Approximation properties of torsion classes

TL;DR

This paper explores the deep connections between large cardinal axioms and torsion classes in abelian groups, establishing new implications and distinctions in their properties and the principles governing them.

Contribution

It strengthens existing results linking large cardinals to torsion classes and introduces the Maximum Deconstructibility principle, clarifying its place among large cardinal assumptions.

Findings

Maximum Deconstructibility requires large cardinals

Deconstructibility implies precovering, but they are not equivalent

The concepts are non-equivalent even for certain classes of abelian groups

Abstract

We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in \cite{Cox_MaxDecon} requires large cardinals; it sits, implication-wise, between Vop\v{e}nka's Principle and the existence of an $\omega_1$-strongly compact cardinal. (2) While deconstructibility of a class of modules always implies the precovering property by \cite{MR2822215}, the concepts are (consistently) non-equivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.