A note on CM-triviality and the geometry of forking

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.

Geometry of *-Finite Types

Assume T is a superstable theory with < 2ℵ0 countable models. We prove that any *- algebraic type of -rank > 0 is m-nonorthogonal to a *-algebraic type of -rank 1. We study the geometry induced by m-dependence on a *-algebraic type p* of -rank 1. We prove that after some localization this geometry becomes projective over a division ring . Associated with p* is a meager type p. We prove that p is determined by p* up to nonorthogonality and that underlies also the geometry induced by forking dependence on any stationarization of p. Also we study some *-algebraic *-groups of -rank 1 and prove that any *-algebraic *-group of -rank 1 is abelian-by-finite.

Flat Morley sequences

Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p, in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ-atomic over {Q}. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I, with I ⊂ dcl(Q), such that M is prime over I.

On the number of nonisomorphic models of size |T|

Let T be an uncountable, superstable theory. In this paper we proveTheorem A. If T has finite rank, then I(|T|, T) ≥ ℵ0.Theorem B. If T is trivial, then I(|T|, T) ≥ ℵ0.

Some trivial considerations

At the source of what is now known as “geometric stability theory” was Zil'ber's intuition that the essential properties of an aleph-one-categorical theory were controlled by the geometries of its minimal types. (However, the situation is much more complex than was assumed in Zil'ber [1984], since the main conjecture of that paper has been disproved by Hrushovski.) This is not unnatural in this unidimensional case, where all these geometries have isomorphic contractions, but it was even realized later, in Cherlin, Harrington and Lachlan [1985] and Buechler [1986], that, for any superstable theory with finite ranks, a certain “local” property, i.e. a property satisfied by the geometry of each type of rank one (namely: to have a projective contraction), was equivalent to a “global” one (the theory is one-based, hence satisfies a coordinatization lemma). Then it was shown, in Pillay [1986], that this situation does not generalize to the infinite rank case, that, even for a theory of rank omega, the (local) assumption of projectivity for all the regular types of the theory does not have an exact global counterpart.To clarify this kind of phenomena, I suggest here the elimination of their geometrical aspect, considering only the case where all of the geometries are degenerate. I will study various notions of triviality, which make sense in a stable context, and turn out to be equivalent in the finite rank case; some of them have a definite global flavour, others are of local character.

A note on subgroups of the automorphism group of a saturated model, and regular types

Let M be a saturated model of a superstable theory and let G = Aut(M). We study subgroups H of G which contain G(A), A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p-simple types.

Limit ultrapowers and abstract logics

We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L.For every countably generated [ω, ω]-compact logic L, our main applications are:(i) Elementary classes of L can be characterized in terms of ≡L only.(ii) If and are countable models of a countable superstable theory without the finite cover property, then .(iii) There exists the “largest” logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω, ω]-compact and satisfies an interpolation property stronger than unrelativized ⊿-closure.(iv) If L = Lωω(Qx), then cf(ωx) > ω and λω < ωx, for all λ < ωx.We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.

Isolated types in a weakly minimal set

Theorem A. Let T be a small superstable theory, A a finite set, and ψ a weakly minimal formula over A which is contained in some nontrivial type which does not have Morley rank. Then ψ is contained in some nonalqebraic isolated type over A.As an application we proveTheorem B. Suppose that T is small and superstable, A is finite, and there is a nontrivial weakly minimal type p ∈ S(A) which does not have Morley rank. Then the prime model over A is not minimal over A.

A note on nonmultidimensional superstable theories

In this paper we prove that if T is the complete elementary diagram of a countable structure and is a theory as in the title, then Vaught's conjecture holds for T. This result is Theorem 7, below. In the process of establishing this proposition, in Theorem 3 we give a sufficient condition for a superstable theory having only countably many types without parameters to be ω-stable. Familiarity with the rudiments of stability theory, as presented in [3] and [4], will be supposed throughout. The notation used is, by now, standard.We begin by giving a new proof of a lemma due to J. Saffe in [6]. For T stable, recall that the multiplicity of a type p over a set A ⊆ ℳ ⊨ T is the cardinality of the collection of strong types over A extending p.Lemma 1 (Saffe). Let T be stable, A ⊆ ℳ ⊨ T. If t(b̄, A) has infinite multiplicity and t(c̄, A) has finite multiplicity, then t(b̄, A ∪ {c̄}) has infinite multiplicity.Proof. We suppose not and work for a contradiction. Let ‹b̄γ:γ ≤ α›, α ≥ ω, be a list of elements so that t(b̄γ, A) = t(b̄, A) for all γ ≤ α, and st(b̄γ, A) ≠ st(b̄δ, A) for γ ≠ δ. Furthermore, let c̄γ satisfy t(b̄γ∧c̄γ, A) = t(b̄ ∧ c̄, A) for each γ < α.Since t(c̄, A) has finite multiplicity, we may assume for all γ, δ < α. that st(c̄γ, A) = st(c̄δ, A). For each γ < α there is an automorphism fγ of the so-called “monster model” of T (a sufficiently large, saturated model of T) that preserves strong types over A and is such that f(c̄γ) = c̄0.