Remarks on classification theory for abstract elementary classes with applications to abelian group theory and ring theory

 

This thesis has two parts. The first part deals with the classification theory of abstract elementary classes and the second part deals with links and applications of this theory to algebra.

Part I: Remarks on classification theory for abstract elementary classes 

This part of the thesis is made up of three chapters based on the corresponding papers: [Ch. 2], [Ch. 3] (a joint work with S. Vasey), and [Ch. 4] (a joint work with R. Grossberg).

Chapter 2, Non-forking w-good frames. We introduce and study the notion of a w-good λ-frame which is a weakening of Shelah’s notion of a good λ-frame. W-good λ-frames are useful as they imply the existence of larger models. We show that if K has a w-good λ-frame, then K has a model of size λ ++. This result extends [Sh:h, §II.4.13.3], [JaSh13, 3.1.9], and [Vas16a, 8.9].

Chapter 3, Universal classes near ℵ1 (a joint work with S. Vasey). Shelah has provided sufficient conditions for an L_ω1,-ωsentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ℵ_n’s. Using tools of Boney, Shelah, and Vasey, we give assumptions on ℵ₀ and ℵ₁ which suffice when ψ is restricted to be universal.

Chapter 4, Simple-like independence relations in abstract elementary classes (a joint work with R. Grossberg). We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. We show that if K has a simple independence relation with the (< ℵ₀)-witness property for singletons, then K does not have the tree property. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

Part II: Applications to abelian group theory and ring theory

This part of the thesis is made up of seven chapters based on the corresponding papers: [Ch. 5], [Ch. 6], [Ch. 7] (a joint work with T.G. Kucera), [Ch. 8], [Ch. 9], [Ch. 10], and [Ch. 11]. Chapters 5 and 6 deal with abelian groups and Chapters 7 - 11 with modules over associative rings with unity.

Chapter 5, Algebraic description of limit models in classes of abelian groups. We study limit models in the class of abelian groups with the subgroup relation and in iii the class of torsion-free abelian groups with the pure subgroup relation. We show that the former are divisible groups. As for the latter, we show that long limit models are pure-injective while short ones are not pure-injective. This is the first place where explicit examples of limit models are studied.

Chapter 6, A model theoretic solution to a problem of László Fuchs. Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals λ such that there is a universal abelian p-group for purity of cardinality λ, i.e., an abelian p-group U_λ of cardinality λ such that every abelian p-group of cardinality ≤ λ purely embeds in U_λ. In this chapter we use ideas from the theory of abstract elementary classes to show:

Theorem. Let p be a prime number. If λ ℵ0 = λ or ∀µ < λ(µ ℵ0 < λ), then there is a universal abelian p-group for purity of cardinality λ. Moreover for n ≥ 2, there is a universal abelian p-group for purity of cardinality ℵn if and only if 2ℵ0 ≤ ℵn.

Chapter 7, On universal modules with pure embeddings (a joint work with T.G. Kucera). We show that if T is a first-order theory (not necessarily complete) with an infinite model extending the theory of R-modules and K^T = (Mod(T), ≤p) has joint embedding and amalgamation, then K^T has a universal model of cardinality λ if λ ^|T| = λ or ∀µ < λ(µ ^|T| < λ). A corollary of this result is [Sh820, 1.2] which asserts the existence of universal models in the class of reduced torsion-free groups with pure embeddings.

Chapter 8, Superstability, noetherian rings and pure-semisimple rings. We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We show:

Theorem. Let R be an associative ring with unity.

1. R is left noetherian if and only if the class of left R-modules with embeddings

is superstable.

2. R is left pure-semisimple if and only if the class of left R-modules with pure embeddings is superstable.

Chapter 9, On superstability in the class of flat modules and perfect rings. We study the notion of superstability in the class of flat modules with pure embeddings.

We show:

Theorem. Let R be an associative ring with unity. R is left perfect if and only if the class of flat left R-modules with pure embeddings is superstable. 

It is worth mentioning that the class of flat left R-modules is not first-order axiomatizable for most rings.

Chapter 10, A note on torsion modules with pure embeddings. We study s-tosion modules with pure embeddings as an abstract elementary class. We analyse its limit models and determine when the class is superstable under the assumption that the ring is right semihereditary. In order to fulfill this goal, we develop relative notions of pure-injectivity and Σ-pure-injectivity. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable, i.e., stable not superstable.

Chapter 11, Some stable non-elementary classes of modules. We address the question of whether every AEC of modules with pure embeddings is stable. We show that many non-elementary classes are stable, among them: absolutely pure modules, locally pure-injective modules, flat modules, and s-torsion modules. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, dedekind domains, and fields via superstability. We also show that these results can be used to obtain universal models.