Vop\v{e}nka's Principle, Maximum Deconstructibility, and singly-generated torsion classes

TL;DR

This paper establishes the equivalence between Vopěnka's Principle, Maximum Deconstructibility, and the single-generation of torsion classes, revealing deep connections in module theory and set-theoretic principles.

Contribution

It proves that Maximum Deconstructibility is equivalent to Vopěnka's Principle and the single-generation of torsion classes, linking set theory with module theory.

Findings

MD is equivalent to Vopěnka's Principle.

MD implies classes like Gorenstein Projective modules are deconstructible.

Each torsion class of abelian groups is generated by a single group.

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 Vop\v{e}nka's Principle and imply the existence of an $\omega_1$-strongly compact cardinal. We prove that MD is equivalent to Vop\v{e}nka'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 G\"obel and Shelah).