Vopěnka's Principle, Maximum Deconstructibility, and singly-generated torsion classes

TL;DR

This paper proves the equivalence $VP \iff MD$, and shows this is equivalent to every torsion class of abelian groups being deconstructible and singly-generated. It builds a transfinite filtration framework $T^{\mathcal{C}}_{\alpha}$ culminating in $T^{\mathcal{C}}_{\infty}=T(\mathcal{C})$, and uses Przeździecki's functor $\mathcal{F}: \mathbf{Graphs} \to \mathbf{Ab}$ to translate a potential failure of $VP$ into a rigid, ungenerateable torsion class, yielding a contradiction. By tying set-theoretic principles to module-theoretic deconstructibility, the results unify conditions under which Gorenstein Projective and Ding Projective classes are deconstructible and relate to torsion theory. The work also clarifies the relationship between torsion-class generation by a single abelian group and $VP$, revealing precise cardinal-strength connections and influencing precovering phenomena in module theory.

Abstract

Deconstructibility is an often-used sufficient condition on a class $\mathcal{C}$ of modules that allows one to carry out homological algebra \emph{relative to $\mathcal{C}$}. The principle \textbf{Maximum Deconstructibility (MD)} asserts that a certain necessary condition for a class to be deconstructible is also sufficient. MD implies, for example, that the classes of Gorenstein Projective modules, Ding Projective modules, their relativized variants, and all torsion classes are deconstructible over any ring. MD was known to follow from Vopěnka's Principle and imply the existence of an $ω_1$-strongly compact cardinal. We prove that MD is equivalent to Vopěnka's Principle, and to the assertion that each torsion class of abelian groups is generated by a single group within the class (yielding the converse of a theorem of Göbel and Shelah).

Theorems & Definitions (10)

• Theorem 1.1
• proof
• Proposition 3.1
• proof
• Definition 3.2
• Lemma 3.3
• proof
• Theorem 3.4: Przeździecki MR3187657
• Claim 3.0.1
• proof