Superstability from categoricity in abstract elementary classes
AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E05.
Abstract.
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstabilitylike property for nonsplitting, a particular notion of independence. We generalize their result as follows: given any abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the ShelahVillaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the ShelahVillaveces proof.
Key words and phrases:
Abstract elementary classes; Categoricity; Superstability; Splitting; Coheir; Independence; Forking1. Introduction
1.1. General motivation and history
Forking is one of the central notions of model theory, discovered and developed by Shelah in the seventies for stable and NIP theories [She78]. One way to extend Shelah’s firstorder stability theory is to move beyond firstorder. In the mid seventies, Shelah did this by starting the program of classification theory for nonelementary classes focusing first on classes axiomatizable in [She75] and later on the more general abstract elementary classes (AECs) [She87a]. Roughly, an AEC is a pair satisfying some of the basic categorytheoretic properties of (but not the compactness theorem). Among the central problems, there are the decadesold categoricity and eventual categoricity conjectures of Shelah. In this paper, we assume that the reader has a basic knowledge of AECs, see for example [Gro02] or [Bal09].
One key shift in this program is the move away from syntactic types (studied in the context by [She72, GS86b, GS86a] and others) and towards a semantic notion of type, introduced in [She87b] and named Galois type by Grossberg [Gro02].^{1}^{1}1Shelah uses the name orbital types in some later papers. This has an easy definition when the class has amalgamation, joint embedding and no maximal models, as these properties allow us to assume that all the elements of we would like to discuss are substructures of a “monster” model . In that case, is defined as the orbit of under the action of the group on . One can also develop the notion of Galois type without the above assumption, however then the definition is more technical.
1.2. Independence, superstability, and no long splitting chains in AECs
In [She99] a first candidate for an independence relation was introduced: the notion of splitting (for both in , splits over provided there are , and such that ).
This notion was used by Shelah to establish a downward version of his categoricity conjecture from a successor for classes having the amalgamation property. Later similar arguments [GV06b, GV06a] were used to derive a strong upward version of Shelah’s conjecture for classes satisfying the additional locality property of (Galois) types called tameness.
In Chapter II of [She09], Shelah introduced good frames: an axiomatic definition of forking on Galois types over models of size . The notion is, by definition, required to satisfy basic properties of forking in superstable firstorder theories (e.g. symmetry, extension, uniqueness, and local character). The theory of good frames is welldeveloped and has had several applications to the categoricity conjecture (see Chapters III and IV of [She09] and recent work of the fourth author [Vasb, Vasc, Vasa, Vas17]).
Constructions of good frames rely on weaker independence notions like nonsplitting, see e.g. [Vas16, VV]. A key property of splitting in these constructions is that there is “no long splitting chains in ”: if is an increasing continuous chain in (so is a limit ordinal) and is universal over for each , then for any there exists so that does not split over (this is called strong universal local character at in the present paper, see Definition 6). This can be seen as a replacement for the statement “every type does not fork over a finite set”. The property is already studied in [She99], and has several nontrivial consequences: for example (assuming amalgamation, joint embedding, no maximal models, stability in , and tameness), no long splitting chains in implies that is stable everywhere above [Vas16, Theorem 5.6] and has a good frame on the subclass of saturated models of cardinality [VV, Corollary 6.14]. No long splitting chains has consequences for the uniqueness of limit models, another superstabilitylike property saying in essence that saturated models can be built in few steps (see for example [SV99, Van06, Van13, Van16]).
The first and second authors have explored another approach to independence by adapting the notion of coheir to AECs. They have shown that for classes satisfying amalgamation which are also tame and short (a strengthening of tameness, using the variables of a type instead of its parameters), failure of a certain order property implies that coheir has some basic properties of forking from a stable firstorder theory. There the “no long coheir chain” property also has strong consequences (for example on the uniqueness of limit models [BG, Corollary 6.18]).
1.3. No long splitting chains from categoricity
It is natural to ask whether no long splitting chains (or no long coheir chains) in follows from categoricity above . Shelah has shown that this holds for splitting (assuming amalgamation and no maximal models) if the categoricity cardinal has cofinality greater than [She99, Lemma 6.3]. Without any cofinality restriction, a breakthrough was made in a paper of Shelah and Villaveces when they proved no long splitting chains assuming no maximal models and instances of diamond [SV99, Theorem 2.2.1]. Later, Boney and Grossberg used the ShelahVillaveces argument to derive the result in their context also for coheir [BG, Theorem 6.8]. It was also observed that the ShelahVillaveces argument does not need diamond if one assumes full amalgamation [GV, 5.3]. In conclusion we have:
Fact 1.
Let be an AEC with no maximal models. Let and assume that is categorical in .
Remark 2.
Fact 1 has applications to more “concrete” frameworks than AECs. One can deduce from it (and the aforementioned fact that no long splitting chains implies stability on a tail in the presence of tameness) an alternate proof that a firstorder theory categorical above is superstable. More generally, one obtains the same statement for the class of models of a homogeneous diagram in [She70]. The later was open for uncountable and categorical in (see [Vasa, Section 4]).
1.4. Gaps in the ShelahVillaveces proof
In a preliminary version of [BG], the proof of Theorem 6.8 referred to the argument used in [SV99, Theorem 2.2.1]. The referee of [BG] insisted that the full argument necessary for Theorem 6.8 be included. After looking closely at the argument in [SV99], we concluded that there was a small gap in the division of cases and a need to specify the exact use of the club guessing principle that they imply.
More specifically, Shelah and Villaveces [SV99, Theorem 2.2.1] assume for a contradiction that no long splitting chains fails and can divide the situation into three cases, (a), (b), and (c). In the division into cases [SV99, Claim 2.2.3], just after the statement of property , Shelah and Villaveces claim that they can “repeat the procedure above” on a certain chain of models of length . However the “procedure above” was used on a chain of length , where is a regular cardinal and regularity was used in the proof. As is a potentially singular cardinal, there is a problem.
Once the division of cases is done, Shelah and Villaveces prove that cases (a), (b), (c) contradict categoricity. When proving this for (b), they use a clubguessing principle for on the stationary set of points of cofinality (see Fact 14). The principle only holds when , so the case is missing.
1.5. Statement and discussion of the main theorem
In this paper, we give a generalized, detailed, and corrected proof of Fact 1 that does not rely on any of the material in [SV99]. The key definitions are given at the start of the next section and the first seven hypotheses are collected in Hypothesis 8.
Theorem 3 (Main Theorem).
If:

is an AEC.

.

For every , there exists an amalgamation base such that .

For every amalgamation base , there exists an amalgamation base such that is universal over .

Every limit model in is an amalgamation base.

is as in Definition 6 with the class of amalgamation bases in (ordered with the strong substructure relation inherited from ).

satisfies invariance and monotonicity .

has weak universal local character at some cardinal .

has an EhrenfeuchtMostowski (EM) blueprint with such that every embeds inside (where we write ).
Then has strong universal local character at all limit ordinals .
Remark 4.
As in [SV99], when we say that is an amalgamation base we mean that it is an amalgamation base in the class , i.e. we do not require that larger models can be amalgamated over .
Some of the hypotheses of Theorem 3 may appear technical. Let us give a little more motivation.
How are the gaps mentioned in Section 1.4 addressed in our proof of Theorem 3? The first gap (in the division into cases) is fixed in Lemma 11.(4). The second gap (in the use of the club guessing principle) is addressed here by a division into cases in the proof of Theorem 3 at the end of this paper: there we use Lemma 13 only when .
Before starting to prove Theorem 3, we give several contexts in which its hypotheses hold. This shows in particular that Fact 1 follows from Theorem 3.
Corollary 5.
Let be an AEC with arbitrarily large models. Let and assume that is categorical in and has no maximal models. Then:
Proof.
Fix an EM blueprint for (with ). We first show that there exists an EM blueprint with such that any embeds inside . Let . Using no maximal models and categoricity, embeds inside , and hence inside for some with . Therefore also embeds inside , where . Now it is well known (see e.g. [Bal09, Claim 15.5]) that embeds inside . The class is an AEC, therefore by composing EM blueprints there exists an EM blueprint for such that and for any linear order . In particular, embeds inside , as desired.
As for the hypotheses on density of amalgamation bases, existence of universal extension, and limit models being amalgamation bases, in the first context this is proven in [SV99] (note that implies ). When has full amalgamation, existence of universal extension is due to Shelah. It is stated (but not proven) in [She99, Lemma 2.2]; see [Bal09, Lemma 10.5] for a proof.
In all the contexts given, it is trivial that satisfies and . In the first context, it can be shown that non splitting has weak universal local character at any such that (see the proof of case (c) in [SV99, Theorem 2.2.1] or [Bal09, Lemma 12.2]). Of course, this also holds when has full amalgamation. As for coheir, it has weak universal local character at any such that . This is given by the proof of [BG, Theorem 6.8] (note that using a back and forth argument, one can assume without loss of generality that any in the chain is saturated). ∎
1.6. Other advantages of the main theorem
As should be clear from Corollary 5, another advantage of the main theorem is that it separates the combinatorial set theory from the model theory (it holds in ZFC) and also shows that there is nothing special about splitting in [SV99].
Some results here are of independent interest. For example, any independence relation satisfying invariance and monotonicity has (assuming categoricity) a certain continuity property (see Lemma 13).
1.7. Acknowledgments
We thank the referee for comments that helped improve the presentation of this work.
This paper was written while the fourth author was working on a Ph.D. thesis under the direction of the second author at Carnegie Mellon University and he would like to thank Professor Grossberg for his guidance and assistance in his research in general and in this work specifically.
2. Proof of the main theorem
We now define the weak framework for independence that we use.
Definition 6.
Let be an abstract class^{2}^{2}2That is, a partial order such that is a class of structures in a fixed vocabulary closed under isomorphisms, is invariant under isomorphisms, and implies that is a substructure of . and be a 4ary relation such that if holds, then are all in and .

The following are several properties we will assume about (but we will always mention when we assume them).

has invariance if it is preserved under isomorphisms: if and , then .

has monotonicity if:

If , , and , then ; and:

If , is such that and , then .



and mean that this relation is really about Galois types, so we write does not fork over for .

For a limit ordinal , has weak universal local character at if for any increasing continuous sequence and any type , if is universal over for each , then there is some such that does not fork over .

For a limit ordinal , has strong universal local character at if for any increasing continuous sequence and any type , if is universal over for each , then there is some such that does not fork over .
Remark 7.

If are limit ordinals and has weak universal local character at , then has weak universal local character at , but this need not hold for strong universal local character (if say ).

If has and has strong universal local character at , then has strong universal local character at .

If has , strong universal local character at implies weak universal local character at .

If (as will be the case in this note) is a class of structures of a fixed size , then we only care about the properties when .
We collect the first seven hypotheses of Theorem 3 into a hypothesis that will be assumed for the rest of the paper.
Hypothesis 8.

is an AEC.

.

For every , there exists an amalgamation base such that .

For every amalgamation base , there exists an amalgamation base such that is universal over .

Every limit model in is an amalgamation base.

is as in Definition 6 with the class of amalgamation bases in (ordered with the strong substructure relation inherited from ).

satisfies invariance and monotonicity .
The proof of Theorem 3 can be decomposed into two steps. First, we study two more variations on local character: continuity and absence of alternations. We show that if strong local character fails but enough weak local character holds, then there must be some failure of continuity, or some alternations. Second, we show that categoricity (or more precisely the existence of a universal EM model in ) implies continuity and absence of alternations. The first step uses the weak local character (but not categoricity, it is essentially forking calculus) but the second does not (but does use categoricity).
The precise definitions of continuity and alternations are as follows.
Definition 9.
Let and be as in Definition 6 and let be a limit ordinal.

has universal continuity at if for any increasing continuous sequence and any type , if for each is universal over and does not fork over , then does not fork over .

For a limit, has no limit alternations at if for any increasing continuous sequence with limit over for all and any type , there exists such that the following fails: forks over and does not fork over . If this fails, we say that has limit alternations at .
Note that the failure of universal continuity and no limit alternation correspond respectively to cases (a) and (b) in the proof of [SV99, Theorem 2.2.1]. Case (c) there corresponds to failure of weak universal local character at (which is assumed to hold here, see (8) of Theorem 3).
The following technical lemmas and proposition implement the first step described after the statement of Hypothesis 8. In particular, Proposition 12 below says that if we can prove weak local character at some , continuity and no alternations at all , then strong local character at all follows. Lemma 11 is a collection of preliminary steps toward proving Proposition 12. Lemma 10 is used separately in the proof of the main theorem (it says that weak universal local character implies the absence of alternations). Throughout, recall that we are assuming Hypothesis 8.
Lemma 10.
Let be a (not necessarily regular) cardinal and be a limit ordinal. If has weak universal local character at , then has no limit alternations at .
Proof.
Fix , , as in the definition of having no limit alternations. Apply weak universal local character to the chain . ∎
We now outline the proof of Proposition 12. Again, it may be helpful to remember that we will later prove that (in the context of Theorem 3) continuity holds at all lengths and that there are no alternations.
Two important basic results are
The first of these is proven by contradiction, and the second is a straightforward argument using universality.
Assume for a moment we have strong universal local character at some limit length . Let us try to prove weak universal local character at (say) (then we can use the first basic result to get the strong version, assuming continuity). By the second basic result, we can assume we are given an increasing continuous sequence with limit over for all and . By the strong universal local character assumption we know that does not fork over some intermediate model between and , so if we assume that forks over for all , we will end up getting alternations. This is the essence of Lemma 11.(5).
Thus to prove strong universal local character at all cardinals, it is enough to obtain it at some cardinal. Fortunately in the hypothesis of Proposition 12, we are already assuming weak universal local character at some . If is regular we are done by the first basic result, but unfortunately could be singular. In this case Lemma 11.(4) (using Lemma 11.(3) as an auxiliary claim) shows that failure of strong universal local character at implies alternations, even when is singular.
Lemma 11.
Let be a regular cardinal, be a (not necessarily regular) cardinal, and be a limit ordinal.

If has universal continuity at and weak universal local character at , then has strong universal local character at .

Assume that has weak universal local character at . Let be increasing continuous in with universal over for all . For any there exists a successor such that does not fork over .

If has universal continuity at , weak universal local character at , and no limit alternations at , then has strong universal local character at .

Assume that has strong universal local character at . If does not have weak universal local character at , then has limit alternations at .
Proof.

Suppose that , is a counterexample.
Claim: For each , there exists such that forks over .
Proof of Claim: If is such that for all , does not fork over , then applying universal continuity at on the chain we would get that does not fork over , contradicting the choice of , .
Now define inductively for , , , and when is limit . Note that is strictly increasing continuous and implies (this uses regularity of ; when is singular, see (4)).
Apply weak universal local character to the chain and the type . We get that there exists such that does not fork over . This is a contradiction since and we chose so that forks over .

We prove the result for weak universal local character, and the proof for the strong version is similar. Fix , witnessing failure of weak universal local character at . We build a witness of failure , such that , and is limit over for each . Using existence of universal extensions, we can extend each to that is limit over . Since is universal over , we can find . Since limit models are amalgamation bases, is an amalgamation base. Now set for limit or and . This is an increasing continuous chain of amalgamation bases with limit over . Let .
This works: if there was an such that does not fork over , this would mean that does not fork over , but since , we have by that does not fork over , a contradiction.

Apply weak universal local character to the chain to get such that does not fork over . By monotonicity, this implies that does not fork over . Let .

Suppose not, and let be a counterexample. By (2), without loss of generality is limit over for all . As in the proof of (1), for each , there exists such that forks over . On the other hand, applying (3) to the chain , for each , there exists a successor ordinal such that does not fork over . Define by induction on , , , , and . By construction, the sequence witnesses that has limit alternations at , a contradiction.

Let . By (2), there exists , witnessing failure of weak universal local character at such that for all , is limit over . Let witness that is limit over (i.e. it is increasing continuous with universal over for all , , and ). By strong universal local character at , for all , there exists such that does not fork over . By replacing by if necessary we can assume without loss of generality that .
Observe also that for any , forks over (using and the assumption that forks over ). Therefore , witness that has limit alternations at .
∎
Proposition 12.
Let be a regular cardinal and be a (not necessarily regular) cardinal. Assume that has weak universal local character at . If has universal continuity at and , has no limit alternations at , and has no limit alternations at , then has strong universal local character at .
Proof.
The next lemma corresponds to the second step outlined at the beginning of this section. Note that the added assumption is (9) from the hypotheses of Theorem 3 and recall we are assuming Hypothesis 8 throughout.
Lemma 13.
Assume has an EM blueprint with such that every embeds inside . Let be a regular cardinal. Then:

has universal continuity at .

If in addition , then for any limit , has no limit alternations at .
Proof.
Let and be as in the definition of universal continuity or limit alternations. Let . We say that is an club sequence if each is club. Clearly, club sequences exist: just take (this will be enough for proving universal continuity). Shelah [She94] proves the existence of clubguessing club sequences in ZFC under various hypotheses (the specific result that we use will be stated later, see Fact 14). We will describe a construction of a sequence of models based on a club sequence and then plug in the necessary club sequence in each case.
Given an club sequence , enumerate in increasing order as .
Claim: Let be a limit ordinal. We can build increasing, continuous such that for all :

is limit over ;

when , there is such that for all ; and:

when , there is that realizes .
Proof of Claim: Build the increasing continuous chain of models as follows: start with an amalgamation base , which exists by Hypothesis 8.(3). Given an amalgamation base , build to be limit over it. This exists by Hypothesis 8.(4) of Theorem 3), and is an amalgamation base by Hypothesis 8.(5). At limits, it also guarantees we have an amalgamation base.
At limits of cofinality , use the uniqueness of limits models to find the desired isomorphisms: the weak version gives , and the strong (over the base) version allows this isomorphism to be extended to get an isomorphism between and as described. Since is universal over , we there is some that realizes .
By assumption, we may assume that . Thus, we can write with:
Now we begin to prove each part of the lemma. In each, we will find such that and are both the same (because of the EM structure) and different (because they exhibit different forking behavior), which is our contradiction.

Assume that does not fork over , for all .
Let be an club sequence, and set as in the Claim (the value of doesn’t matter here, e.g. take ). By Fodor’s Lemma, there is a stationary subset , a term , and ordinals , such that:

For every , we have ; ; ; for ; and .
Set is limit and . This is a club. Let both be in . Then we have:
where all the types are computed inside . This is because the only differences between and lie entirely above .
We have that and that forks over . Thus, forks over . On the other hand, is cofinal in , so there is such that and, thus, . Again, and does not fork over by assumption. Thus, does not fork over . By monotonicity (M), does not fork over . Thus, , a contradiction.


Let be a bigenough cardinal and create an increasing, continuous elementary chain of models of set theory such that for all :

;

;

contains, as elements^{3}^{3}3When we say that contains a sequence as an element, we mean that it contains the function that maps an index to its sequence element., , , , , , , , and each ; and

is an ordinal.
We will use the following fact which was originally proven in [She94, III.2] (or see [AM10, Theorem 2.17] for a short proof).
Fact 14.
Let be a cardinal such that for some regular and let be stationary. Then there is a club sequence such that, if is club, then there are stationarily many such that .
We have that , so we can apply Fact 14 with there standing for here. Let be the club sequence that the fact gives. Let be as in the Claim. Note that is a club. By the conclusion of Fact 14, there is some such that . We have , with:
Since the ’s enumerate a cofinal sequence in , we can find such that . Recall that we have does not fork over by assumption. Then satisfies the following formulas with parameters exactly the objects listed in item (2c) above and ordinals below :
This is witnessed by and . By elementarity, satisfies this formula as it contains all the parameters. Let ^{4}^{4}4The equality here is the key use of club guessing. witness this, along with . Then we have:
