Salce's problem on cotorsion pairs is undecidable

TL;DR

This paper demonstrates that the question of whether all cotorsion pairs of abelian groups are complete is independent of ZFC axioms, assuming the consistency of Vopěnka's Principle, highlighting the problem's undecidability.

Contribution

It proves the independence of Salce's problem from ZFC, contingent on Vopěnka's Principle, connecting set-theoretic assumptions with algebraic properties.

Findings

Salce's problem is independent of ZFC.

Under Vopěnka's Principle, the answer is affirmative.

The result links set theory with algebraic structures.

Abstract

Salce \cite{MR565595} introduced the notion of a \emph{cotorsion pair} of classes of abelian groups, and asked whether every such pair is \emph{complete} (i.e., has enough injectives and projectives). We prove that it is consistent, relative to the consistency of Vop\v{e}nka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof-Shelah \cite{MR2031314}, this shows that Salce's Problem is independent of the ZFC axioms (modulo the consistency of VP).