[Paper Review] GCH implies the existence of many rigid almost free abelian groups

Rüdiger Göbel, Saharon Shelah|ArXiv.org|Nov 22, 2000

Advanced Topology and Set Theory29 references22 citations

TL;DR

This paper establishes that under the Generalized Continuum Hypothesis (G.C.H.), there exist strongly λ-free abelian groups of uncountable cardinality λ ≥ ℵ₁ with prescribed countable, free endomorphism rings and trivial duals. Using a novel Step-Lemma and combinatorial tools from Shelah’s forcing and ladder systems, the authors construct such rigid almost-free groups by lifting properties through successive cardinal stages, extending previous ZFC results to larger cardinals.

ABSTRACT

We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal and such groups of cardinality aleph_omega do not exist by the well-known Singular Compactness Theorem (see [Sh:52]).

Motivation & Objective

• To extend ZFC results on ℵ₁-free abelian groups with prescribed endomorphism rings to larger uncountable cardinals.
• To investigate whether such rigid almost-free abelian groups can exist without assuming V = L or Martin’s Axiom, under weaker set-theoretic assumptions.
• To determine the limits of G.C.H. in realizing endomorphism rings for strongly free abelian groups beyond ℵω.
• To develop new combinatorial and algebraic techniques—particularly a generalized Step-Lemma and u-freeness conditions—for constructing such groups at limit stages.
• To show that the existence of these groups is consistent with ZFC + G.C.H., even at singular limit cardinals like ℵω+1, under specific freeness type constraints.

Proposed method

• Constructing strongly λ-free abelian groups via continuous λ-filtrations with free submodules and free quotients at successor stages.
• Applying a new Step-Lemma to extend modules at successor stages while preserving endomorphism ring structure and trivial duals.
• Using n-ladder systems and stationary subsets to control freeness and endomorphism behavior at limit ordinals.
• Employing generalized u-freeness conditions (Proposition 7.3) to ensure freeness of unions at limit ordinals cofinal to certain cardinals.
• Leveraging Shelah’s combinatorial tools on ladder systems and prediction principles to guide the construction of the correct endomorphism ring.
• Verifying that the constructed group G satisfies EndZG = A and G∗= 0 by ensuring that extensions of homomorphisms vanish or lie in A via relations involving basic elements and fixed generators.

Experimental results

Research questions

• RQ1 Can strongly λ-free abelian groups with prescribed countable free endomorphism rings be constructed for λ > ℵ₁ under weaker set-theoretic assumptions than V = L?
• RQ2 What role does G.C.H. play in enabling the existence of such rigid almost-free groups beyond the ℵ₁ case?
• RQ3 How can freeness be preserved at limit ordinals in the construction of strongly free abelian groups when the cofinality is uncountable and not cofinal to ω?
• RQ4 To what extent can the construction be extended past ℵω, and what are the limitations imposed by singular compactness?
• RQ5 Is it possible to realize arbitrary countable free R-algebras as endomorphism rings of strongly free abelian groups of uncountable cardinality under G.C.H.?

Key findings

• Under ZFC + G.C.H., for any countable free R-algebra A, there exists a strongly ℵω+1-free R-module G of cardinality ℵω+1 with EndRG = A and freeness type ⟨ℵω+1, ℵ1⟩.
• The existence of strongly λ-free R-modules with trivial dual G∗= 0 and |G| = λ = 2µ = µ+ is established for regular uncountable µ, provided such a module H of size µ exists with H∗= 0.
• For any regular uncountable cardinal λ = µ+ and a strongly µ-free A-module H with EndRH = A and freeness type u = ⟨µ1, ..., µn⟩, a strongly λ-free G with EndRG = A exists if λ admits a stationary, non-small subset S with appropriate ladder systems.
• The construction succeeds at limit stages by using the generalized u-freeness condition (Proposition 7.3), which ensures that unions at limit ordinals remain free.
• The paper shows that the results cannot be extended beyond ℵω+1 in general, as models of ZFC + G.C.H. exist where ℵω2+1-free groups of size ℵω2+1 are actually free.
• The key technical innovation is a new Step-Lemma that allows control over endomorphism ring and dual structure during the inductive construction of the group at successor stages.