction

In the last twenty years most of Shelah’s effort in model theory was in developing classification theory for abstract elementary classes (Some important mile stones are: [Sh 87a], [Sh 87b],[Sh 88], [Sh 300], [Sh 394], [Sh 576], [Sh 600] and [Sh 705]). An abstract elementary class (AEC) is a class of structures of the same similarity type endowed with a morphism satisfying natural properties such as closure under directed limits (see [Gr1] for an introduction). The notion of an AEC (introduced by Shelah in [Sh 88]) is

Date: June 6, 2006. AMS Subject Classification: Primary: 03C45, 03C52, 03C75. Secondary: 03C05,

03C55 and 03C95.

This paper is part of the second author’s Ph.D. thesis written under the guidance of Rami Grossberg.

broad enough to capture classes axiomatized by non-first-order logics including L_{ω_{1},ω}(Q). To date, there is no known stability theory or even a categoricity theorem for abstract elementary classes without some additional strong assumptions.

The most general of Shelah’s attempts to develop a stability theory is [Sh 600] where Shelah works in the context of AECs with the amalgamation and joint embedding properties and a very nicely behaved forking-like relation. He introduces the notion of good-frame which is a generalization of a first-order superstable theory (the definition alone needs a couple of pages). Most of his 100-page paper is dedicated to the derivation of structural results under the assumption that the AEC has a forking-like relation.

In our paper we deal with a much wider class of AECs than Shelah in [Sh 600] as we do not assume as part of the setting built-in forking and existence of regular types. We deal with stable but tame classes (see Definitions 1.7 and 3.2), in other words, stable classes where inequality of types behaves locally. Tameness is a general property capturing finite diagrams (homogeneous model theory). While it is not difficult to see that excellent classes of atomic models of a first-order theory (from [Sh 87a] and [Sh 87b]) are tame, Grossberg and Alexei Kolesnikov show that every abstract excellent class is also tame (see [GrKo]). Very recently using ideas from [GrKo], Andrés Villaveces and Pedro Zambrano discovered that the class of structures obtained from Hrushovski’s fusion of strongly minimal first-order theories is another source of examples of tame AECs. See [ViZa] and [Ba].

Despite the generality of our context, recent progress on a Morley’s Theorem for tame abstract elementary classes provides evidence that a stabilitytheory can be developed. A short time after submitting this paper for publication we realized that using the splitting machinery developed here, we can prove in ZFC an instance of Shelah’s categoricity conjecture for tame classes (see [GrVa1] and [GrVa2]).

Theorem 0.3 (Grossberg and VanDieren 2003). Let K be an AEC, κ := ℶ_{(2^{LS(K)})^{+}}. Denote by μ_{0} := ℶ_{(2^{κ})^{+}}. Suppose that K_{>κ} has the amalgamation property and is κ-tame. If K is categorical in some λ^{+} > μ_{0} then K is categorical in every μ ≥ μ_{0}.

This work removes the set-theoretic assumption of the existence of a strongly compact cardinal from the work of Makkai and Shelah [MaSh] replacing it with a weaker model theoretic notion.

In addition to examining splitting in this paper, we prove the existence of Morley sequences from the assumption of stability in tame AECs. We plan to use Morley sequences for non-splitting to define a dividing-like concept for these classes and to prove a stability spectrum theorem for AECs. More work on the stability spectrum theorem (extension of our results under the additional assumption of LS(K) = ℵ_{0}) appears in [BaKuVa].

We are grateful to John Baldwin for reading several preliminary versions and suggesting us to consider the stability spectrum problem for AECs.

The referee’s comments and questions led to significant improvement of the presentation.

1. Background

Much of the necessary background for this paper can be found in [Gr1]. We will use α, β, γ, i, j to denote ordinals and κ, λ, μ, χ will be used for cardinals. We will use (K, ≺_{K}) to denote an abstract elementary class and K_{μ} is the subclass of models in K of cardinality μ. Similarly we define K_{<μ} and K_{≥μ}. Models are denoted by M, N and may be decorated with superscripts and subscripts. Sequences of elements from M are written as ā, b̄, c̄, d̄ and we write ^{β}M to denote the collection of all sequences of length β made up of elements from M. The letters e, f, g, h are reserved for Kmappings and id is the identity mapping.

For the remainder of this paper we will fix (K, ≺_{K}) to be an abstract elementary class.

We begin by reviewing the definition of Galois-type, since we will be considering variations of the underlying equivalence relation E later in this paper.

Definition 1.1. Let β > 0 be an ordinal. For triples (ā_{ℓ}, M, N_{l}) where ā_{ℓ} ∈ ^{β}N_{ℓ} and M_{ℓ} ≺_{K} N_{ℓ} ∈ K for ℓ = 0, 1, we define a binary relation E as follows: (ā_{0}, M, N_{0})E(ā_{1}, M, N_{1}) iff and there exists N ∈ K and K-mappings f_{0}, f_{1} such that f_{ℓ} : N_{ℓ} → N, f_{ℓ} ↾ M = id_{M} for ℓ = 0, 1 and f_{0}(ā_{0}) = f_{1}(ā_{1}):

\xymatrix{N_{1} \ar[r]_{f_{1}} & N \\ M \ar[u]^{id} \ar[r]_{id} & N_{0} \ar[u]_{f_{0}}}

The relation E is used to define Galois-types in classes that satisfy the amalgamation property.

Definition 1.2. (1) Let μ ≥ LS(K). We say that K has the μ-amalgamation property iff for any M_{ℓ} ∈ K_{μ} (for ℓ ∈ {0, 1, 2}) such that M_{0} ≺_{K} M_{1} and M_{0} ≺_{K} M_{2} there are N ∈ K_{μ} and Kembeddings f_{ℓ} : M_{ℓ} → N such that f_{ℓ} ↾ M_{0} = id_{M_{0}} for ℓ = 1, 2.

(2) A model M_{0} ∈ K_{μ} satisfying the above requirement is called an amalgamation base.

(3) K has the amalgamation property iff K has the μ-amalgamation property for all μ ≥ LS(K).

Remark 1.3. E is an equivalence relation on the class of triples of the form (ā, M, N) where M ≺_{K} N, ā ∈ N and both M and N are amalgamation bases. When N is not an amalgamation base, E may fail to be transitive, but the transitive closure of E could be used instead.

Remark 1.4. While, the focus of this paper is on classes with the amalgamation property, several of the proofs in this paper can be adjusted to the context of abstract elementary classes with density of amalgamation bases as in [ShVi] and [Va].

We will make the following assumption for the remainder of the paper:

Assumption 1.5. K satisfies the amalgamation property.

Under the assumption of the amalgamation property, there exists a large model-homogeneous model, which we will denote by ℭ and call the monster model. A μ-model homogeneous model is a model M in which for every N ≺_{K} M of cardinality μ and every extension N_{1} of cardinality μ, there exists a K-embedding f : N_{1} → M with f ↾ N = id_{N}. M is model homogeneous when it is μ-model homogeneous for every μ < ∥ℭ∥. All models and sequences of elements will be assumed to come from ℭ. Thus, E is an equivalence relation and we can now define types as equivalence classes of E.

In order to avoid confusing this non-elementary notion of “type” with the conventional first-order one (i.e. set of formulas) we will follow the notation of [Gr1] and [Gr2] and introduce it below under the name of Galois-type. This concept was introduced by Shelah in [Sh 394] using the usual notation of first-order logic.

Definition 1.6. Let β be a positive ordinal (can be one).

(1) For M, N ∈ K and ā ∈ ^{β}N . The Galois-type of ā in N over M, written ga-tp(ā/M, N), is defined to be (ā, M, N)/E.

(2) We abbreviate ga-tp(ā/M, N) by ga-tp(ā/M).

(3) For M ∈ K,

ga-S^{β}(M) := {ga-tp(ā/M, N) | M ≺ N ∈ K_{∥M∥}, ā ∈ ^{β}N}.

We write ga-S(M) for ga-S^{1}(M).

(4) Let p := ga-tp(ā/M′, N) for M ≺_{K} M′ we denote by p ↾ M the type ga-tp(ā/M, N). The domain of p is denoted by dom(p) and it is by definition M′.

(5) Let p = ga-tp(ā/M, N), suppose that M ≺_{K} N′ ≺_{K} N and let b̄ ∈ ^{β}N′ we say that b̄ realizes p iff ga-tp(b̄/M, N′) = p ↾ M.

(6) For types p and q, we write p ≤ q if dom(p) ⊆ dom(q) and there exists ā realizing p in some N extending dom(p) such that (ā, dom(p), N) = q ↾ dom(p).

Definition 1.7. We say that K is β-Galois-stable in μ if for every M ∈ K_{μ}, | ga-S^{β}(M)| = μ. The class K is Galois stable in μ iff K is 1-stable in μ.

Remark 1.8. Let T be a stable, countable, first-order, complete theory. Set K := Mod(T) and ≺_{K} the usual elementary submodel relation. Take μ = 2^{ℵ_{0}}. While K is 1-Galois-stable in μ (in fact it is n-Galois-stable in μ for all n < ω), it can be shown that K is ω-Galois-stable in μ iff K has the ndop and notop.

While there is a nice relationship between β-stability and β′-stability for first order theories, the corresponding relationship between β-stability and β′-stability for AECs in general is unknown and probably fails.

Notation 1.9. Although not standard in AECs with the amalgamation property, we will use the notation ga-tp(ā/∅) to denote the orbit of ā under Aut(ℭ). This notation is only used as a device in the proof of Theorem 4.13.

2. Saturated and Limit Models

In this section we recall that limit models (see Definition 2.6) can be used as a substitute for saturated models. Later, we will see that limit models will be more convenient in situations where the first order case would call for saturated models.

Definition 2.1. We say that M ∈ K_{>LS(K)} is Galois-saturated if for every N ≺_{K} M of cardinality < ∥M∥, and every p ∈ ga-S(N), we have that M realizes p.

Remark 2.2. When K = Mod(T) for a first-order T, using the compactness theorem one can show (Theorem 2.2.3 of [Gr1]) that for M ∈ K, the model M is Galois-saturated iff M is saturated in the first-order sense.

We begin with universal extensions which are used to build limit models. A universal extension captures some properties of saturated models without referring explicitly to types. The notion of universality over countable models was first analyzed by Shelah in Theorem 1.4(3) of [Sh 87a].

Definition 2.3. (1) Let κ be a cardinal ≥ LS(K). We say N is κuniversal over M iff for every M′ ∈ K_{κ} with M ≺_{K} M′ there exists a K-embedding g : M′ → N such that g ↾ M = id_{M}:

\xymatrix{M′ \ar[dr]^{g} \\ M \ar[u]^{id} \ar[r]_{id} & N}

(2) We say N is universal over M or N is a universal extension of M whenever N is ∥M∥-universal over M.

Remark 2.4. Notice that the definition of N universal over M requires all extensions of M of cardinality ∥M∥ to be embeddable into N. Variants of this definition in the literature often involve ∥M∥ < ∥N∥. We would like to emphasize that in this paper we are concentrating on the case when ∥M∥ = ∥N∥.

Examples 2.5. (1) Suppose that T is a first-order complete theory that is stable in some regular μ. Then every model M of T of cardinality μ has an elementary extension N of cardinality μ which is universal over M. To see this define an elementary-increasing and continuous chain of models of T of cardinality μ, ⟨N_{i} | i < μ⟩ such that N_{i+1} realizes all types over N_{i}. Let N = ⋃_{i<μ} N_{i}. By a back-and-forth construction, one can show that N is universal over M.

(2) Let K be the class of atomic models of a complete first-order theory in a countable language and ≺_{K} the usual notion of first-order elementary submodel. If 1 ≤ I(ℵ_{1}, K) < 2^{ℵ_{1}} and 2^{ℵ_{0}} < 2^{ℵ_{1}} then for every M ∈ K_{ℵ_{0}} there exists a proper extension N ∈ K_{ℵ_{0}} universal over M, by Theorem 1.4(3) of [Sh 87a].

Definition 2.6. For M′, M ∈ K_{μ} and σ a limit ordinal with σ < μ^{+}, we say that M′ is a (μ, σ)-limit over M iff there exists a ≺_{K}-increasing and continuous sequence of models ⟨M_{i} ∈ K_{μ} | i < σ⟩ such that

(1) M = M_{0},

(2) M′ = ⋃_{i<σ} M_{i} and

(3) M_{i+1} is universal over M_{i}.

Remark 2.7. Notice that in Definition 2.6, for i < σ and i a limit ordinal, M_{i} is a (μ, i)-limit model.

Definition 2.8. We say that M′ is a (μ, σ)-limit iff there is some M ∈ K such that M′ is a (μ, σ)-limit over M.

When K = Mod(T) for a first-order and stable T then automatically (by Theorem III.3.12 of [Shc]):

M ∈ K_{μ} is saturated ⟹ M is (μ, σ)-limit for all σ < μ^{+} of cofinality ≥ κ(T).

The uniqueness of (μ, σ)-limit models is an interesting problem: The statement is

(∗) If N_{ℓ} are (μ, σ_{ℓ})-limit models over M then N_{1} ≅_{M} N_{2}.

A standard back and forth construction gives a uniqueness proof when cf(σ_{1}) = cf(σ_{2}).

Problem (∗) for σ_{1} ≠ σ_{2} is important. Let’s make some basic observations:

1. Suppose K = Mod(T) and T is a complete first-order theory. If μ is regular and σ = μ then the (μ, σ)-limit model (if it exists) is the saturated model of T of cardinality μ.

2. Showing that two (μ, σ)-limit models are isomorphic for different σs is a central (dichotomy) property even for first-order theories: E.g. Let T be a countable (for simplicity), stable, non-superstable first-order theory. Suppose T is stable in μ. By Harnik’s theorem [Ha] T has a saturated model of cardinality μ, in fact for σ < μ^{+} the theory has (μ, σ)-limit models. Since T is not superstable one can modify the construction in Thm 13.1 of [AlGr] to construct a (μ, ω)-limit model which is not saturated. there exists a union of an ω-sequence of saturated models which is not saturated (see [AlGr]). The union of such as sequence is not isomorphic to a (μ, ω_{1})-limit model (that was obtained by taking a union of saturated models, see Theorem III.3.12 of [Shc]). Thus, when T is countable and stable but not superstable, the saturated model of cardinality μ is a (μ, ω_{1})-limit but not a (μ, ω)-limit model.

We will see that under the amalgamation property, the existence of limit models follows from stability. Limit models have been applied in several contexts including [Sh 394],[Sh 576] and [Sh 600]. Even without the amalgamation property, the existence [ShVi] and uniqueness [Va] have been shown for categorical AECs under some mild set-theoretic assumptions.

Lately Grossberg, VanDieren and Villaveces proved uniqueness of limit models under mild model-theoretic assumptions on the AEC without assuming categoricity or working outside of ZFC (see [GVV]).

We now describe how one proves the existence of universal extensions and limit models in stable AECs. It turns out that for first-order theories stable in μ any model of cardinality μ has a universal extension of cardinality μ. The result for AECs is claimed in [Sh600] without a proof.

Claim 2.9 (Claim 1.15.1 from [Sh 600]). Suppose K is an abstract elementary class with the amalgamation property. If K is Galois-stable in μ, then for every M ∈ K_{μ}, there exists M′ ∈ K_{μ} such that M′ is universal over M. Moreover M′ can be chosen to be a (μ, σ)-limit over M for any σ < μ^{+}.

We provide a proof of this claim here for completeness.

As it is easy to prove:

Proposition 2.10. Suppose κ ≥ LS(K), K has the κ-amalgamation property and M, N ∈ K_{κ}. If M ⪵_{K} N and N is maximal (i.e. there is no N′ ⪶_{K} N) then N is universal over M.

For the rest of this paper we assume:

Assumption 2.11. K does not have maximal models.

This is a reasonable assumption, since (in the presence of the disjoint amalgamation property) it holds automatically for categorical classes or classes with the joint embedding property.

Theorem 2.12. Let K be an AEC which is 1-Galois-stable in μ. If K has the amalgamation property then for every M ∈ K_{μ} there exists M^{∗} ⪶_{K} M, universal over M of cardinality μ.

Proof. Define an increasing continuous chain ⟨M_{i} | i < μ⟩ ⊆ K_{μ} such that M_{0} := M and every p ∈ ga-S(M_{i}) is realized in M_{i+1}. Let M^{∗} := ⋃_{i<μ} M_{i}.

We now show that M^{∗} is universal over M. Let N ≻_{K} M be a given model of cardinality μ, we’ll construct an embedding f : N → M^{∗} s.t. f ↾ M = id_{M}.

Fix {a_{i} | i < μ} an enumeration of |N| − |M|.

By induction on i < μ define ≺_{K}-increasing and continuous chains {N_{ℓ}^{i} | i < μ, ℓ = 0, 1} ⊆ K_{μ} and {f_{i} | i < μ} such that

(1) N_{0}^{i} ≺_{K} N_{1}^{i},

(2) N_{0}^{0} = M, N_{1}^{0} = N

(3) a_{i} ∈ N_{0}^{i+1}

(4) f_{i} : N_{0}^{i} → M_{i}.

Clearly this is sufficient since (⋃_{i<μ} f_{i}) ↾ N is as required. By continuity it is enough to describe the construction at successor stages.

If a_{j} ∈ N_{0}^{j} let N_{ℓ}^{j+1} := N_{ℓ}^{j}.

Otherwise, denote by M_{0}^{j} ≺_{K} M_{j} the image of N_{0}^{j} under f_{j}, let g ⊇ f_{j} and \overline{M}_{1}^{j} ≻_{K} M_{0}^{j} be such that g : N_{1}^{j} ≅ \overline{M}_{1}^{j}.

By the amalgamation property there are M_{1}^{j} and \overline{g} : \overline{M}_{1}^{j} → M_{1}^{j} whose restriction is the identity over M_{1}^{j} such that the diagram

\xymatrix{N_{1}^{j} \ar[r]^{g} & \overline{M}_{1}^{j} \ar[r]^{\overline{g}} & M_{1}^{j} \\ N_{0}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M_{0}^{j} \ar[u]_{id} \ar[r]_{id} & M_{j} \ar[r]_{id} \ar[u]_{id} & M}

commutes.

Since a_{j} ∈ N_{1}^{j} − N_{0}^{j}, g(a_{j}) ∈ \overline{M}_{1}^{j} − M_{0}^{j}. We can choose ḡ such that \overline{g}(g(a_{j})) ∉ M_{j}. Let p := ga-tp(\overline{g}(g(a_{j}))/M_{j}, M_{1}^{j}). By the construction of ⟨M_{i}⟩_{i<μ} there exists b ∈ M_{j+1} realizing the type p. By definition of Galoistypes we have (M_{j}, M_{1}^{j}, \overline{g}(g(a_{j})))E(M_{j}, M_{j+1}, b). Using the definition of E (Definition 1.1), there are N^{∗∗} ∈ K_{μ} and mappings h_{1}, h_{2} such that the diagram

\xymatrix{M_{1}^{j} \ar[r]^{h_{2}} & N^{∗∗} \\ M_{j} \ar[u]^{id} \ar[r]_{id} & M_{j+1} \ar[u]_{h_{1}}}

commutes and

(∗) h_{2}(ḡ(g_{j}(a_{j}))) = h_{1}(b).

We may assume that h_{2} = id_{M_{1}^{j}}. Thus by gluing the last two diagrams together we get that the diagram

\xymatrix{N_{1}^{j} \ar[r]^{g∘\overline{g}} & M_{1}^{j} \ar[r]^{id} & N^{∗∗} \\ N_{0}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M_{j} \ar[u]_{id} \ar[r]_{id} & M_{j+1} \ar[u]_{h_{1}} \ar[r]_{id} & M}

commutes. Now pick N_{1}^{j+1} ≻_{K} N_{1}^{j} and h ⊇ g ∘ \overline{g} such that h : N_{1}^{i} ≅ N^{∗∗}.

So we have that

\xymatrix{N_{1}^{i+1} \ar[dr]^{h} \\ N_{1}^{j} \ar[u]^{id} \ar[r]_{g∘\overline{g}} & N^{∗∗} \\ N_{0}^{j} \ar[u]^{id} \ar[r]_{f_{j}} & M_{j+1} \ar[u]_{h_{1}}}

commutes. Let N_{0}^{j+1} := h^{−1}[h_{1}[M_{j+1}]]. Notice that by the above diagram, we get that N_{0}^{j+1} ⊇ N_{0}^{j}. Now we are ready to define the mapping f_{i}, let f_{j+1} := h_{1}^{−1} ∘ (h ↾ N_{0}^{j}). It is a Kembedding that extends f_{j} as required.

One can easily verify that ⋃_{i<μ} f_{i} is an embedding of N_{2} over N_{1} into M. ⊣

The following is now immediate

Corollary 2.13. Let K be an AEC without maximal models and suppose it is Galois-stable in μ. If K has the amalgamation property and arbitrarily large models then for every M ∈ K_{μ} and every σ < μ^{+} there exists N ∈ K_{μ} a (μ, σ)-limit model over M.

Remark 2.14. In a preliminary draft of this paper we included the μ-Disjoint Amalgamation Property as one of the assumptions of Theorem 2.12. We thank John Baldwin and Alexei Kolesnikov for analyzing our proof and bringing to our attention that we actually didn’t use disjoint amalgamation.

3. Tame Abstract Elementary Classes

In AECs we view types as equivalences classes of triples (Definition 1.6) instead of sets of formulas. However, a localized equivalence relation (denoted E_{μ}) is eventually utilized in various partial solutions to Shelah’s Categoricity Conjecture (see [Sh 394] and [Sh 576]).

Shelah identified E_{μ} in [Sh 394]. Here we recall the definition.

Definition 3.1. Triples (ā_{1}, M, N_{1}) and (ā_{2}, M, N_{2}) are said to be E_{μ}- related provided that for every M′ ≺_{K} M with M′ ∈ K_{≤μ},

(ā_{1}, M′, N_{1})E(ā_{2}, M′, N_{2}).

Notice that in first order logic, the finite character of consistency implies that two types are equal if and only if they are E_{<ω}-related.

Main Claim 2.3 of part II on page 288 [Sh 394] Shelah ultimately proves that, under categoricity in some λ > ℶ_{(2^{Hanf(K)})^{+}} and under the assumption that K has the amalgamation property, E-equivalence is the same as E_{μ} equivalence for 1-types over saturated models, for some μ < λ.

We now define a context for abstract elementary classes where consistency has small character.

Definition 3.2. Let χ be a cardinal number. For β > 0, we say K is χ-tame for β-types provided that for every M ∈ K_{>χ}, p ≠ q ∈ ga-S^{β}(M) implies existence of N ≺_{K} M of cardinality χ such that p ↾ N ≠ q ↾ N. We say K is χ-tame if it is χ-tame for 1-types.

Notice that in [Sh 394] Shelah defines a relative to tameness which considers only those types over saturated models. Here we are interested in types over arbitrary models.

Definition 3.3. K is tame iff there exists a χ < Hanf(K) such that K is χ-tame.

The relationship between tameness and χ-tameness is not completely clear. It is plausible that the following holds:

Conjecture 3.4. Let K be an AEC. If K is χ-tame for some χ then it is tame.

Remark 3.5. In this paper, we will actually only use that E-equivalence is the same as E_{χ}-equivalence for types over limit models.

Notice that if K is a finite diagram in the sense of [Sh3] (i.e. we have amalgamation not only over all models but also over all subsets of every model), then K is a tame AEC.

There are tame AECs with amalgamation which are not finite diagrams. In fact Leo Marcus in [Ma] constructed an L_{ω_{1},ω}-sentence which is categorical in every cardinality but does not have an uncountable sequentially homogeneous model. Recently Boris Zilber found a mathematically more natural example [Zi] motivated by Schanuel’s Conjecture. His example is not homogeneous nor L_{ω_{1},ω}-axiomatizable.

Shelah proves excellence (from V=L) for countable L_{ω_{1},ω}-theories satisfying I(ℵ_{n+1}, K) < 2^{ℵ_{n+1}} for all n < ω ([Sh 87a] and [Sh 87b]).

In [GrKo] it is shown that excellence implies that K is ℵ_{0}-tame. In fact, a certain two dimensional property weaker than (< χ, P^{−}(3))-AP already implies χ-tameness. Even more examples of tame classes which are not finite diagrams can be found from the work of Villaveces and Zambrano (see [ViZa] or [Ba]).

Let K be an AEC, suppose κ ≥ LS(K) is a strongly compact cardinal. If K is categorical in some λ^{+} > ℶ_{(2^{κ})^{+}} then by Proposition 1.13 of [MaSh] K_{≥κ} has the amalgamation property and since Galois types can be identified with sets of formulas in L_{κ,κ} (see [MaSh]). Thus classes satisfying the above assumptions (categoricity above the Hanf number of a strongly compact cardinal), by Makkai and Shelah’s results fit into our framework of κ-tame classes.

While we are convinced that there are examples of arbitrary levels of tameness at the time of submission of this paper we did not have any.

Question 3.6. For μ_{1} < μ_{2} < ℶ_{ω_{1}}, find an AEC which is μ_{2}-tame but not μ_{1}-tame.

Since the submission of our paper, some progress on this question has been made. Back in a discussion with Baldwin in Bogotá in 2001 we suggested that perhaps a non-free but almost free group of cardinality χ could be used to find an example of a class which is not χ-tame. In August 2005 Baldwin informed us that he and Shelah have a work in progress for class which is not ℵ_{1}-tame they obtain this with groups and short exact sequences, by essentially coding the functor Ext(G, Z) and using an ℵ_{1}-free group of cardinality ℵ_{1} which is not free and not Whitehead (see [BaSh]).

Historical Comment 3.7. We introduce tameness as a sufficient assumption to prove Theorem 4.13. Later we realized that a (much) stronger property appears in [Sh 394] where it is proved that categoricity implies that galois types over saturated models are determined by smaller substructures. This paper (our first article containing tameness) was circulated by us in July 2001 (under the title of Morley sequences in AEC). Several authors among them Baldwin, Hyttinen, Kesalla and Lessmann recently published papers using tameness (usually ℵ_{0}-tameness). Unfortunately some of these blur the distinction between tameness and Shelah’s notion of local character from [Sh 394].

4. Bound on κ_{μ}^{β}

In this section we study a non-elementary relative to the first order notion of splitting and we derive bounds for an invariant that corresponds to κ(T) under the assumption of stability and tameness.

First let us recall the notion of μ-splitting introduced by Shelah in [Sh 394].

Definition 4.1. A type p ∈ ga-S^{β}(N) μ-splits over M ≺_{K} N if and only if ∥M∥ ≤ μ, there exist N_{1}, N_{2} ∈ K_{≤μ} and h, a K-embedding such that M ≺_{K} N_{l} ≺_{K} N for l = 1, 2 and h : N_{1} → N_{2} such that h ↾ M = id_{M} and p ↾ N_{2} ≠ h(p ↾ N_{1}).

Remark 4.2. If T is a first-order theory stable in μ and M is saturated, then for all N ≺ M of cardinality μ, the first order type, tp(a/M), does not split (in the first order sense) over N iff ga-tp(a/M) does not μ-split over N.

Notice that non-splitting is monotonic: I.e. If p ∈ ga-S(N) does not μsplit over M (for some M ≺_{K} N) then p does not μ-split over M′ for every M ≺_{K} M′ ≺_{K} N of cardinality μ.

Similar to κ(T) when T is first-order the following is a natural cardinal invariant of K:

Definition 4.3. Let β > 0. We define κ_{μ}^{β}(K) to be the minimal κ < μ^{+} such that for every ⟨M_{i} ∈ K_{μ} | i ≤ κ⟩ which satisfies

(1) κ = cf(κ) < μ^{+},

(2) ⟨M_{i} | i ≤ κ⟩ is ≺_{K}-increasing and continuous and

(3) for every i < κ, M_{i+1} is a (μ, θ)-limit over M_{i} for some θ < μ^{+}, and for every p ∈ ga-S^{β}(M_{κ}), there exists i < κ such that p does not μ-split over M_{i}. If no such κ exists, we say κ_{μ}^{β}(K) = ∞.

Another relative of κ(T) is the following

Definition 4.4. For β > 0, κ̄_{μ}^{β}(K) is the minimal cardinal κ̄ such that for every N ∈ K_{μ} and every p ∈ ga-S^{β}(N) there are λ < κ̄ and M ∈ K_{λ} such that p does not λ-split over M.

It is not difficult to verify that

Proposition 4.5. For μ with cf(μ) > κ̄_{μ}^{β}(K), we have κ_{μ}^{β}(K) ≤ κ̄_{μ}^{β}(K).

While for stable first order theories (when β < ω) both invariants are equal, the situation for non-elementary classes is more complicated. Already in [Sh 394] it was observed that κ_{μ}^{β}(K) is better behaved than κ̄_{μ}^{β}(K) when a bound for κ_{μ}^{β}(K) was found. A corresponding bound for κ̄_{μ}^{β}(K) is unknown. We will defer dealing with the invariant κ̄_{μ}^{β}(K) to a future paper.

Notice that Theorem 2.2.1 of [ShVi] states that certain categorical abstract elementary classes satisfy κ_{μ}^{1}(K) = ω, for various μ. A slight modification of the argument of Claim 3.3 from [Sh 394] can be used to prove a related result using the weaker assumption of Galois-stability only:

Fact 4.6. Let β > 0. Suppose that K is β-Galois-stable in μ. For every p ∈ ga-S^{β}(N) there exists M ≺_{K} N of cardinality μ such that p does not μ-split over M. Thus κ̄_{μ}^{β}(K) ≤ μ.

For the sake of completeness an argument for Fact 4.6 is included:

Proof. Suppose p = ga-tp(ā/N, ℭ) with ā ∈ ^{β}| ℭ| and p μ-splits over M, for every M ≺_{K} N of cardinality μ. We may assume that ā ∉ N, since it is routine to check that an algebraic type never splits over its domain.

Let χ := min{χ | 2^{χ} > μ}. Notice that χ ≤ μ and 2^{<χ} ≤ μ.

First we will define ⟨M_{α}, N_{α,ℓ} ≺ N | α < χ, ℓ = 1, 2⟩ ⊆ K_{μ} which will then be used to construct another sequence ⟨M_{α}^{∗} ∈ K_{μ} | α ≤ χ⟩ so that

| ga-S^{β}(M_{χ}^{∗})| ≥ 2^{χ} > μ will witness Galois-instability in μ.

The definition of ⟨M_{α}, N_{α,1}, N_{α,2} | α < χ⟩ follows by induction on α and our choice of p. Let M_{0} ≺ M be any model of cardinality μ.

For α = γ + 1, by our choice of p, we know p must μ-split over M_{γ}. Then, there are N_{γ,ℓ} ≺_{K} M of cardinality μ for ℓ = 1, 2 and there is F_{γ} : N_{γ,1} ≅_{M_{γ}} N_{γ,2} such that F_{γ}(p ↾ N_{γ,1}) ≠ p ↾ N_{γ,2}. Pick M_{α} ≺_{K} M of cardinality μ containing the set |N_{γ,1}| ∪ |N_{γ,2}|.

When α is a limit ordinal we define M_{α} := ⋃_{γ<α} M_{γ}.

Now we use our sequence ⟨M_{α} | α < χ⟩ to define for α ≤ χ another sequence of models M_{α}^{∗} ∈ K_{μ} and a tree of K-embeddings ⟨h_{η} | η ∈ ^{α}2⟩ satisfying

(1) γ < α ⟹ M_{γ}^{∗} ≺_{K} M_{α}^{∗},

(2) for α limit let M_{α}^{∗} = ⋃_{γ<α} M_{γ}^{∗},

(3) γ < α ∧ η ∈ ^{α}2 ⟹ h_{η↾γ} ⊆ h_{η},

(4) η ∈ ^{α}2 ⟹ h_{η} : M_{α} → M_{α}^{∗} and

(5) α = γ + 1 ∧ η ∈ ^{α}2 ⟹ h_{ηˆ 0}(N_{γ,1}) = h_{ηˆ 1}(N_{γ,2})

The construction is possible by using the μ-amalgamation property several times. For α = 0, define M_{0}^{∗} = M and h_{⟨⟩} := id_{M_{0}^{∗}}. When α is a limit ordinal we define M_{α}^{∗} := ⋃_{γ<α} M_{γ} and for η ∈ ^{α}2, h_{η} := ⋃_{γ<α} h_{η↾γ}.

Suppose we have completed the construction for γ and α = γ + 1. Given η ∈ ^{γ}2, let h̄_{η} be an extension of h_{η} to an automorphism of ℭ. Notice that h̄_{η} ∘ F_{γ}(N_{γ,1}) = h̄_{η}(N_{γ,2}) by our choice of F_{γ}, N_{γ,1} and N_{γ,2}. Let h_{ηˆ ⟨0⟩} be some extension of (h̄_{η} ↾ N_{γ,2}) ∘ F_{γ} to M_{γ+1} and let h_{ηˆ ⟨1⟩} be h̄_{η} ↾ M_{γ+1}. Let M_{γ+1}^{∗} be a model of cardinality μ extending M_{γ}^{∗} and containing h_{ηˆ ⟨0⟩}(M_{γ+1}) and h_{ηˆ ⟨1⟩}(M_{γ+1}) for each η ∈ ^{γ}2.

This completes the construction and for each η ∈ ^{χ}2 we can fix H_{η} ∈ Aut(ℭ) extending h_{η}.

Claim 4.7. η ≠ ν ∈ ^{χ}2 ⟹ ga-tp(H_{η}(ā)/M_{χ}^{∗}) ≠ ga-tp(H_{ν}(ā)/M_{χ}^{∗}).

Proof. Let ρ := η ∧ ν and suppose that ρˆ ⟨0⟩ < η and ρˆ ⟨1⟩ < ν. Let γ be the length of ρ. Suppose for the sake of contradiction that

ga-tp(H_{η}(ā)/M_{χ}^{∗}) = ga-tp(H_{ν}(ā)/M_{χ}^{∗}).

In particular,

ga-tp(H_{η}(ā)/H_{ρˆ ⟨0⟩}(N_{γ,1})) = ga-tp(H_{ν}(ā)/H_{ρˆ ⟨1⟩}(N_{γ,2})).

Referring back to our choice of h̄_{ρ} and h_{ρˆ ⟨0⟩}, we can apply H_{ν}^{−1} to obtain

ga-tp(H_{ν}^{−1} ∘ H_{η}(ā)/F_{γ}(N_{γ,1}) = ga-tp(ā/N_{γ,2}).

All we need to verify in order to contradict our original choices of N_{γ,1}, N_{γ,2} and F_{γ} is that for some extension of F_{γ} to an automorphism of ℭ we have that

(∗) ga-tp(F_{γ}(ā)/N_{γ,2}) = ga-tp(H_{ν}^{−1} ∘ H_{η}(ā)/N_{γ,2}).

Notice that by our choice of H_{ν} and H_{η} we have that H_{ν}^{−1}∘H_{η} ↾ N_{γ,1} = F_{γ}. Thus H_{ν}^{−1} ∘ H_{η} ⊇ F_{γ}, so by definition of the image of a type, F_{γ}(p ↾ N_{γ,1}) = ga-tp(H_{ν}^{−1}∘H_{η}(ā)/N_{γ,2}). Combining this with (∗) we obtain a contradiction to our choice of N_{γ,1}, N_{γ,2} and F_{γ} which witness the splitting of p over M_{γ}. ⊣

Thus Claim 4.7 gives us that | ga-S(M_{χ}^{∗})| ≥ 2^{χ} > μ and K is not β-Galois-stable in μ. ⊣

In Theorem 4.13 below we present an improvement of Fact 4.6 for tame AECs: In case K is β-stable in μ for some μ above its second Hanf number then κ_{μ}^{β}(K) is bounded by the second Hanf number. Notice that the bound does not depend on μ.

We will need to relate Galois-stability with the failure of a certain infinitary order property in order to find a bound for κ_{μ}^{β}(K). The order property, defined next, is an analog of the first order definition using formulas. This order property for non-elementary classes was introduced by Shelah in [Sh 394].

Definition 4.8. K is said to have the χ-order property provided that for every α, there exists ⟨d̄_{i} | i < α⟩ and where d̄_{i} ∈ ^{χ}ℭ such that if i_{0} < j_{0} < α and i_{1} < j_{1} < α,

(∗) then for no f ∈ Aut(ℭ) do we have f(d̄_{i_{0}} ˆ d̄_{j_{0}}) = d̄_{j_{1}} ˆ d̄_{i_{1}}.

Example 4.9. Let T be a first order theory in a countable language. Set K := Mod(T) and ≺_{K} the usual elementary submodel relation. K has the ℵ_{0}-order property iff T is unstable of T has dop or otop.

Remark 4.10 (Trivial monotonicity). Notice that for κ_{1} < κ_{2} if a class has the κ_{1}-order property then it has the κ_{2}-order property.

Fact 4.11 (Claim 4.6.3 of [Sh 394]). We may replace the phrase every α in Definition 4.8 with every α < ℶ_{(2^{χ+LS(K)})^{+}} and get an equivalent definition.

Fact 4.12 (Claim 4.8.2 of [Sh 394]). If K has the χ-order property and μ ≥ χ, then for some M ∈ K_{μ} we have that | ga-S^{χ}(M)/E_{χ}| ≥ μ^{+}. Moreover, we can conclude that K is not χ-Galois-stable in μ.

The following is a generalization of an old theorem (Theorem 4.17 in [GrLe2]) of Shelah from [Sh3]. A special case of this theorem is Theorem 0.1 in the abstract.

Theorem 4.13. Let β > 0. Suppose that K is χ-tame for β-types. If K is β-stable in μ with ℶ_{(2^{χ+LS(K)})^{+}} ≤ μ, then κ_{μ}^{β}(K) < ℶ_{(2^{χ+LS(K)})^{+}}.

Proof. Let χ′ := ℶ_{(2^{χ+LS(K)})^{+}}. Suppose that the conclusion of the theorem does not hold. Let ⟨M_{i} ∈ K_{μ} | i ≤ χ′⟩ and p ∈ ga-S^{β}(M_{χ′}) witness the failure. Namely, the following hold:

(1) ⟨M_{i} | i ≤ χ′⟩ is ≺_{K}-increasing and continuous,

(2) for every i < χ′, M_{i+1} is a (μ, θ)-limit over M_{i} for some θ < μ^{+} and (3) for every i < μ^{+}, p μ-splits over M_{i}. For every i < χ′ let f_{i}, N_{i}^{1} and N_{i}^{2} witness that p μ-splits over M_{i}. Namely,

M_{i} ≺_{K} N_{i}^{1}, N_{i}^{2} ≺_{K} M,
f_{i} : N_{i}^{1} ≅ N_{i}^{2} with f_{i} ↾ M_{i} = id_{M_{i}}
and f_{i}(p ↾ N_{i}^{1}) ≠ p ↾ N_{i}^{2}.

By χ-tameness, there exist models B_{i} and A_{i} := f_{i}^{−1}(B_{i}) of size < χ such that

f_{i}(p ↾ A_{i}) ≠ p ↾ B_{i}.

By renumbering our chain of models, we may assume that

(4) A_{i}, B_{i} ⊂ M_{i+1}. Since M_{i+1} is a limit model over M_{i}, we can additionally fix

(5) c̄_{i} ∈ M_{i+1} realizing p ↾ M_{i}. For each i < μ, let d̄_{i} := A_{i}ˆ B_{i} ˆ c̄_{i}.

Claim 4.14. ⟨d̄_{i} | i < χ′⟩ witnesses the χ-order property.

Proof. Suppose for the sake of contradiction that there exist g ∈ Aut(ℭ), i_{0} < j_{0} < χ′ and i_{1} < j_{1} < χ′ such that

g(d̄_{i_{0}} ˆ d̄_{j_{0}}) = d̄_{j_{1}} ˆ d̄_{i_{1}}.

Notice that since i_{0} < j_{0} < α we have that c̄_{i_{0}} ∈ M_{j_{0}}. So f_{j_{0}}(c̄_{i_{0}}) = c̄_{i_{0}}. Recall that f_{j_{0}}(A_{j_{0}}) = B_{j_{0}}. Thus, f_{j_{0}} witnesses that

(∗) ga-tp(c̄_{i_{0}}ˆ A_{j_{0}}/∅) = ga-tp(c̄_{i_{0}}ˆ B_{j_{0}}/∅).

Applying g to (∗) we get

(∗∗) ga-tp(c̄_{j_{1}}ˆ A_{i_{1}}/∅) = ga-tp(c̄_{j_{1}}ˆ B_{i_{1}}/∅).

Applying f_{i_{1}} to the RHS of (∗∗), we notice that

(♯) ga-tp(f_{i_{1}}(c̄_{j_{1}})ˆ B_{i_{1}}/∅) = ga-tp(c̄_{j_{1}}ˆ B_{i_{1}}/∅).

Because i_{1} < j_{1}, we have that c̄_{j_{1}} realizes p ↾ M_{i_{1}}. Thus, (♯) implies

f_{i_{1}}(p ↾ A_{i_{1}}) = p ↾ B_{i_{1}},

which contradicts our choice of f_{i_{1}}, A_{i_{1}} and B_{i_{1}}.

⊣

By Claim 4.11 and Fact 4.12, we have that K is β-unstable in μ, contradicting our hypothesis.

⊣

5. Morley sequences

We have now developed enough splitting machinery to derive the existence of Morley sequences for tame, Galois-stable AECs.

The following is a new Galois-theoretic notion of indiscernible sequence.

Definition 5.1. (1) ⟨ā_{i} | i < i^{∗}⟩ is a Galois-indiscernible sequence over M iff for every i_{1} < ⋯ < i_{n} < i^{∗} and every j_{1} < ⋯ < j_{n} < i^{∗}, ga-tp(ā_{i_{1}} … ā_{i_{n}}/M) = ga-tp(ā_{j_{1}} … ā_{j_{n}}/M).

(2) ⟨ā_{i} | i < i^{∗}⟩ is a Galois-indiscernible sequence over A iff for every i_{1} < ⋯ < i_{n} < i^{∗} and every j_{1} < ⋯ < j_{n} < i^{∗}, there exists M_{i}, M_{j}, M^{∗} ∈ K and ≺_{K}-mappings f_{i}, f_{j} such that

(a) A ⊆ M_{i}, M_{j};
(b) f_{ℓ} : M_{ℓ} → M^{∗}, for ℓ ∈ {i, j};
(c) f_{i}(ā_{i_{0}}, …, ā_{i_{n}}) = f_{j}(ā_{j_{0}}, …, ā_{j_{n}}) and
(d) and f_{i} ↾ A = f_{j} ↾ A = id_{A}.

Remark 5.2. This is on the surface a weaker notion of indiscernible sequence than is presented in [Sh 394]. However, under the amalgamation property, this definition and the definition in [Sh 394] are equivalent.

Lemma 5.3 provides us with sufficient conditions to find an indiscernible sequence.

Lemma 5.3. Let μ ≥ LS(K), κ, λ be ordinals and β a positive ordinal. Suppose that ⟨M_{i} | i < λ⟩ and ⟨ā_{i} | i < λ⟩ satisfy

(1) ⟨M_{i} ∈ K_{μ} | i < λ⟩ are ≺_{K}-increasing;

(2) M_{i+1} is a (μ, κ)-limit over M_{i};

(3) ā_{i} ∈ ^{β}M_{i+1};

(4) p_{i} := ga-tp(ā_{i}/M_{i}, M_{i+1}) does not μ-split over M_{0} and

(5) for i < j < λ, p_{i} ≤ p_{j}. Then, ⟨ā_{i} | i < λ⟩ is a Galois-indiscernible sequence over M_{0}.

Definition 5.4. A sequence ⟨ā_{i}, M_{i} | i < λ⟩ satisfying conditions (1) − (5) of Lemma 5.3 is called a Morley sequence. It is a Morley sequence for p ∈ ga-S(⋃_{i<λ} M_{i}) if p extends each ga-tp(ā_{i}/M_{i}). For A ⊆ M_{0}, such a sequence is a Morley sequence over A.

Remark 5.5. Notice that our definition of Morley sequence varies from some literature. An alternative name for our sequences (suggested by John Baldwin) is a coherent non-splitting sequence.

Remark 5.6. While the statement of the lemma is similar to Shelah’s first order Lemma I.2.5 in [Shc], the proof differs, since types are not sets of formulas.

Proof. We prove that for i_{0} < ⋯ < i_{n} < λ and j_{0} < ⋯ < j_{n} < λ, ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n+1}}) = ga-tp(ā_{j_{0}}, …, ā_{j_{n}}/M_{0}, M_{j_{n+1}}) by induction on n < ω.

n = 0: Let i_{0}, j_{0} < λ be given. Condition 5, gives us

ga-tp(ā_{i_{0}}/M_{0}, M_{i_{0}+1}) = ga-tp(ā_{j_{0}}/M_{0}, M_{j_{0}+1}).

n > 0: Suppose that the claim holds for all increasing sequences ī and j̄ ∈ λ of length n. Let i_{0} < ⋯ < i_{n} < λ and j_{0} < ⋯ < j_{n} < λ be given. Without loss of generality, i_{n} ≤ j_{n}. Define M^{∗} := M_{1}. From condition 2 and uniqueness of (μ, ω)-limits, we can find a ≺_{K}-isomorphism, g : M_{j_{n}} → M_{i_{n}} such that g ↾ M_{0} = id_{M_{0}}. Moreover we can extend g to g : M_{j_{n}+1} → M_{i_{n}+1}. Denote by b̄_{j_{l}} := g(ā_{j_{l}}) for l = 0, …, n. Notice that b_{j_{l}} ∈ M_{i_{n}} for l < n. Since ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(ā_{j_{0}}, …, ā_{j_{n}}/M_{0}, M_{j_{n}+1}) it suffices to prove that ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n}+1}).

Also notice that our ≺_{K}-mapping g preserves some properties of p_{j}. Namely, since p_{j} does not μ-split over M_{0}, g(p_{j} ↾ M_{j_{n}}) = p_{j} ↾ M_{i_{n}}. Thus, ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}) = ga-tp(ā_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}). In particular we have that ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}) does not μ-split over M_{0}.

By the induction hypothesis

ga-tp(b̄_{j_{0}}, …, b̄_{j_{n−1}}/M_{0}, M_{i_{n}}) = ga-tp(ā_{i_{0}}, …, ā_{i_{n−1}}/M_{0}, M_{i_{n}}).

Thus we can find h_{i} : M_{i_{n}+1} → M^{∗} and h_{j} : M_{i_{n}+1} → M^{∗} such that h_{i}(ā_{i_{0}}, …, ā_{i_{n−1}}) = h_{j}(b̄_{j_{0}}, …, b̄_{j_{n−1}}). Let us abbreviate b̄_{j_{0}}, …, b̄_{j_{n−1}} by b̄j̄. Similarly we will write āī for ā_{i_{0}}, …, ā_{i_{n−1}}.

By appealing to condition 4, we derive several equalities that will be useful in the latter portion of the proof. Since p_{j} does not μ-split over M_{0}, we have that p_{j} ↾ h_{j}(M_{i_{n}}) = h_{j}(p_{j} ↾ M_{i_{n}}), rewritten as

(∗) ga-tp(b̄_{j_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{j}(b̄_{j_{n}})/h_{j}(M_{i_{n}}), M^{∗}).

Similarly as p_{i} does not μ-split over M_{0}, we get p_{i} ↾ h_{j}(M_{i_{n}}) = h_{j}(p_{i} ↾ M_{i_{n}}) and p_{i} ↾ h_{i}(M_{i_{n}}) = h_{i}(p_{i} ↾ M_{i_{n}}). These equalities translate to

(∗∗)_{j} ga-tp(ā_{i_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}) and

(∗∗)_{i} ga-tp(ā_{i_{n}}/h_{i}(M_{i_{n}}), M_{i_{n}+1}) = ga-tp(h_{i}(ā_{i_{n}})/h_{i}(M_{i_{n}}), M^{∗}), respectively.

Finally, from condition 5, notice that

(∗ ∗ ∗) ga-tp(ā_{i_{n}}/M_{i_{n}}, M_{i_{n}+1}) = ga-tp(b̄_{j_{n}}/M_{i_{n}}, M_{i_{n}+1}).

Applying h_{j} to (∗ ∗ ∗) yields

(†) ga-tp(h_{j}(b̄_{j_{n}})/h_{j}(M_{i_{n}}), M^{∗}) = ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}).

Since h_{i}(āī) = h_{j}(b̄j̄) ∈ h_{j}(M_{i_{n}}), we can draw from (†) the following:

(1) ga-tp(h_{j}(b̄_{j_{n}})ˆ h_{j}(b̄j̄)/M_{0}, M^{∗}) = ga-tp(h_{j}(ā_{j_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}).

Similarly, equality (∗∗)_{i} allows us to see

(2) ga-tp(ā_{i_{n}}ˆ h_{i}(āī)/M_{0}, M^{∗}) = ga-tp(h_{i}(ā_{i_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}).

Since ga-tp(h_{j}(ā_{i_{n}})/h_{j}(M_{i_{n}}), M^{∗}) = ga-tp(ā_{i_{n}}/h_{j}(M_{i_{n}}), M_{i_{n}+1}) (equality (∗∗)_{j})) and h_{i}(āī) = h_{j}(b̄j̄) ∈ h_{j}(M_{i_{n}}), we get that

(3) ga-tp(h_{j}(ā_{i_{n}})ˆ h_{i}(āī)/M_{0}, M^{∗}) = ga-tp(ā_{i_{n}}ˆ h_{i}(āī)/M_{0}, M^{∗}).

Combining equalities (1), (2) and (3), we get

(††) ga-tp(h_{i}(āī)ˆ h_{i}(ā_{i_{n}})/M_{0}, M^{∗}) = ga-tp(h_{j}(b̄j̄)ˆ h_{j}(b̄_{j_{n}})/M_{0}, M^{∗}).

Recall that h_{i} ↾ M_{0} = h_{j} ↾ M_{0} = id_{M_{0}}. Thus (††), witnesses that

ga-tp(ā_{i_{0}}, …, ā_{i_{n}}/M_{0}, M_{i_{n}+1}) = ga-tp(b̄_{j_{0}}, …, b̄_{j_{n}}/M_{0}, M_{i_{n}+1}).
⊣

Theorem 5.7. Fix β > 0. Suppose K is χ-tame for β-types with χ < ℶ_{(2^{Hanf(K)})^{+}}. Suppose μ ≥ ℶ_{(2^{Hanf(K)})^{+}}. Let M ∈ K_{>μ}, A ⊂ M and I ⊆ ^{β}M be given such that |I| ≥ μ^{+} > |A|. If K is Galois β-stable in μ, then there exists J ⊂ I of cardinality μ^{+}, Galois indiscernible over A. Moreover J can be chosen to be a Morley sequence over A.

Proof. Fix κ = κ_{μ}^{β}. By Theorem 4.13, κ < ℶ_{(2^{Hanf(K)})^{+}}. Let {ā_{i} ∈ I | i < μ^{+}} be given. Define ⟨M_{i} ∈ K_{μ} | i < μ^{+}⟩, ≺_{K}-increasing and continuous, satisfying

(1) A ⊆ |M_{0}|

(2) M_{i+1} is a (μ, κ)-limit over M_{i}

(3) ā_{i} ∈ M_{i+1}

Let p_{i} := ga-tp(ā_{i}/M_{i}, M_{i+1}) for every i < μ^{+}. Define f : S_{κ}^{μ^{+}} → μ^{+} by

f(i) := min{j < μ^{+} | p_{i} does not μ-split over M_{j}}.

By our choice of κ, f is regressive. Thus by Fodor’s Lemma, there are a stationary set S ⊆ S_{κ}^{μ^{+}} and j_{0} < μ^{+} such that for every i ∈ S,

(†) p_{i} does not μ-split over M_{j_{0}}.

By stability and the pigeon-hole principle there exists p^{∗} ∈ ga-S(M_{j_{0}}) and S^{∗} ⊆ S of cardinality μ^{+} such that for every i ∈ S^{∗}, p^{∗} = p_{i} ↾ M_{j_{0}}. Let M^{∗} := M_{j_{0}}. Enumerate and ⟨M_{j_{i}} | i ∈ S^{∗}⟩. Let M^{∗∗} := M_{j_{1}} in this enumeration. Again, by stability we can find S^{∗∗} ⊂ S^{∗} of cardinality μ^{+} such that for every i ∈ S^{∗∗}, p^{∗∗} = p_{i} ↾ M^{∗}. Notice that M^{∗}, M^{∗∗} ≺_{K} M_{i} for every i ∈ S^{∗∗}.

Subclaim 5.8. For i < j ∈ S^{∗∗}, p_{i} = p_{j} ↾ M_{i}.

Proof. Let 0 < i < j ∈ S^{∗∗} be given. Since M_{i+1} and M_{j+1} are (μ, κ)- limits over M_{i}, there exists an isomorphism g : M_{j+1} → M_{i+1} such that g ↾ M_{i} = id_{M_{i}}. Let b̄_{j} := g(ā_{j}). Since the type p_{j} does not μ-split over M^{∗}, g cannot witness the splitting. Therefore, it must be the case that ga-tp(b̄_{j}/M_{i}, M_{i+1}) = p_{j} ↾ M_{i}. Then, it suffices to show that p_{i} = ga-tp(b̄_{j}/M_{i}, M_{i+1}).

Since p_{i} ↾ M^{∗} = p_{j} ↾ M^{∗}, we can find ≺_{K}-mappings witnessing the equality. Furthermore since M^{∗∗} is universal over M^{∗}, we can find h_{l} : M_{l+1} → M^{∗∗} such that h_{l} ↾ M^{∗} = id_{M^{∗}} for l = i, j and h_{i}(ā_{i}) = h_{j}(b̄_{j}).

We will use (†) to derive several inequalities. Consider the following possible witness to splitting. Let N_{1} := M_{i} and N_{2} := h_{i}(M_{i}). Since p_{i} does not μ-split over M^{∗}, we have that p_{i} ↾ N_{2} = h_{i}(p_{i} ↾ N_{1}), rewritten as

(∗) ga-tp(ā_{i}/h_{i}(M_{i}), M_{i+1}) = ga-tp(h_{i}(ā_{i})/h_{i}(M_{i}), M^{∗∗}).

Similarly we can conclude that

(∗∗) ga-tp(b̄_{j}/h_{j}(M_{i}), M_{i+1}) = ga-tp(h_{j}(b̄_{j})/h_{j}(M_{i}), M^{∗∗}).

By choice of S^{∗∗}, we know that

(∗ ∗ ∗) ga-tp(b̄_{j}/M^{∗∗}) = ga-tp(ā_{i}/M^{∗∗}).

Now let us consider another potential witness of splitting. N_{1}^{∗} := h_{i}(M_{i}) and N_{2}^{∗} := h_{j}(M_{i}) with H^{∗} := h_{j} ∘ h_{i}^{−1} : N_{1}^{∗} → N_{2}^{∗}. Since p_{j} ↾ M_{i} does not μ-split over M_{0}, p_{j} ↾ N_{2}^{∗} = H^{∗}(p_{j} ↾ N_{1}^{∗}). Thus by (∗∗) we have

(♯) H^{∗}(p_{j} ↾ N_{1}^{∗}) = ga-tp(h_{j}(b̄_{j})/h_{j}(M_{i}), M^{∗∗}).

Now let us translate H^{∗}(p_{j} ↾ N_{1}^{∗}). By monotonicity and (∗ ∗ ∗), we have that p_{j} ↾ N_{1}^{∗} = ga-tp(b̄_{j}/h_{i}(M_{i}), M_{i+1}) = ga-tp(ā_{i}/h_{i}(M_{i}), M_{i+1}). We can then conclude by (∗) that p_{j} ↾ N_{1}^{∗} = ga-tp(h_{i}(ā_{i})/h_{i}(M_{i}), M_{i+1}). Applying H^{∗} to this equality yields

(♯♯) H^{∗}(p_{j} ↾ N_{1}^{∗}) = ga-tp(h_{j}(ā_{i})/h_{j}(M_{i}), M^{∗∗)}.

By combining the equalities from (♯) and (♯♯) and applying h_{j}^{−1} we get that

ga-tp(b̄_{j}/M_{i}, M_{i+1}) = ga-tp(ā_{i}/M_{i}, M_{i+1}).
⊣

Notice that by Subclaim 5.8 and our choice of S^{∗∗}, ⟨M_{i} | i ∈ S^{∗∗}⟩ and J := ⟨ā_{i} | i ∈ S^{∗∗}⟩ satisfy the conditions of Lemma 5.3. Applying Lemma 5.3, we get that ⟨ā_{i} | i ∈ S^{∗∗}⟩ is a Morley sequence over M^{∗}. In particular, since A ⊂ M^{∗}, we have that ⟨ā_{i} | i ∈ S^{∗∗}⟩ is a Morley sequence over A.

⊣

Remark 5.9. Our effort to prove the results of this section in summer 2001 lead us to the notion of tameness as defined in this paper. A preliminary version of this paper containing the results of this section was posted by us on the web already in July 27th, 2001 with the title Morley Sequences in AECs.

6. Towards a Stability Spectrum Theorem

In this section we begin work towards a stability spectrum theorem for tame classes. Other partial results in this direction appear in [BaKuVa].

Theorem 6.1 (Extension property for non-μ-splitting types). Let K be an AEC. Suppose N ≺_{K} M′ ≺_{K} M are such that M′ is universal over N and all three models have cardinality μ. Then for every p′ ∈ ga-S(M′) which does not μ-split over N, there exists p ∈ ga-S(M) extending p′ such that p does not μ-split over N.

Proof. Let N, M′, M be as in the statement of the theorem. Suppose p = ga-tp(a′/M′) does not μ-split over N. Since M′ is universal over N, there exists f : M′ → M with f ↾ N = id_{N}. Notice that by monotonicity of nonμ-splitting, we have that ga-tp(a′/f(M)) does not μ-split over N. Then, by invariance we have that also

(∗) ga-tp(f(a′)/M) does not μ-split over N.

By one more appeal to non-splitting, we can conclude that ga-tp(f(a′)/M′) = ga-tp(a′/M′). Otherwise these two types would contradict (∗). ⊣

Theorem 6.2 (Uniqueness of non-splitting extensions). Let K be χ-tame abstract elementary class and μ a cardinal with μ ≥ χ. Let M ∈ K_{>μ} and M′, N ∈ K_{μ} with N ≺_{K} M′ ≺_{K} M. If M′ is universal over N, then for every p′ ∈ ga-S(M′) which does not μ-split over N, if there are q, p ∈ ga-S(M) such that q and p both extend p′ and do not μ-split over N, then p = q.

Proof. Suppose for the sake of contradiction that there exists q ≠ p ∈ ga-S(M) extending p′ ∈ ga-S(M′) such that both p and q do not μ-split over N. Let a, b be such that p = ga-tp(a/M) and q = ga-tp(b/M). By tameness there exists M^{∗} ∈ K_{μ} such that M′ ≺_{K} M^{∗} ≺_{K} M and p ↾ M^{∗} ≠ q ↾ M^{∗}.

Since M′ is universal over N, there exists a ≺_{K}-mapping f : M^{∗} → M′ with f ↾ N = id_{N}. Since p and q do not μ-split over N we have

(∗)_{a} ga-tp(a/f(M^{∗})) = ga-tp(f(a)/f(M^{∗})) and
(∗)_{b} ga-tp(b/f(M^{∗})) = ga-tp(f(b)/f(M^{∗})).

On the other hand, since p ↾ M^{∗} ≠ q ↾ M^{∗}, we have that their images are also not equal

(∗) ga-tp(f(a)/f(M^{∗})) ≠ ga-tp(f(b)/f(M^{∗})).

Combining (∗)_{a}, (∗)_{b} and (∗), we get

ga-tp(a/f(M^{∗})) ≠ ga-tp(b/f(M^{∗})).

Since f(M^{∗}) ≺_{K} M′, this inequality witnesses that

ga-tp(a/M′) ≠ ga-tp(b/M′),

contradicting our choice of p and q both extending p′.

⊣

Corollary 6.3. Let K be χ-tame abstract elementary class and let μ be a cardinal with μ > χ. Let M ∈ K_{>μ}, N ≺_{K} M with M universal over N. If K is Galois-stable in μ then

|{p ∈ ga-S(M) : p does not μ-split over N}| ≤ μ.

Proof. Since M is universal over N, there is a M′ ∈ K_{μ} with N ≺_{K} M′ ≺_{K} M and M′ universal over N. By μ-stability there are only μ-many types over M′ and by Theorem 6.2 each type over N has at most one non-μ-splitting extension to M. ⊣

Corollary 6.4 (Partial stability spectrum). Suppose that K is a tame abstract elementary class satisfying the amalgamation property. If K is Galoisstable in some μ > Hanf(K), then K is stable in every κ with κ^{μ} = κ.

Proof. Let κ = κ^{μ}. Consider M ∈ K_{κ} and let M′ be a μ-saturated extension of M. Such an extension exists by stability in μ and our cardinal arithmetic assumption. Suppose that ga-tp(a/M, ℭ) ≠ ga-tp(b/M, ℭ), then surely ga-tp(a/M′, ℭ) ≠ ga-tp(b/M′, ℭ). So the number of Galois-types over M is ≤ the number of Galois-types over M′.

Thus we may assume that M is μ-saturated. Again by our assumption that κ^{μ} = κ, we can enumerate ⟨N_{i} | i < κ⟩ all the ≺_{K}-submodels of M of cardinality μ. Since M is μ-saturated, for each one of the N_{i} there is M′_{i} of cardinality μ such that N_{i} ≺_{K} M′_{i} ≺_{K} M with M′_{i} universal over N_{i}. By Fact 4.6 for every p ∈ ga-S(M), there exists N_{i} ≺_{K} M of cardinality μ such that p does not μ-split over N_{i}. Thus

| ga-S(M)| = | ⋃_{i<κ}{p ∈ ga-S(M) | p does not μ-split over N_{i}}|.

By monotonicity of non-splitting we have that for each p ∈ ga-S(M) that does not μ-split over N_{i}, also p ↾ M′_{i} does not μ-split over N_{i}. By Corollary 6.3, for each i < κ,

|{p ∈ ga-S(M) | p does not μ-split over N_{i}}|
≤ |{p ∈ ga-S(M′_{i}) | p does not μ-split over N_{i}}|.

Now by stability in μ and our assumption that κ^{μ} = κ, we can conclude that | ga-S(M)| ≤ κ.

⊣

7. Exercise on Dividing

With the existence of Morley sequences a natural extension is to study the following dependence relation to determine whether or not it satisfies properties such as transitivity, symmetry or extension. Here we derive the existence property for types over saturated models.

Definition 7.1. Let p ∈ ga-S(M) and N ≺_{K} M. We say that p divides over N iff there are a ∈ |M|\|N| and a Morley sequence, {a_{n} | n < ω} for the ga-tp(a/N, M) such that for some collection {f_{n} ∈ Aut_{M}ℭ | n < ω} with f_{n}(a) = a_{n} we have

{f_{n}(p) | n < ω} is inconsistent.

I.e. there is no common realization to the above mentioned ω-many types.

Theorem 7.2 (Existence). Suppose that K is Galois-stable in μ and χ-tame for some χ < μ. For every μ-model homogeneous model M ∈ K_{≥μ} and ever p ∈ ga-S(M), there exists N ≺_{K} M of cardinality μ such that p does not divide over N.

Proof. Suppose that p and M form a counter-example. We will find N_{i}, N_{i}^{1}, N_{i}^{2} and h_{i} for i < μ satisfying:

(1) ⟨N_{i} ∈ K_{μ} | i ≤ μ⟩ is a ≺_{K}-increasing and continuous sequence of models;

(2) N_{i} ≺_{K} N_{i}^{l} ≺_{K} N_{i+1} for i < μ and l = 1, 2;

(3) for i < μ, h_{i} : N_{i}^{1} ≅ N_{i}^{2} and h_{i} ↾ N_{i} = id_{N_{i}} and

(4) p ↾ N_{i}^{2} ≠ h_{i}(p ↾ N_{i}^{1}). Suppose that N_{i} has been defined. Since p divides over every substructure

of cardinality μ, we may find ā, {ā_{n} | n < ω} and {f_{n} | n < ω} witnessing that p divides over N_{i}. Namely, we have that {f_{n}(p) | n < ω} is inconsistent. Let n < ω be such that f_{0}(p) ≠ f_{n}(p). Then p ≠ f_{0}^{−1} ∘ f_{n}(p). By χtameness, we can find N^{∗} ≺_{K} M of cardinality μ containing N such that p ↾ N^{∗} ≠ (f_{0}^{−1} ∘ f_{n}(p)) ↾ N^{∗}. WLOG f_{0}^{−1} ∘ f_{n} ∈ Aut_{N}N^{∗}.

Let h_{i} := f_{0}^{−1} ∘ f_{n}, N_{i}^{1} := N^{∗} and N_{i}^{2} := N^{∗}. Choose N_{i+1} ≺_{K} M to be an extension of N^{∗} of cardinality μ.

Now we can use ⟨N_{i}, N_{i}^{1}, N_{i}^{2}, h_{i} | i < μ⟩ to contradict Galois-stability in μ by repeating the argument from the proof of Fact 4.6, (starting at the second paragraph) we construct an increasing chain of models and a tree of K-embedding which by Claim 4.7 gives many Galois-types contradicting stability in μ. ⊣

Remark 7.3. In a previous version of this paper, we stated for the existence property for arbitrary models M ∈ K_{≥μ}. We do not know how to remove the assumption of μ-model homogeneity and we suspect that this is not possible with only the assumption of μ-stability and tameness.

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E-mail address, Rami Grossberg: rami@andrew.cmu.edu

Department of Mathematics, Carnegie Mellon University, Pittsburgh PA 15213

E-mail address, Monica VanDieren: mvd@umich.edu

Department of Mathematics, University of Michigan, Ann Arbor MI 48109-