Tameness from two successive good frames
Sebastien Vasey
math.LO 03C48 (Primary), 03C45, 03C52, 03C55, 03C75 (Secondary)



  We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality λ  and a superstable-like forking notion for models of cardinality λ^+, then orbital types over models of cardinality λ^+ are determined by their restrictions to submodels of cardinality λ. By a superstable-like forking notion, we mean here a good frame, a central concept of Shelah's book on AECs. 

  It is known that locality of orbital types together with the existence of a superstable-like notion for models of cardinality λ implies the existence of a superstable-like notion for models of cardinality λ^+, and here we prove the converse. An immediate consequence is that forking in λ^+ can be described in terms of forking in λ.


[
    L. N. Pfeiffer
    December 30, 2023
=====================







§ INTRODUCTION


Good frames are the central notion of Shelah's two volume book <cit.> on classification theory for abstract elementary classes (AECs). Roughly speaking, an AEC is a concrete category (whose objects are structures) satisfying several axioms (for example, morphisms must be monomorphisms and the class must be closed under directed colimits). It generalizes the notion of an elementary class (i.e. a class axiomatized by an _ω, ω-theory, where the morphisms are elementary embeddings) and also encompasses many infinitary logics such as _∞, ω (i.e. disjunctions and conjunctions of arbitrary, possibly infinite, length are allowed). An AEC  has a good λ-frame if its restriction to models of cardinality λ is reasonably well-behaved (e.g. has amalgamation, no maximal models, and is stable) and it admits an abstract notion of independence (for orbital types of elements over models of cardinality λ) that satisfies some of the basic properties of forking in a superstable elementary class: monotonicity, existence, uniqueness, symmetry, and local character. Here, local character is described not as “every type does not fork over a finite set” but as “every type over the union of an increasing continuous chain of models of cardinality λ does not fork over some member of the chain”. The theory of good frames is used heavily in several recent results of the classification theory of AECs, including the author's proof of Shelah's eventual categoricity conjecture in universal classes <cit.>, see also <cit.> for a survey.

The reason for restricting oneself to models of cardinality λ is that the compactness theorem fails in general AECs, and so it is much easier in practice to exhibit a local notion of forking than it is to define forking globally for models of all sizes. In fact, Shelah's program is to start with a good λ-frame and only then try to extend it to models of bigger sizes. For this purpose, he describes a dividing line[The nonstructure side is described by <cit.>, showing that failure of successfulness implies the AEC has many non-isomorphic models.], being successful, and shows that if a good λ-frame is successful, then there is a good λ^+-frame on an appropriate subclass of _λ^+.

A related approach is to outright assume some weak amount of compactness. Tameness <cit.> was proposed by Grossberg and VanDieren to that end: λ-tameness says that orbital types are determined by their restrictions of cardinality λ. This is a nontrivial assumption, since in AECs syntactic types are not as well-behaved as one might wish, so one defines types purely semantically (roughly, as the finest notion of type preserving isomorphisms and the -substructure relation). It is known that tameness follows from a large cardinal axiom (see Fact <ref>) and some amount of it can be derived from categoricity (see Fact <ref>). The present paper gives another way to derive some tameness.

Grossberg conjectured in 2006 that[See the introduction of <cit.> for a detailed history.], assuming amalgamation in λ^+, a good λ-frame extends to a good λ^+-frame if the class is λ-tame. He told his conjecture to Jarden who could prove all the axioms of a good λ^+-frame, except symmetry. Boney <cit.> then proved symmetry assuming a slightly stronger version of tameness, Jarden <cit.> proved symmetry from a certain continuity assumption, and Boney and the author finally settled the full conjecture <cit.>, see Fact <ref>. In this context, forking in the good λ^+-frame can be described in terms of forking in the good λ-frame. Let us call this result the upward frame transfer theorem.

This paper discusses the converse of the upward frame transfer theorem. Consider the following question: if there is a good λ-frame and a good λ^+-frame, can we say anything on how the two frames are related (i.e. can forking in λ^+ be described in terms of forking in λ?) and can we conclude some amount of tameness? We answer positively by proving the following converse to the upward frame transfer theorem:

Corollary <ref>.
Let  be an AEC and let λ≥ (). Assume 2^λ < 2^λ^+. If there is a categorical good λ-frame  on _λ and a good λ^+-frame  on _λ^+, then  is (λ, λ^+)-tame. Moreover, _λ^+λ^+_λ^+ = λ^+_λ^+ (see Definition <ref>).

In the author's opinion, Corollary <ref> is quite a surprising result since it shows that we cannot really study superstability in λ and λ^+ “independently”: the two levels must in some sense be connected (this follows from a canonicity result for good frames, see the paragraph after next). Put another way, two successive local instances of superstability already give a nontrivial amount of compactness. In fact after the initial submission of this article, Corollary <ref> was used by the author <cit.> as a key tool to prove a class has some amount of tameness and deduce strong versions of Shelah's eventual categoricity conjecture in several types of AECs.

In Corollary <ref>, “categorical” simply means that  is assumed to be categorical in λ. We see it as a very mild assumption, since we can usually restrict to a subclass of saturated models if this is not the case, see the discussion after Definition <ref>. As for (λ, λ^+)-tameness, it means that types over models of cardinality λ^+ are determined by their restrictions of cardinality λ. In fact, it is possible to obtain a related conclusion by starting with a good λ^+-frame on the class of saturated models in  of cardinality λ^+. In this case, we deduce that  is (λ, λ^+)-weakly tame, i.e. only types over saturated models of cardinality λ^+ are determined by their restrictions to submodels of cardinality λ. We deduce that weak tameness is equivalent to the existence of a good λ^+-frame on the saturated models of cardinality λ^+, see Corollary <ref>.

An immediate consequence of Corollary <ref> is that forking in λ^+ (at least over saturated models) can be described in terms of forking in λ. Indeed, the upward frame extension theorem gives a good λ^+-frame with such a property, and good frames on subclass of saturated models are canonical: there can be at most one, see Fact <ref>. In fact, assuming that forking in λ^+ is determined by forking in λ is equivalent to tameness (see <cit.>) because of the uniqueness and local character properties of forking. In Corollary <ref> we of course do not start with such an assumption: forking in λ^+ is any abstract notion satisfying some superstable-like properties for models of cardinality λ^+.

The proof of Corollary <ref> goes as follows: we use 2^λ < 2^λ^+ to derive that the good λ-frame is weakly successful (a dividing line introduced by Shelah in Chapter II of <cit.>). This is the only place where 2^λ < 2^λ^+ is used. Being weakly successful imply that we can extend the good λ-frame from types of singletons to types of models of cardinality λ. We then have to show that the good λ-frame is also successful. This is equivalent to a certain reflecting down property of nonforking of models. Jarden <cit.> has shown that successfulness follows from (λ, λ^+)-weak tameness and amalgamation in λ^+, and here we push Jarden's argument further by showing that having a good λ^+-frame suffices, see Theorem <ref>. A key issue that we constantly deal with is the question of whether a union of saturated models of cardinality λ^+ is saturated. In Section <ref>, we introduce a new property of forking, being decent, which characterizes a positive answer to this question and sheds further light on recent work of VanDieren <cit.>. The author believes it has independent interest.

Tameness has been used by Grossberg and VanDieren to prove an upward categoricity transfer from categoricity in two successive cardinals <cit.>. In Section <ref>, we revisit this result and show that tameness is in some sense needed to prove it. Although this could have been derived from the results of Shelah's books, this seems not to have been noticed before. Nevertheless, the results of this paper show that if an AEC is categorical λ and λ^+ and has a good frame in both λ and λ^+, then it is categorical in λ^++, see Corollary <ref>.

To read this paper, the reader should preferably have a solid knowledge of good frames, including knowing Chapter II of <cit.>, <cit.>, as well as <cit.>. Still, we have tried to give all the definitions and relevant background facts in Section <ref>.

The author thanks John T. Baldwin, Will Boney, Rami Grossberg, Adi Jarden, Marcos Mazari-Armida, and the referee for comments that helped improve the quality of this paper.



§ PRELIMINARIES




 §.§ Notational conventions


Given a structure M, write write |M| for its universe and M for the cardinality of its universe. We often do not distinguish between M and |M|, writing e.g. a ∈ M or ∈<αM instead of a ∈ |M| and ∈<α|M|. We write M ⊆ N to mean that M is a substructure of N.



 §.§ Abstract elementary classes

An abstract class is a pair = (K, ), where K is a class of structures in a fixed vocabulary τ = τ () and  is a partial order, M  N implies M ⊆ N, and both K and  respect isomorphisms (the definition is due to Grossberg). Any abstract class admits a notion of -embedding: these are functions f: M → N such that f: M ≅ f[M] and f[M]  N.

We often do not distinguish between K and . For λ a cardinal, we will write _λ for the restriction of  to models of cardinality λ. Similarly define _≥λ or more generally _S, where S is a class of cardinals. We will also use the following notation:


  For  an abstract class and N ∈, write (N) for the set of M ∈ with M  N. Similarly define λ(N), <λ(N), etc.


For an abstract class , we denote by () the number of models in  up to isomorphism (i.e. the cardinality of /_≅). We write (, λ) instead of (_λ). When () = 1, we say that  is categorical. We say that  is categorical in λ if _λ is categorical, i.e. (, λ) = 1.

We say that  has amalgamation if for any M_0  M_ℓ, ℓ = 1,2 there is M_3 ∈ and -embeddings f_ℓ : M_ℓ→ M_3, ℓ = 1,2.  has joint embedding if any two models can be -embedded in a common model.  has no maximal models if for any M ∈ there exists N ∈ with M  N and M ≠ N (we write M  N). Localized concepts such as amalgamation in λ mean that _λ has amalgamation.

The definition of an abstract elementary class is due to Shelah <cit.>:


  An abstract elementary class (AEC) is an abstract class  in a finitary vocabulary satisfying:

  
  
  * Coherence: if M_0, M_1, M_2 ∈, M_0 ⊆ M_1  M_2 and M_0  M_2, then M_0  M_1.
  
  * Tarski-Vaught chain axioms: if M_i : i ∈ I is a -directed system and M := ⋃_i ∈ I M_i, then:
    
    
  * M ∈.
    
  * M_i  M for all i ∈ I.
    
  * Smoothness: if N ∈ is such that M_i  N for all i ∈ I, then M  N.
    
  
  * Löwenheim-Skolem-Tarski (LST) axiom: there exists a cardinal λ≥ |τ ()| + ℵ_0 such that for any N ∈ and any A ⊆ |N|, there exists M ∈≤ (|A| + λ) (N) with A ⊆ |M|. We write () for the least such λ.
  

  We say that an abstract class  is an AEC in λ if  satisfies coherence, = _λ, λ≥ |τ ()|, and  satisfies the Tarski-Vaught chain axioms whenever |I| ≤λ.


  If  is an AEC in λ, then by <cit.> there exists a unique AEC ^∗ such that _λ = _λ^∗, _<λ^∗ = ∅, and (^∗) = λ. We call ^∗ the AEC generated by .




 §.§ Types


In any abstract class , we can define a semantic notion of type, called Galois or orbital types in the literature (such types were introduced by Shelah in <cit.>). For M ∈, A ⊆ |M|, and ∈<∞M, we write _ ( / A; M) for the orbital type of  over A as computed in M (usually  will be clear from context and we will omit it from the notation). It is the finest notion of type respecting -embeddings, see <cit.> for a formal definition. When  is an elementary class, ( / A; M) contains the same information as the usual notion of _ω, ω-syntactic type, but in general the two notions need not coincide <cit.>.

The length of ( / A; M) is the length of . For M ∈ and α a cardinal, we write _^α (M) = ^α (M) for the set of types over M of length α. Similarly define ^<α (M). When α = 1, we just write (M). We define naturally what it means for a type to be realized inside a model, to extend another type, and to take the image of a type by a -embedding.



 §.§ Stability and saturation


We say that an abstract class , is stable in λ (for λ an infinite cardinal) if | (M)| ≤λ for any M ∈_λ. If  is an AEC, λ≥ (),  is stable in λ and  has amalgamation in λ, then we will often use without comments the existence of universal extension <cit.>: for any M ∈_λ, there exists N ∈_λ universal over M. This means that M  N and any extension of M of cardinality λ -embeds into N over M.

For  an AEC and λ >  (), a model N ∈ is called λ-saturated if for any M ∈_<λ with M  N, any p ∈ (M) is realized in N. N is called saturated if it is N-saturated.

We will often use without mention the model-homogeneous = saturated lemma <cit.>: it says that when _<λ has amalgamation, a model N ∈ is λ-saturated if and only if it is λ-model-homogeneous. The latter means that for any M ∈<λ (N), any M' ∈_<λ -extending M can be -embedded into N over M. In particular, assuming amalgamation and joint embedding, there is at most one saturated model of a given cardinality. We write λ for the abstract class of λ-saturated models in  (ordered by the appropriate restriction of ).



 §.§ Frames


Roughly, a frame consists of a class of models of the same cardinality together with an abstract notion of nonforking. The idea is that if the frame is “sufficiently nice”, then it is possible to extend it to cover bigger models as well. This is the approach of Shelah's book <cit.>, where good frames were introduced. There the abstract notion of nonforking is required to satisfy some of the basic properties of forking in a superstable elementary class. We redefine here the definition of a frame (called pre-frame in <cit.>, <cit.>). We give a slightly different definition, as we do not include certain monotonicity axioms as part of the definition. Shelah assumes that nonforking is only defined for a certain class of types he calls the basic types. For generality, we will work in the same setup (except that we define the basic types from the nonforking relation), but the reader will not lose much by pretending that all types are basic (In Shelah's terminology, that the frame is type-full). Note that any weakly successful good frame can be extended to a type-full one, see <cit.>.


  Let λ be an infinite cardinal, and α≤λ^+ be a non-zero cardinal. A (<α, λ)-frame consists of a pair = (, ), where:

  
  
  *  is an abstract class with = _λ.
  
  *  is a 4-ary relation on pairs (, M_0, M, N), where M_0  M  N and ∈<αN. We write M_0MN instead of (, M_0, M, N).
  
  *  respects -embeddings: if f: N → N' is a -embedding and M_0  M  N, ∈<αN, then M_0MN if and only if f[M_0]f ()f[M]N'.
  

  We may write _ = (K_, ) for  and _ for . We say that  is on ^∗ if _ = ^∗. A (≤β, λ)-frame (for β≤λ) is a (<β^+, λ)-frame. A λ-frame is a (≤ 1, λ)-frame.


  We say a (<α, λ)-frame  extends a (<β, λ)-frame  if β≤α, _ = _, and for M_0  M  N and ∈<βN, _ (M_0, , M, N) if and only if _ (M_0, , M, N).


Since  respects -embeddings, it respects types. Therefore we can define:


  Let  be a (<α,λ)-frame. For M_0  M and p ∈^<α (M), we say that p does not -fork over M_0 if M_0MN whenever p =  ( / M; N). When  is clear from context, we omit it and just say that p does not fork over M_0.


We will call a type basic if it interacts with the nonforking relation, i.e. if it does not fork over a -substructure of its base. Note that basic types are respected by -embeddings (by the definition of a frame).


  Let  be a (<α, λ)-frame. For M ∈_, a type p ∈^<α (M) will be called -basic (or just basic when  is clear from context) if there exists M_0  M such that p does not -fork over M_0. We say that  is type-full if for any M ∈_, every type in ^<α (M) is basic.

  

We will often consider frames whose underlying class is categorical:


  We say that a (<α, λ)-frame  is categorical if _ is categorical (i.e. it contains a single model up to isomorphism).


We will consider the following properties that forking may have in a frame:


  Let  be a (<α, λ)-frame.

  
  
  *  has non-order if whenever M_0  M  N, , ∈<αM and A :=   () =  (), then M_0MN if and only if M_0MN. In this case we will write M_0AMN.
  
  *  has monotonicity if whenever M_0  M_0'  M  N  N', ∈<αN', I ⊆ (), and M_0NN', we have that M_0' IMN'.
  
  *  has density of basic types if whenever M  N and β < α, there exists ∈βN \ M such that ( / M; N) is basic.
  
  *  has disjointness if M_0MN and ∈<αM imply ∈<αM_0.
  
  *  has existence if whenever M  N, any basic p ∈^<α (M) has a nonforking extension to ^<α (N).
  
  *  has uniqueness if whenever M  N, if p, q ∈^<α (N) both do not fork over M and p  M = q  M, then p = q.
  
  *  has local character if for any β≤min (α, λ), any limit ordinal δ < λ^+ with δ≥β, any -increasing continuous chain M_i : i ≤δ, and any basic p ∈^<β (M_δ), there exists i < δ such that p does not fork over M_i.
  
  *  has symmetry if the following are equivalent for any M  N, , ∈<αN such that ( / M; N) and ( / M; N) are both basic:
    
    
    
  * There exists M_, N_ such that N  N_, M  M_ N_, ∈<αM_, and ( / M_; N_) does not fork over M.
    
  * There exists M_, N_ such that N  N_, M  M_ N_, ∈<αM_, and ( / M_; N_) does not fork over M.
    
  
  * When α = λ^+,  has long transitivity if it has non-order and whenever γ < λ^+ is an ordinal (not necessarily limit), M_i : i ≤γ, N_i : i ≤γ are -increasing continuous, and M_iN_iM_i + 1N_i + 1 for all i < γ, we have that M_0N_0M_γN_γ.
    


Shelah's definition of a good frame (for types of length one) says that a frame must have all the properties above and its underlying class must be “reasonable” <cit.>. The prototypical example is the class of models of cardinality λ of a superstable elementary class which is stable in λ. Taking this class with nonforking gives a good λ-frame (even a good (≤λ, λ)-frame). We use the definition from <cit.> (we omit the continuity property since it follows, see <cit.>). We add the long transitivity property from <cit.> when the types have length λ.


  We say a (<α, λ)-frame  is good if:

  
  
  * _ is an AEC in λ (see the end of Definition <ref>), and _≠∅. Moreover _ is stable in λ, has amalgamation, joint embedding, and no maximal models.
  
  *  has non-order, monotonicity, density of basic types, disjointness, existence, uniqueness, local character, symmetry, and (when α = λ^+) long transitivity.
  


  We also call the AEC generated by _ (see Remark <ref>) the AEC generated by .


We will use the following conjugation property of good frames at a crucial point in the proof of Theorem <ref>:

[III.1.21 in <cit.>]
  Let  be a categorical good λ-frame (see Definition <ref>). Let M  N and let p ∈ (N). If p does not fork over M, then p and p  M are conjugate. That is, there exists an isomorphism f: N ≅ M such that f (p) = p  M.



  The results of this paper also carry over (with essentially the same proofs) in the slightly weaker framework of semi-good λ-frames introduced in <cit.>, where only “almost stability” (i.e. | (M)| ≤λ^+ for all M ∈_λ) and the conjugation property are assumed. For example, in Corollary <ref> we can assume only that there is a semi-good λ-frame on _λ with conjugation (but we still should assume there is a good λ^+-frame on λ^+_λ^+).


Several sufficient conditions for the existence of a good frame are known. Assuming GCH, Shelah showed that the existence of a good λ-frame follows from categoricity in λ, λ^+, and a medium number of models in λ^++ (see Fact <ref>). Good frames can also be built using a small amount of tameness  that follows from amalgamation, no maximal models, and categoricity in a sufficiently high cardinal (see Section <ref> and Fact <ref>).

Note that on a categorical good λ-frame, there is only one possible notion of nonforking with the required properties. In fact, nonforking can be given an explicit description, see <cit.>. We will use this without comments:

[Canonicity of categorical good frames]
  If  and  are categorical good λ-frames with the same basic types and _ = _, then =.




 §.§ Superlimits


As has been done in several recent papers (e.g. <cit.>), we have dropped the requirement that  has a superlimit in λ from Shelah's definition of a good frame:

[I.3.3 in <cit.>]
  Let  be an AEC and let λ≥ (). A model N is superlimit in λ if:

  
  
  * N ∈_λ and N has a proper extension.
  
  * N is universal: any M ∈_λ -embeds into N.
  
  * For any limit ordinal δ < λ^+ and any increasing chain N_i : i < δ, if N ≅ N_i for all i < δ, then N ≅⋃_i < δ N_i.
  

  We say that M is superlimit if it is superlimit in M.


Again, for a prototypical example consider a superstable elementary class which is stable in a cardinal λ and let M be a saturated model of cardinality λ. Then because unions of chains of λ-saturated are λ-saturated, M is superlimit in λ. In fact, an elementary class has a superlimit in some high-enough cardinal if and only if it is superstable <cit.>.

There are no known examples of good λ-frames that do not have a superlimit in λ. In fact most constructions of good λ-frames give one, see for example <cit.>. When a good λ-frame  has a superlimit, we can restrict  to the AEC generated by this superlimit (i.e the unique AEC ^∗ such that _λ consists of isomorphic copies of the superlimit and is ordered by the appropriate restriction of ) and obtain a new good frame that will be categorical in the sense of Definition <ref>. Thus in this paper we will often assume that the good frame is categorical to start with.

When one has a good λ-frame, it is natural to ask whether the frame can be extended to a good λ^+-frame. It turns out that the behavior of the saturated model in λ^+ can be crucial for this purpose. Note that amalgamation in λ and stability in λ indeed imply that there is a unique, universal, saturated model in λ^+ (amalgamation in λ^+ is not needed for this purpose). It is also known that there are no maximal models in λ^+ (see <cit.> or <cit.>). A key property is whether union of chains of saturated models of cardinality λ^+ are saturated. In fact, it is easy to see that this is equivalent to the existence of a superlimit in λ^+. We will use it without comments and leave the proof to the reader:


  Let  be a good λ-frame generating an AEC . The following are equivalent:

  
  
  *  has a superlimit in λ^+.
  
  * The saturated model in  of cardinality λ^+ is superlimit.
  
  * For any limit δ < λ^++ and any increasing chain M_i : i < δ of saturated model in _λ^+, ⋃_i < δ M_i is saturated.
  




 §.§ Tameness


In an elementary class, types coincides with sets of formulas so are in particular determined by their restrictions to small subsets of their domain. One may be interested in studying AECs where types have a similar behavior. Such a property is called tameness. Tameness was extracted from an argument of Shelah <cit.> and made into a definition by Grossberg and VanDieren <cit.> who used it to prove an upward categoricity transfer theorem <cit.>. Here we present the definitions we will use, and a few sufficient conditions for tameness (given as motivation but not used in this paper). We refer the reader to the survey of Boney and the author <cit.> for more on tame AECs.


  For an abstract class , a class of types Γ, and an infinite cardinal χ, we say that  is (<χ)-tame for Γ if for any p, q ∈Γ over the same set B, p  A = q  A for all A ⊆ B with |A| < χ implies that p = q. We say that  is χ-tame for Γ if it is (<χ^+)-tame for Γ.


We will use the following variation from <cit.>:


  An AEC  is (<χ, λ)-weakly tame if  is (<χ)-tame for the class of types of length one over saturated models of cardinality λ. When we omit the weakly, we mean that  is (<χ)-tame for the class of types of length one over any model of cardinality λ. Define similarly variations such as (χ, <λ)-weakly tame, or χ-tame (which means (χ, λ)-tame for all λ≥χ).


A consequence of the compactness theorem is that any elementary class is (<ℵ_0)-tame (for any class of types). Boney <cit.>, building on work of Makkai and Shelah <cit.> showed that tameness follows from a large cardinal axiom:


  If  is an AEC and χ >  () is strongly compact, then  is (<χ)-tame for any class of types.


Recent work of Boney and Unger <cit.> show that this can, in a sense, be reversed: the statement “For every AEC  there is χ such that  is χ-tame” is equivalent to a large cardinal axiom. It is however known that when the AEC is stability-theoretically well-behaved, some amount of tameness automatically holds. This was observed for categoricity in a cardinal of high-enough cofinality by Shelah <cit.> and later improved to categoricity in any big-enough cardinal by the author <cit.>:


  Let  be an AEC with arbitrarily large models and let λ≥ (). If _<λ has amalgamation and no maximal models and  is categorical in λ, then there exists χ <  () such that  is (χ, <λ)-weakly tame.


Very relevant to this paper is the fact that tameness allows one to transfer good frames up <cit.>:


  Let  be a good λ-frame generating an AEC . If  is (λ, λ^+)-tame and has amalgamation in λ^+, then there is a good λ^+-frame  on _λ^+. Moreover -nonforking can be described in terms of -nonforking as follows: for M  N, p ∈ (N) does not -fork over M if and only if there exists M_0 ∈λM so that for all N_0 ∈λN with M_0  N_0, p  N_0 does not -fork over M_0.

[<cit.>]
  Let  be a good λ-frame generating an AEC . We write _λ^+ for the frame on _λ^+ with nonforking relation described by Fact <ref>. We write λ^+_λ^+ for its restriction to λ^+_λ^+.




§ WHEN IS THERE A SUPERLIMIT IN Λ^+?


Starting with a good λ-frame generating an AEC , it is natural to ask when  has a superlimit in λ^+, i.e. when the union of any increasing chains of λ^+-saturated models is λ^+-saturated. We should say that there are no known examples when this fails, but we are unable to prove it unconditionally. We give here the following condition on forking characterizing the existence of a superlimit in λ^+. The condition is extracted from the property (∗∗)_M_1^∗, M_2^∗ in <cit.>. After the initial submission of this article, John T. Baldwin pointed out that the condition is similar, in the first-order case, to the fact (valid in any stable theory) that λ^+-saturated models are strongly λ^+-saturated (in Shelah's terminology, F_λ^+^a-saturated), see <cit.>.

Recall that whenever  is a good λ-frame, there is always a unique saturated model in _λ^+ (amalgamation in λ^+ is not needed to build it), so the definition below always makes sense.


  Let  be a good λ-frame generating an AEC . We say that  is decent if for any two saturated model M  N both in _λ^+, any two λ-sized models M_0  N_0  N with M_0  M, and any basic p ∈ (M_0), the nonforking extension of p to N_0 is already realized inside M. In other words, there exists a ∈ M such that (a / N_0; N) is the nonforking extension of p to N_0.


Note that the type p in the above definition will of course be realized inside N (since N is saturated and contains N_0). The point is that it will also be realized inside the smaller model M, even though M itself will not necessarily contain N_0.


  Let  be a good λ-frame generating an AEC . The following are equivalent:

  
  
  *   has a superlimit model in λ^+.
  
  *  There exists a partial order ≤^∗ on Kλ^+_λ^+ such that:
    
      
  * Whenever M_0  M_1 are both in λ^+_λ^+, there exists M_2 ∈λ^+_λ^+ such that M_1  M_2 and M_0 ≤^∗ M_2
      
  * For any increasing chain M_i : i < ω in λ^+_λ^+ such that M_i ≤^∗ M_i + 1 for all i < ω, ⋃_i < ω M_i is saturated.
    
  
  *   is decent.
  

   
  
  * (<ref>) implies (<ref>): Trivial (take ≤^∗ to be ).
  
  * (<ref>) implies (<ref>): Assume (<ref>). Let δ < λ^++ be a limit ordinal and let M_i : i < δ be an increasing chain of saturated models in _λ^+. We want to show that M_δ := ⋃_i < δ M_i is saturated. Without loss of generality, δ = δ < λ^+. Let M^0 ∈λ(M_δ). Build M_i^0 : i < δ increasing chain in _λ such that for all i < δ, M_i^0  M_i and |M^0| ∩ |M_i| ⊆ |M_i + 1^0|. This is possible using that each M_i is saturated. Let M_δ^0 := ⋃_i < δ M_i^0. Then M^0  M_δ^0. Let p ∈ (M_δ^0). By local character, there exists i < δ such that p does not fork over M_i^0. Since  is decent, p is realized in M_i, hence in M_δ. This shows that M_δ realizes all basic types over its -substructures of cardinality λ, which in turns implies that M_δ is saturated <cit.>.

  
  * (<ref>) implies (<ref>): Assume (<ref>) and suppose for a contradiction that  is not decent. We build M_i : i ≤ω, M_i^0 : i ≤ω increasing continuous and a type p ∈ (M_0^0) such that for all i < ω: (writing p_M for the nonforking extension of p to (M)):

  
  
  * M_i ∈λ^+_λ^+.
  
  * M_i≤^∗ M_i + 1.
  
  * M_i^0 ∈λ(M_i).
  
  * p_M_i + 1^0 is not realized in M_i.
  

  This is enough: By assumption, M_ω is saturated. Therefore p_M_ω^0 is realized inside M_ω, and therefore inside M_i for some i < ω. This means in particular that p_M_i + 1 is realized in M_i, a contradiction.
  
  This is possible: For i = 0, pick M_0, M_1', M_0^0, M_1^0, p ∈ (M_0^0) witnessing the failure of being decent (i.e. M_0  M_1' are both saturated in _λ^+, M_0^0  M_0, M_0^0  M_1^0  M_1', M_0^0, M_1^0 ∈_λ, and p_M_1^0 is omitted in M_)), and then use the properties of ≤^∗ to obtain M_1 ∈λ^+_λ^+ such that M_0 ≤^∗ M_1 and M_1'  M_1. Now given M_j^0 : j ≤ i + 1, M_j : j ≤ i + 1, M_i and M_i + 1 are both saturated and so must be isomorphic over any common submodel of cardinality λ. Let f: M_i ≅_M_i^0 M_i + 1 and let g : M_i + 1≅ M_i + 2' be an extension of f (in particular M_i + 1 M_i + 2'). Since g is an isomorphism, g (p_M_i + 1^0) is not realized in g[M_i] = M_i + 1. Pick M_i + 2^0 in λ (M_i + 2') such that M_i + 1^0  M_i + 2^0 and g[M_i + 1^0]  M_i + 2^0. Finally, pick some M_i + 2∈λ^+_λ^+ such that M_i + 1≤^∗ M_i + 2 and M_i + 2'  M_i + 2. This is possible by the assumed properties of ≤^∗.
  


  Theorem <ref> can be generalized to the weaker framework of a λ-superstable AEC (implicit for example in <cit.>; see <cit.> for what forking is there and what properties it has). For this we ask in the definition of decent that M be limit, and we ask for example that M_i + 1 is limit over M_i in the proof of (<ref>) implies (<ref>) of Theorem <ref>. VanDieren <cit.> has shown that λ-symmetry (a property akin to the symmetry property of good λ-frame, see <cit.>) is equivalent to the property that reduced towers are continuous, and if there is a superlimit in λ^+, then reduced towers are continuous. Thus decent is to “superlimit in λ^+” what λ-symmetry is to “reduced towers are continuous”. In particular, being decent is a strengthening of the symmetry property. Note also that taking ≤^∗ in (<ref>) of Theorem <ref> as “being universal over”, we obtain an alternate proof of the main theorem of <cit.> which showed that λ and λ^+-superstability together with the uniqueness of limit models in λ^+ imply that the union of a chain of λ^+-saturated models is λ^+-saturated (see also the proof of Lemma <ref> here).


A property related to being decent is what Shelah calls . We show that  implies decent. We do not know whether the converse holds but it seems (see Fact <ref>) that wherever Shelah uses , he only needs decent.

[III.1.3 in <cit.>]
  Let  be a good λ-frame generating an AEC . We say that  is  if the following is impossible:
    
    There exists increasing continuous chains in _λ M_i : i < λ^+, N_i : i < λ^+, a basic type p ∈ (M_0), and a_i : i < λ^+ such that for any i < λ^+:

  
  
  * M_i  N_i.
  
  * a_i + 1∈ |M_i + 2| and (a_i + 1, M_i + 1, M_i + 2) is a nonforking extension of p, but (a_i + 1, N_0, N_i + 2) forks over M_0.
  
  * ⋃_j < λ^+ M_j is saturated.
  



  Let  be a good λ-frame. If  is , then  is decent.


  Let  be the AEC generated by . Suppose that  is not decent. Fix witnesses M, N, M^0, N^0, p. We build increasing continuous chains in _λ, M_i : i < λ^+, N_i : i < λ^+ and a_i :i < λ^+ such that for all i < λ^+:

  
  
  * M_0 = M^0, N_0 = N^0.
  
  * M_i  M, N_i  N.
  
  * M_i  N_i.
  
  * M_i + 1 is universal over M_i.
  
  * a_i ∈ M_i + 1.
  
  * (a_i / M_i; M_i + 1) is a nonforking extension of p.
  

  This is possible since both M and N are saturated. This is enough: M_λ^+ := ⋃_i < λ^+ M_i is saturated (we could also require that M_λ^+ = M but this is not needed) and for any i < λ^+, (a_i + 1 / N_0; N_i + 2) forks over M_0. If not, then by the uniqueness property of nonforking, we would have that a_i + 1 realizes the nonforking extension of p to N_0 = N^0, which we assumed was impossible.




§ FROM WEAK TAMENESS TO GOOD FRAME


In this section, we briefly investigate how to generalize Fact <ref> to AECs that are only (λ, λ^+)-weakly tame. The main problem is that the class λ^+_λ^+ may not be closed under unions of chains (i.e. it may not be an AEC in λ^+). Indeed, this is the only difficulty. 


  Let  be a good λ-frame generating an AEC . If  is (λ, λ^+)-weakly tame, the following are equivalent:

  
  
  *   is decent and λ^+_λ^+ has amalgamation.
  
  *  _λ^+λ^+_λ^+ (see Definition <ref>) is a good λ^+-frame on λ^+_λ^+.
  
  *  There is a good λ^+-frame on λ^+_λ^+.
  

  Moreover in (<ref>), if  is type-full, then we can conclude that the good λ^+-frame described in (<ref>) is also type-full.


  (<ref>) implies (<ref>) is exactly as in the proof of Fact <ref>. (<ref>) implies (<ref>) is trivial. (<ref>) implies (<ref>) is by Fact <ref> and Theorem <ref>, since by definition the existence of a good λ^+-frame on λ^+_λ^+ implies that λ^+_λ^+ has amalgamation and λ^+_λ^+ is an AEC in λ^+, hence that the union of an increasing chain of λ^+-saturated models is λ^+-saturated, i.e. that  has a superlimit in λ^+.


  A note on the definitions: statements of the form “there is a good λ^+-frame on λ^+_λ^+” imply in particular (from the definition of a good frame) that λ^+_λ^+ is an AEC in λ^+, i.e. that the union of an increasing chain of λ^+-saturated models is λ^+-saturated. We will often use this without comments.


It is natural to ask whether the good λ^+-frame in Theorem <ref> is itself decent. We do not know the answer, but can answer positively assuming 2^λ < 2^λ^+ (see Corollary <ref>) and in fact in this case  is also . We can show in ZFC that being  transfers up. The proof adapts an argument of Shelah <cit.>.


  Let  be a  λ-frame generating an AEC . If  is (λ, λ^+)-weakly tame and λ^+_λ^+ has amalgamation, then _λ^+λ^+_λ^+ is a  λ^+-frame on λ^+_λ^+.


  By Lemma <ref>,  is decent, so by Theorem <ref> := _λ^+λ^+_λ^+ is a good λ^+-frame on λ^+_λ^+. Moreover by definition -nonforking is described in terms of -nonforking as in the statement of Fact <ref>.

  Suppose that  is not . Let M_i : i < λ^++, N_i : i < λ^++, p, a_i : i < λ^++ witness that  is not . Let M_j' : j < λ^+ be increasing continuous in _λ such that M_0 = ⋃_j < λ^+ M_j' and let N_j' : j < λ^+ be increasing continuous in _λ such that N_0 = ⋃_j < λ^+ N_j' and M_j'  N_j' for all j < λ^+.

  By a standard pruning argument, there is j^∗ < λ^+ and an unbounded S ⊆λ^++ of successor ordinals such that for all i ∈ S and all M' ∈λ (M_i) with M_j^∗'  M', (a_i / M'; M_i + 1) does not -fork over M_j^∗'. Now by assumption for all i ∈ S, (a_i / N_0; N_i + 1) -forks over M_0, so by a pruning argument again, there is an unbounded S' ⊆ S and j^∗∗∈ [j^∗, λ^+) such that for all i ∈ S', (a_i / N_j^∗∗'; N_i + 1) -forks over M_j^∗'. We build i_j : j < λ^+ and M_j^∗ : j < λ^+, N_j^∗ : j < λ^+ increasing continuous in _λ such that for all j < λ^+:

    
    
  * i_j ∈ S'.
    
  * M_j^∗ N_j^∗.
    
  * M_j + 1^∗ is universal over M_j^∗.
    
  * M_0^∗ = M_j^∗', N_0^∗ = N_j^∗∗'.
    
  * M_j^∗ M_i_j, N_j^∗ N_i_j.
    
  * a_i_j∈ M_j + 1^∗.
    

    This is enough: Then by construction of S', j^∗, and j^∗∗, M_j^∗ : j < λ^+, N_j^∗ : j < λ^+, (a_i_0 / M_0^∗; M_1^∗) and a_i_j : j < λ^+ witness that  is not .

    This is possible: Let M_λ^++ := ⋃_i < λ^++ M_i, N_λ^++ := ⋃_i < λ^++ N_i. We are already given M_0^∗ and N_0^∗ and for j limit we take unions. Now assume inductively that M_k^∗ : k ≤ j, N_k^∗ : k ≤ j and i_k : k < j are already given, with M_j^∗ M_λ^++ and N_j^∗ N_λ^++. Let i_j ∈ S' be big-enough such that N_i_j contains N_j^∗, M_i_j contains M_j^∗, and i_k < i_j for all k < j. Such an i_j exists since S' is unbounded. Now let M^∗∈λ^+ M_λ^++ contain M_j^∗ and a_i_j and be saturated. Such an M^∗ exists since M_λ^++ is saturated by assumption. Now pick M_j + 1^∗∈λ (M^∗) so that a_i_j∈ M_j + 1^∗ and M_j + 1^∗ is universal over M_j^∗. This is possible since M^∗ is saturated. Finally, pick any N_j + 1^∗∈λ (N_λ^++) containing M_j + 1^∗ and N_j^∗.


We end this section by noting that in the context of Fact <ref>,  is decent and hence by Theorem <ref> there is a good λ^+-frame on λ^+_λ^+. In fact:


  Let  be a good λ-frame generating an AEC . If there is a good λ^+-frame on _λ^+, then  is decent.


The proof uses the uniqueness of limit models in good frames, due to Shelah <cit.> (see <cit.> for a proof): 


  Let  be a good λ-frame, δ_1, δ_2 < λ^+ be limit ordinals. Let M_i^ℓ : i ≤δ_ℓ, ℓ = 1,2, be increasing continuous. If for all ℓ = 1,2, i < δ_ℓ, M_i + 1^ℓ is universal over M_i, then M_δ_1^1 ≅ M_δ_2^2.



  By Theorem <ref>, it suffices to show that  has a superlimit in λ^+. We could apply two results of VanDieren <cit.> but we prefer to give a more explicit proof here.

  Let N_i : i ≤λ^+ be increasing continuous in _λ^+ such that N_i + 1 is universal over N_i for all i < λ^+. This is possible since by definition of a good λ^+-frame,  is stable in λ^+ and has amalgamation in λ^+. Clearly, N_λ^+ is saturated. Moreover by Fact <ref>, N_λ^+≅ N_ω. Thus N_ω is also saturated. We chose N_i : i ≤ω arbitrarily, therefore (<ref>) of Theorem <ref> holds with ≤^∗ being “universal over or equal to”. Thus (<ref>) there holds:  is decent, as desired.


There are other variations on Lemma <ref>. For example, if  is a good λ-frame generating an AEC ,  has amalgamation in λ^+, and  is (λ, λ^+)-weakly tame, then  is decent (to prove this, we transfer enough of the good λ-frame up, then apply results of VanDieren <cit.>). 



§ FROM GOOD FRAME TO WEAK TAMENESS


In this section, we look at a sufficient condition (due to Shelah) implying that a good λ-frame can be extended to a good λ^+-frame and prove its necessity.

It turns out it is convenient to first extend the good λ-frame to a good (≤λ, λ)-frame. For this, the next technical property is of great importance, and it is key in Chapter II and III of <cit.>. The definition below follows <cit.> (but as usual, we work only with type-full frames). Note that we will not use the exact content of the definition, only its consequence. We give the definition only for the benefit of the curious reader. 


  Let  be an abstract class and λ be a cardinal.
  
  
  
  * amalgam For M_0  M_ℓ all in _λ, ℓ = 1,2, an amalgam of M_1 and M_2 over M_0 is a triple (f_1, f_2, N) such that N ∈_λ and f_ℓ : M_ℓ N.
  
  * equivalence of amalgam Let (f_1^x, f_2^x, N^x), x = a,b be amalgams of M_1 and M_2 over M_0. We say (f_1^a, f_2^a, N^a) and (f_1^b, f_2^b, N^b) are equivalent over M_0 if there exists N_∗∈_λ and f^x : N^x → N_∗ such that f^b ∘ f_1^b = f^a ∘ f_1^a and f^b ∘ f_2^b = f^a ∘ f_2^a, namely, the following commutes:

  
    N^a @.>[r]^f^a    N_∗
    
        M_1 [ru]^f_1^a[rr]|>>>>>f_1^b       N^b @.>[u]_f^b
    
        M_0 [u] [r]     M_2 [uu]|>>>>>f_2^a[ur]_f_2^b


  Note that being “equivalent over M_0” is an equivalence relation (<cit.>).

  * Let  be a good (<α, λ)-frame.
  
    
  * A uniqueness triple in  is a triple (, M, N) such that M  N, ∈<αN and for any M_1  M, there exists a unique (up to equivalence over M) amalgam (f_1, f_2, N_1) of N and M_1 over M such that (f_1 (a) / f_2[M_1] ; N_1) does not fork over M.
    
  *  has the existence property for uniqueness triples if for any M ∈_ and any basic p ∈^<α (M), one can write p =  ( / M; N) with (, M, N) a uniqueness triple.
    
  * We say that  is weakly successful if its restriction to types of length one has the existence property for uniqueness triples.
  
  


As an additional motivation, we mention the closely related notion of a domination triple:


  Let  be a good (≤λ, λ)-frame. A domination triple in  is a triple (, M, N) such that M  N, ∈≤λN, and whenever M'  N' are such M  M', N  N', then MM'N' implies MNM'N'.


The following fact shows that domination triples are the same as uniqueness triples once we have managed to get a type-full good (≤λ, λ)-frame. The advantage of uniqueness triples is that they can be defined already inside a good λ-frame.

[11.7, 11.8 in <cit.>]
  Let  be a type-full good (≤λ, λ)-frame. Then in  uniqueness triples and domination triples coincide.


The importance of weakly successful good frames is that they extend to longer types. This is due to Shelah:


  Let  be a categorical good λ-frame. If  is weakly successful, then there is a unique type-full good (≤λ, λ)-frame extending .


  The uniqueness is <cit.>. Existence is the main result of <cit.>, although there Shelah only builds a nonforking relation for models satisfying the axioms of a good (≤λ, λ)-frame. How to extend this to all types of length at most λ is done in <cit.>. An outline of the full argument is in the proof of <cit.>.


It is not clear whether the converse of Fact <ref> holds, but see Fact <ref>.

Shelah proved <cit.> that a categorical good λ-frame (generating an AEC ) is weakly successful whenever 2^λ < 2^λ^+ < 2^λ^++ and  has a “medium” number of models in λ^++. Shelah has also shown that being weakly successful follows from some stability in λ^+ and 2^λ < 2^λ^+, see <cit.> for a proof:


  Let  be a categorical good λ-frame generating an AEC . Assume 2^λ < 2^λ^+. If for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M, then  is weakly successful.


Once we have a good (≤λ, λ)-frame, we can define a candidate for a good λ^+-frame on the saturated models in _λ^+. Nonforking in λ^+ is defined in terms of nonforking in λ (in fact one can make sense of this definition even if we only start with a good λ-frame), but the problem is that we do not know that the class of saturated models in _λ^+ has amalgamation. To achieve this, the ordering  is changed to a new ordering ^+ so that nonforking “reflects down”. The definition is due to Shelah <cit.> but we follow <cit.>:


  Let = (_, ) be a type-full good (≤λ, λ)-frame generating an AEC . We define a pair ^+ = (_^+, _^+) as follows:

  
  
  * _^+ = (K_^+, ^+), where:
    
    
  * K_^+ is the class of saturated models in _λ^+.
    
  * For M, N ∈ K_^+, M ^+ N holds if and only if there exists increasing continuous chains M_i : i < λ^+, N_i : i < λ^+ in _λ such that:

  
  
  * M = ⋃_i < λ^+ M_i.
  
  * N = ⋃_i < λ^+ N_i.
  
  * For all i < j < λ^+, M_iN_iM_jN_j.
  
    
  
  * For M_0 ^+ M ^+ N and a ∈ N, _^+ (M_0, a, M, N) holds if and only if there exists M_0' ∈λ (M_0) such that for all M' ∈λ (M) and all N' ∈λ (N), if M_0'  M'  N' and a ∈ N', then _ (M_0', a, M', N').
  

  For a weakly successful categorical good λ-frame , we define ^+ := ^+, where  is the unique extension of  to a type-full good (≤λ, λ)-frame (Fact <ref>).


As a motivation for the definition of ^+, observe that if M_i : i < λ^+, N_i : i < λ^+ are increasing continuous in _λ such that M_i  N_i for all i < λ^+, then it is known that there is a club C ⊆λ^+ such that M_j ∩ N_i = M_i for all i ∈ C and all j > i. We would like to conclude the stronger property that M_iN_iM_jN_j for i ∈ C and j > i. If we are working in a superstable elementary class this is true as nonforking has a strong finite character property, but in the more general setup of good frames this is not clear. Thus the property is built into the definition by changing the ordering. This creates a new problem: we do not know whether _^+ is an AEC (smoothness is the problematic axiom). Note that there are no known examples of weakly successful good λ-frame  where ^+ is not .

The following general properties of ^+ are known. They are all proven in <cit.> but we cite from <cit.> since the proofs there are more detailed:


  Let  be a weakly successful categorical good λ-frame generating an AEC .

  
  
  * <cit.> If M, N ∈_^+ and M ^+ N, then M  N.
  
  *  <cit.>, _^+ is an abstract class which is closed under unions of chains of length strictly less than λ^++.
  
  *  <cit.> _^+ satisfies the following strengthening of the coherence axiom: if M_0, M_1, M_2 ∈_^+ are such that M_0 ^+ M_2 and M_0  M_1  M_2, then M_0 ^+ M_1.
  
  * <cit.> _^+ has no maximal models and amalgamation.

  
  *  <cit.> Let M ^+ N_ℓ and let a_ℓ∈ N_ℓ, ℓ = 1,2. Then (a_1 / M; N_1) =  (a_2 / M; N_2) if and only if _^+ (a_1 / M; N_1) = _^+ (a_2 / M; N_2). In particular, ^+ is a λ^+-frame.
  


We will use the following terminology, taken from <cit.>:


  Let  be a type-full good (≤λ, λ)-frame generating an AEC . We say that  reflects down if M  N implies M ^+ N for all M, N ∈_^+. We say that  almost reflects down if _^+ is an AEC in λ^+ (see the bottom of Definition <ref>). We say that a weakly successful good λ-frame [almost] reflects down if its extension to a type-full good (≤λ, λ)-frame [almost] reflects down, see Fact <ref>.


Shelah uses the less descriptive “successful”:

[III.1.1 in <cit.>]
  We say that a good λ-frame  is successful if it is weakly successful and almost reflects down.


The point of this definition is that if it holds, then  can be extended to a good λ^+-frame. Moreover  is successful when there are few models in λ^++:

[III.1.9 in <cit.>]
  If  is a successful categorical good λ-frame, then ^+ is a  λ^+-frame.


[II.8.4, II.8.5 in <cit.>, or see 7.1.3 in <cit.>]
  Let  be a weakly successful categorical good λ-frame generating the AEC . If (, λ^++) < 2^λ^++, then  is successful.


The downside of working only with a successful good λ-frame is that the ordering of _^+ may not be  anymore. Thus the AEC generated by _^+ may be different from the one generated by _. For example it may fail to have arbitrarily large models even if the original one does. This is why we will focus on good frames that reflect down, i.e. ^+ is just . Several characterizations of this situation are known. (<ref>) implies (<ref>) below is due to Jarden and all the other implications are due to Shelah (but as usual we mostly quote from <cit.>):


  Let  be a categorical good λ-frame. The following are equivalent:

  
  
  *   extends to a type-full good (≤λ, λ)-frame which reflects down.
  
  *   is successful and  (see Definition <ref>).
  
  *   is successful and decent (see Definition <ref>).
  
  *   is weakly successful and generates an AEC  which is (λ, λ^+)-weakly tame and so that λ^+_λ^+ has amalgamation.
  

   
  
  * (<ref>) implies (<ref>): By <cit.>,  is weakly successful. Since it reflects down, it is successful and ^+ is the restriction of  to λ^+_λ^+. By adapting the proof of <cit.> (see <cit.>), we get that  is .
  
  * (<ref>) implies (<ref>): By Lemma <ref>.
  
  * (<ref>) implies (<ref>): By definition of successful,  is weakly successful, so by Fact <ref> extends to a unique type-full good (≤λ, λ)-frame. Since  is decent, we have that the ordering ≼_λ^+^⊗ from <cit.> is the same as . By assumption  is successful, so all of the equivalent conditions of <cit.> are false, so in particular for M, N ∈_^+, M  N implies M ^+ N. Therefore  reflects down.
  
  * (<ref>) implies (<ref>): we have already argued that  is successful, and by definition ^+ is just the restriction of , i.e. _^+ = λ^+_λ^+. Now weak tameness follows from <cit.> (a similar argument already appears in <cit.>) and amalgamation is because by Fact <ref> ^+ is a good λ^+-frame.
  
  * (<ref>) implies (<ref>): By <cit.>.
  


The aim of this section is to add another condition to Fact <ref>: the existence of a good λ^+-frame on λ^+_λ^+. Toward this, we will use the following ordering on pairs of saturated models, introduced by Jarden <cit.>. The idea is as follows: call a pair (M, N) of saturated models in _λ^+ bad if M  N but M ^+ N. Jarden's ordering of these pairs works in such a way that higher pairs interact non-trivially with lower pairs. Jarden showed that there are maximal such pairs, and we will show that if there is a good λ^+-frame then on the other hand it is always possible to extend these pairs. This will be a contradiction showing that there are no bad pairs after all.


  Let  be a weakly successful categorical good λ-frame generating an AEC . For pairs (M, N), (M', N') in _^+ with M  N, M'  N', we write (M, N)  (M', N') if:

  
  
  * M ^+ M'.
  
  * N  N'.
  
  * M' ∩ N ≠ M.
  

  We write (M, N)  (M', N') if (M, N)  (M', N') or (M, N) = (M', N').





  Let  be a decent weakly successful categorical good λ-frame generating an AEC .

  
  
  *  For any M  N both in _^+, there exists a pair (M', N') such that (M, N)  (M', N') and (M', N') is -maximal.
  
  *  <cit.> M  N are both in _^+, and M ^+ N, then there exists M_i : i ≤λ^+ increasing continuous in _^+ such that M_λ^+ = M and M_i ^+ N for all i < λ^+.
  

   
  
  * By the proof of <cit.>. We use decency here to make sure (via Theorem <ref>) that λ^+_λ^+ is an AEC in λ^+, hence closed under unions of short chains (Jarden works with the whole of _λ^+).
  
  * The definition of decent says that the ordering ≼_λ^+^⊗ from <cit.> is the same as the usual ordering  on λ^+_λ^+. Thus if M  N are both in _^+ but M ^+ N, then condition (2) in the equivalence proven in <cit.> holds, hence condition (3) there also holds, which is what we wanted to prove.
  


We have arrived to the main theorem of this section:


  Let  be a weakly successful categorical good λ-frame generating an AEC . If there is a good λ^+-frame on λ^+_λ^+, then  reflects down.


  Let  be the good λ^+-frame on λ^+_λ^+. Note that since  is good, λ^+_λ^+ must be an AEC in λ^+, i.e. unions of chains of λ^+-saturated models are λ^+-saturated, so by Fact <ref>  has a superlimit in λ^+ which by Theorem <ref> implies that  is decent. Therefore we will later be able to apply Fact <ref>. We first show:

  Claim: If M  N are both in _^+, a ∈ |N| \ |M| and (a / M ; N) is realized in a ^+-extension of M, then (M, N) is not -maximal.

  Proof of Claim: Let p :=  (a / M; N). Let N' be such that M ^+ N' and p is realized by b ∈ N' (i.e. p =  (b / M; N')). By amalgamation in λ^+_λ^+ (which holds since we are assuming there is a good frame on this class) and the fact that (a / M; N) =  (b / M; N'), there exists N”∈λ^+_λ^+ and f: N'  N” with N  N” such that f (b) = a. Consider the pair (f[N'], N”). Since ^+ is invariant under isomorphisms and M ^+ N', M ^+ f[N']. Moreover, a ∈ |N| \ |M|, so b ∈ |N'| \ |M|, and so a = f (b) ∈ |f[N']| \ |M|. Since we also have that a ∈ |N|, this implies that f[N'] ∩ N ≠  M, so (M, N)  (f[N'], N”), hence (M, N) is not -maximal. †_Claim
  
  Let M, N ∈_^+ be such that M  N. We have to show that M ^+ N. Suppose not. By Fact <ref>(<ref>), there exists (M', N') such that (M, N)  (M', N') and (M', N') is -maximal. Observe that M' ^+ N': if M' ^+ N', then since also M ^+ M' and _^+ is an abstract class (Fact <ref>(<ref>)), we would have that M ^+ N' so by Fact <ref>(<ref>), M ^+ N, a contradiction. By Fact <ref>(<ref>), there exists M_i : i ≤λ^+ increasing continuous in _^+ such that M_λ^+ = M' and M_i ^+ N' for all i < λ^+. Since M' ^+ N', we have in particular that M' ≠ N'. By density of basic types, let a ∈ N' \ M' and let p :=  (a / M'; N') be basic. By local character in  (the good λ^+-frame on λ^+_λ^+), there is i < λ^+ such that p does not -fork over M_i. By conjugation in  (Fact <ref>), this means that p  M_i and p are isomorphic. However since M_i ^+ N', p  M_i is realized inside some ^+-extension of M_i, hence p is also realized inside some ^+-extension of M'. Together with the claim, this contradicts the -maximality of (M', N').


  We are not using all the properties of the good λ^+-frame. In particular, it suffices that local character holds for chains of length λ^+.


We obtain a new characterization of when a weakly successful frame reflects down. We emphasize that only (<ref>) implies (<ref>) below is new. (<ref>) implies (<ref>) and (<ref>) implies (<ref>) are due to Shelah while (<ref>) implies (<ref>) is due to Jarden.


  Let  be a weakly successful categorical good λ-frame generating the AEC . The following are equivalent:

  
  
  *   is successful and decent.
  
  *   is (λ, λ^+)-weakly tame and λ^+_λ^+ has amalgamation.
  
  *  There is a  good λ^+-frame on λ^+_λ^+.
  

  Moreover, if it exists then the good λ^+-frame  on λ^+_λ^+ is unique, can be enlarged to be type-full, and coincides (on the appropriate class of basic types) with ^+ and _λ^+λ^+_λ^+ (see Definition <ref>).

   
    
  * (<ref>) is equivalent to (<ref>): By Fact <ref>.
    
  * (<ref>) implies (<ref>): By Facts <ref> and <ref>.
    
  * (<ref>) implies (<ref>): By Theorem <ref> and Fact <ref>.
  

  For the moreover part, assume that  is a good λ^+-frame on λ^+_λ^+. By the proof of (<ref>) implies (<ref>), ^+ is in fact a good λ^+-frame on λ^+_λ^+. By weak tameness and Theorem <ref>, _λ^+λ^+_λ^+ is also a good λ^+-frame on λ^+_λ^+. Without loss of generality (Fact <ref>(<ref>)),  is type-full so by the moreover part of Theorem <ref>, _λ^+λ^+_λ^+ is also type-full, and by canonicity (Fact <ref>) is the desired extension of .


We can combine our result with the weak GCH to obtain weak tameness from two successive good frames. This gives a converse to Theorem <ref>.


  Let  be an AEC, let λ≥ () and assume that 2^λ < 2^λ^+. Let  be a categorical good λ-frame on _λ. The following are equivalent:

  
  
  *  There is a  λ^+-frame on λ^+_λ^+.
  
  *  There is a good λ^+-frame on λ^+_λ^+.
  
  *   is (λ, λ^+)-weakly tame,  is decent, and λ^+_λ^+ has amalgamation.
  
  *   is (λ, λ^+)-weakly tame and for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M.
  
  *   is successful and .
  

   
  
  * (<ref>) implies (<ref>): Trivial.
  
  * (<ref>) implies (<ref>): By Fact <ref>,  is weakly successful. Now apply Corollary <ref>.
  
  * (<ref>) implies (<ref>): By Theorem <ref>.
  
  * (<ref>) implies (<ref>): Assume (<ref>). Then by definition of a good λ^+-frame, for every saturated M ∈_λ^+ there is N ∈_λ^+ universal over M. Further we proved already that (<ref>) holds. Therefore  is (λ, λ^+)-weakly tame.
  
  * (<ref>) implies (<ref>): By Fact <ref>,  is weakly successful. Now apply (<ref>) implies (<ref>) in Fact <ref>.
  
  * (<ref>) implies (<ref>): By Fact <ref> and since  reflects down (Fact <ref>).
  




§ FROM WEAK TO STRONG TAMENESS


Assuming amalgamation in λ^+, we are able to conclude (λ, λ^+)-tameness in (<ref>) of Corollary <ref>. To prove this, we recall that the definition of ^+ (Definition <ref>) can be extended to all of _λ in the following way:


  Let = (_, ) be a good (≤λ, λ)-frame generating an AEC . We define a pair ^∗ = (_^∗, _^∗) as follows:

  
  
  * _^∗ = (K_^∗, ^∗), where:
    
    
  * K_^∗ = K_λ^+.
    
  * For M, N ∈ K_^∗, M ^∗ N holds if and only if there exists increasing continuous chains M_i : i < λ^+, N_i : i < λ^+ in _λ such that:

  
  
  * M = ⋃_i < λ^+ M_i.
  
  * N = ⋃_i < λ^+ N_i.
  
  * For all i < j < λ^+, M_iN_iM_jN_j.
  
    
  
  * For M_0 ^∗ M ^∗ N and a ∈ N, _^∗ (M_0, a, M, N) holds if and only if there exists M_0' ∈λ (M_0) such that for all M' ∈λ (M) and all N' ∈λ (N), if M_0'  M'  N' and a ∈ N', then _ (M_0', a, M', N').
  

  For a weakly successful categorical good λ-frame , we define ^∗ := ^∗, where  is the unique extension of  to a good (≤λ, λ)-frame (Fact <ref>).


The parts of <cit.> referenced by Fact <ref> apply to the whole of _λ^+ (i.e. there is no need to restrict to saturated models). Thus we have:


  Let  be a weakly successful categorical good λ-frame.

  
  
  *  <cit.> ^∗ is a λ^+-frame and _^∗ has amalgamation.
  
  *  <cit.> if M_0, M_1, M_2 ∈_^∗ are such that M_0 ^∗ M_2 and M_0  M_1  M_2, then M_0 ^∗ M_1.
  


We will also use that every model of _λ^+ has a saturated ^∗-extension.

[7.1.10, 7.1.12(a) in <cit.>]
  Let  be a weakly successful categorical good λ-frame. For any M ∈_^∗, there exists N ∈_^+ such that M ^∗ N. 


Similarly to Definition <ref>, we name what it means for ^∗ to be trivial:


  Let  be a good (≤λ, λ)-frame generating an AEC . We say that  strongly reflects down if M  N implies M ^∗ N for all M, N ∈_^∗. We say that a weakly successful good λ-frame strongly reflects down if its extension to a good (≤λ, λ)-frame strongly reflects down, see Fact <ref>.


We establish the following criteria for strongly reflecting down:


  Let  be a weakly successful categorical good λ-frame generating an AEC . The following are equivalent:
  
  
  *   reflects down and  has amalgamation in λ^+.
  
  *   strongly reflects down.
  


  (<ref>) implies (<ref>) is Fact <ref>(<ref>). Assume now that (<ref>) holds. Let M, N ∈_^∗ be such that M  N. By Fact <ref>, there exists M' ∈_^+ such that M ^∗ M'. By Facts <ref> and <ref>, ^+ is a good λ^+-frame. In particular, there exists N' ∈_^+ such that M' s^+ N' and N' is universal over M' in _^+. Since  has amalgamation in λ^+ and ^+ is the same as  by definition of reflecting down, N' is universal over M. Moreover, M ^∗ N' by transitivity of ^∗. Now let f: N  N' by a -embedding. By Fact <ref>(<ref>), M ^∗ f[N], so by invariance M ^∗ N, as desired.


It is known that strongly reflecting down gives tameness. This is <cit.>. There is a mistake in the statement given by Jarden and Shelah (≼ there should be ≼_NF) but the proof is still correct.


  Let  be a weakly successful categorical good λ-frame generating an AEC . If  strongly reflects down, then  is (λ, λ^+)-tame.


We obtain that tameness is equivalent to being able to extend the frame. Note that only (<ref>) implies (<ref>) and (<ref>) implies (<ref>) are new.


  Let  be a weakly successful categorical good λ-frame generating an AEC . The following are equivalent:

  
  
  *   strongly reflects down.
  
  *   has amalgamation in λ^+ and is (λ, λ^+)-weakly tame.    
  
  *   has amalgamation in λ^+ and is (λ, λ^+)-tame.
  
  *  There is a good λ^+-frame on _λ^+.
  

   
  
  * (<ref>) implies (<ref>): By Facts <ref>(<ref>) and <ref>.
  
  * (<ref>) implies (<ref>): Trivial.
  
  * (<ref>) implies (<ref>): By Corollary <ref>,  is successful and decent. By Fact <ref>,  reflects down. By Theorem <ref>,  strongly reflects down.
  
  * (<ref>) implies (<ref>): By Fact <ref>.
  
  * (<ref>) implies (<ref>): Assume that (<ref>) holds. By Lemma <ref>,  is decent so by Theorem <ref>,  has a superlimit in λ^+. Thus by Fact <ref> we can restrict the good λ^+-frame on _λ^+ to the class λ^+_λ^+ and still obtain a good λ^+-frame. By Theorem <ref>,  reflects down. Since we are assuming there is a good λ^+-frame on _λ^+,  has amalgamation in λ^+, so by Theorem <ref>,  strongly reflects down.
  


Assuming 2^λ < 2^λ^+, we do not need to assume that  is weakly successful. We do not repeat all the equivalences of Corollary <ref> and only state our main result:


  Let  be an AEC, let λ≥ () and assume that 2^λ < 2^λ^+. If there is a categorical good λ-frame  on _λ and a good λ^+-frame  on _λ^+, then  is (λ, λ^+)-tame. Moreover, λ^+_λ^+ = _λ^+λ^+_λ^+ (see Definition <ref>).


  Let  be a categorical good λ-frame on _λ. By Fact <ref>,  is weakly successful. Now apply Corollary <ref>. The moreover part follows from the moreover part of Corollary <ref>.




§ ON CATEGORICITY IN TWO SUCCESSIVE CARDINALS


Grossberg and VanDieren have shown <cit.> that in a λ-tame AEC with amalgamation and no maximal models, categoricity in λ and λ^+ imply categoricity in all μ≥λ. In <cit.>, the author gave a more local conclusion as well as a more abstract proof using good frames. Here, we give a converse: assuming the weak GCH, if we can prove categoricity in λ^++ from categoricity in λ and λ^+, then we must have some tameness. This follows from combining results of Shelah[Let us sketch how (of course Corollary <ref> gives another proof but it uses the results of the present paper). By Fact <ref>(<ref>), there is a successful good λ-frame  on _λ. In fact, the basic types of this good frames are the minimal types. Thus  is  (as nonforking corresponds to disjoint amalgamation). It then follows from <cit.> and categoricity that  is (λ, λ^+)-tame.] but seems not to have been noticed before. Only (<ref>) implies (<ref>) below really uses the results of this paper.


  Let  be an AEC and let λ≥ (). Assume 2^λ < 2^λ^+ < 2^λ^++ and assume that  is categorical in λ and λ^+. The following are equivalent:

  

  
  *  There is a successful good λ-frame on _λ.
  
  *  There is a good λ-frame on _λ and a good λ^+-frame on _λ^+.
  
  *   is stable in λ,  is (λ, λ^+)-tame,  has amalgamation in λ^+, and _λ^++≠∅.
  
  *   is categorical in λ^++.
  
  *  1 ≤ (, λ^++) < μ_unif (λ^++, 2^λ^+).
  
  *   is stable in λ,  is stable in λ^+, and 1 ≤ (, λ^++) < 2^λ^++.
  


On μ_unif, see <cit.> for a definition and <cit.> for what is known. It seems that for all practical purposes the reader can take μ_unif (λ^++, 2^λ^+) to mean 2^λ^++.

Note that if the AEC  of Corollary <ref> has arbitrarily large models, then stability in λ would follow from categoricity in λ^+ <cit.>. Thus we obtain that  is categorical in λ^++ if and only if  is (λ, λ^+)-tame and has amalgamation in λ^+.

To prove Corollary <ref>, we will use several facts:

[I.3.8 in <cit.>]
  Let  be an AEC and let λ≥ (). Assume 2^λ < 2^λ^+. If  is categorical in λ and (, λ^+) < 2^λ^+, then  has amalgamation in λ.


  Let  be an AEC and let λ≥ (). Assume 2^λ < 2^λ^+ and assume that  is categorical in both λ and λ^+.

  
  
  *  If _λ^++≠∅ and  is stable in λ, then there is an almost good λ-frame (see <cit.>) on _λ.
  
  *  If _λ^++≠∅,  has amalgamation in λ^+,  is stable in λ, and  is stable in λ^+, then there is a weakly successful good λ-frame on _λ.
  
  *  If 2^λ^+ < 2^λ^++, 1 ≤ (, λ^++) < 2^λ^++,  is stable in λ, and  is stable in λ^+, then there is a successful good λ-frame on _λ.
  
  *  If 2^λ < 2^λ^+ < 2^λ^++ and 1 ≤ (, λ^++) < μ_unif (λ^++, 2^λ^+), then there is a successful good λ-frame on _λ.
  

   
  
  * By Fact <ref>,  has amalgamation in λ. We check that the hypotheses of <cit.> are satisfied. The only ones that we are not explicitly assuming are:
    
    
  * The extension property in _λ, i.e. for every M  N both in _λ and every p ∈ (M), if p is not algebraic (i.e. not realized inside M), then p has a nonalgebraic extension to (N): holds by <cit.>.
    
  * The existence of an inevitable type in _λ: holds by <cit.> and the density of minimal types (see the proof of (∗)_5 in <cit.>).
    
  
  * By (<ref>), there is an almost good λ-frame  on _λ. The proof of <cit.> goes through even for almost good λ-frames and gives that  is weakly successful (note that Shelah's proof of Fact <ref> still goes through in almost good λ-frames). By <cit.>,  is in fact a good λ-frame.
  
  * By Fact <ref>,  has amalgamation in λ and λ^+. By (<ref>), there is a weakly successful good λ-frame  on _λ. By Fact <ref>,  is successful.
  
  * By <cit.>, there is an almost good λ-frame  on _λ. By <cit.>,  has existence for a certain relative of uniqueness triples. Thus by <cit.>,  is actually a good λ-frame. By <cit.>,  is weakly successful. By Fact <ref>,  is successful.
  



  Let  be an AEC and let λ≥ (). If  is categorical in both λ and λ^+ and there is a successful good λ-frame on _λ, then  is categorical in λ^++.


  Let  be a successful good λ-frame on _λ. Since  is categorical in λ^+, there is a superlimit in λ^+, hence by Theorem <ref>  is decent, hence by Fact <ref> is . Now combine <cit.> with <cit.>.



  By categoricity in λ and λ^+ all good λ-frames on _λ are categorical, λ^+_λ^+ = _λ^+, and (λ, λ^+)-weak tameness is the same as (λ, λ^+)-tameness. By Theorem <ref>, any good λ-frame on _λ is decent. We will use this without comments.
  
  
  
  * (<ref>) implies (<ref>):    By Fact <ref>, the definition of reflecting down (Definition <ref>) and the definition of ^+ (Definition <ref>), _^+ = λ^+_λ^+ = _λ^+, so the result follows from Fact <ref>.
  
  * (<ref>) implies (<ref>): By definition of a good λ-frame,  is stable in λ and by definition of a good λ^+-frame  has amalgamation in λ^+. Since a good λ^+-frame has no maximal models in λ^+, _λ^++≠∅. By Corollary <ref>,  is (λ, λ^+)-tame.
    
  * (<ref>) implies (<ref>): By Fact <ref>,  has amalgamation in λ. By Fact <ref>, there is an almost good λ-frame on _λ. By the proof of Fact <ref>,  is stable in λ^+. By Fact <ref>, there is a weakly successful good λ-frame  on _λ. By Fact <ref>,  is successful.
    
  * (<ref>) implies (<ref>): By Fact <ref>.
    
  * (<ref>) implies (<ref>): Trivial.
    
  * (<ref>) implies (<ref>): By Fact <ref>.
    
  * (<ref>) implies (<ref>): (<ref>) trivially implies that 1 ≤ (, λ^++) < 2^λ^++. We have also seen that (<ref>) implies (<ref>) implies (<ref>) which by definition implies stability in λ and λ^+.
    
  * (<ref>) implies (<ref>): By Fact <ref>.
      





amsalpha


