Test sets for factorization properties of modules

TL;DR

This paper investigates factorization properties of modules related to injectivity and projectivity, exploring their dependence on set-theoretic axioms and ring structures, with results on independence and accessibility.

Contribution

It generalizes factorization properties using cotorsion pairs and shows independence results for projectivity in certain rings, also establishing conditions for accessibility of projective modules.

Findings

Injectivity is a factorization property w.r.t. a single monomorphism.

The assertion about projectivity as a factorization property is independent of ZFC + GCH.

The category of all projective modules is accessible under the existence of a strongly compact cardinal.

Abstract

Baer's Criterion of injectivity implies that injectivity of a module is a factorization property w.r.t. a single monomorphism. Using the notion of a cotorsion pair, we study generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring $R$ and on additional set-theoretic hypotheses. For $R$ commutative noetherian of Krull dimension $0 < d < \infty$, we show that the assertion `projectivity is a factorization property w.r.t. a single epimorphism' is independent of ZFC + GCH. We also show that if $R$ is any ring and there exists a strongly compact cardinal $\kappa > |R|$, then the category of all projective modules is accessible.