Deconstructible classes of modules and stability

We show that every deconstructible class of modules with all embeddings, all pure embedding and all RD-embeddings is stable. The argument is presented in the context of abstract classes of modules without amalgamation and the key idea is to construct a stable-like independence relation. In particular, the following classes of modules with all embeddings, all pure embedding and all RD-embeddings are shown to be stable: all free and torsion-free modules over any ring, and for each $n \geq 0$, the classes of all modules of projective and flat dimension $\leq n$ over any ring, and the class of all modules of injective dimension $\leq n$ over any right noetherian ring.

💡 Research Summary

The paper “Deconstructible classes of modules and stability” establishes that any deconstructible class of modules is stable when equipped with the class of all embeddings, all pure embeddings, or all RD‑embeddings. The authors work in a setting that does not assume amalgamation, which distinguishes their approach from much of the existing literature on abstract elementary classes (AECs).

The main technical framework begins with the notion of a κ‑deconstructible class D of right R‑modules: D is generated by a filtration whose factors are <κ‑presented modules belonging to D. Standard results such as the Hill Lemma guarantee that any submodule of a D‑filtered module can be approximated by a submodule of size at most |B|+κ, which is crucial for establishing a Löwenheim‑Skolem‑Tarski (LST) number for the abstract class (D, ≤_M), where ≤_M denotes one of the three embedding relations (all, pure, or RD).

The authors then define an abstract class (D, ≤_M) and verify that it satisfies all AEC axioms except possibly closure under unions of ≤_M‑chains. They show that when D is closed under direct summands, this missing condition reduces to closure under direct limits, a property enjoyed by many natural deconstructible classes (e.g., projective, flat, and injective dimension ≤ n).

The core of the stability proof is the construction of a “stable‑like” independence relation ⟂ based on pushouts (Definition 3.2). For submodules A, B ⊆ M, the relation A ⊥_M B holds when a certain pushout diagram yields a strong embedding (a morphism belonging to the chosen class of embeddings). Lemmas 3.13 and 3.15 establish existence and uniqueness of non‑forking extensions with respect to this relation, while Lemma 3.16 proves a local character property for strong embeddings. Together these three properties mimic the classical definition of a stable independence relation in AECs.

The central theorem (Theorem 3.18) states that if λ satisfies λ^κ = λ (where κ is the cardinal witnessing κ⁺‑deconstructibility of D), then (D, ≤_M) is λ‑stable: for every M∈D with |M|≤λ, the number of Galois types over M is at most λ. The proof proceeds by showing that any type over a large model is determined by its restriction to a small strong submodule, using the independence relation constructed earlier.

Concrete applications are given. The authors verify that the following classes are deconstructible and thus satisfy the theorem:

• All free modules and all torsion‑free modules over any ring.
• For each n≥0, the classes P_n (projective dimension ≤ n) and F_n (flat dimension ≤ n) over any ring.
• For each n≥0, the class I_n (injective dimension ≤ n) over any right Noetherian ring.

Consequently, each of these classes is stable with respect to all three embedding notions. This generalizes earlier results that dealt only with pure embeddings (e.g., stability of torsion‑free abelian groups, flat modules) and extends them to the broader context of arbitrary embeddings and RD‑embeddings.

The paper also discusses the relationship with recent work of Paolini and Shelah, who produced a counterexample showing that an AEC of torsion‑free abelian groups with pure subgroups can be unstable. The present work shows that even without amalgamation, deconstructible classes retain stability, highlighting a distinction between the combinatorial properties of deconstructibility and the model‑theoretic property of amalgamation.

In summary, the authors develop a novel independence calculus based on pushouts, prove that it satisfies the key axioms of stability, and apply it to a wide range of natural module classes. Their results demonstrate that deconstructibility is a robust sufficient condition for stability in the absence of amalgamation, thereby broadening the scope of classification theory for modules beyond the traditional AEC framework.