Vopěnka’s Principle, Maximum Deconstructibility, and Singly-Generated Torsion Classes

Deconstructibility is an often-used sufficient condition on a class \(\mathcal {C}\) of modules that allows one to carry out homological algebra relative to \(\mathcal {C}\). The principle 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 \(\omega _1\)-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).