Enochs' conjecture for small precovering classes of modules

TL;DR

This paper proves Enochs' conjecture for small precovering classes of modules under certain set-theoretic assumptions, showing these classes are closed under direct limits and providing conditions for perfect decompositions.

Contribution

It establishes the conjecture for classes of the form Add(M) with new set-theoretic conditions, extending the understanding of module covering classes.

Findings

Enochs' conjecture holds for small precovering classes under specific assumptions.

Modules in these classes have perfect decompositions.

Redundancy of assumptions when modules decompose into sums of finitely generated modules.

Abstract

Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this short paper, we prove the validity of the conjecture for small precovering classes, i.e. the classes of the form $\mathrm{Add}(M)$ where $M$ is any module, under a mild additional set-theoretic assumption which ensures that there are enough non-reflecting stationary sets. We even show that $M$ has a perfect decomposition if $\mathrm{Add}(M)$ is a covering class. Finally, the additional set-theoretic assumption is shown to be redundant if there exists an $n<\omega$ such that $M$ decomposes into a direct sum of $\aleph_n$-generated modules.