Enochs Conjecture for cotorsion pairs and more

Abstract

Enochs Conjecture asserts that each covering class of modules (over any 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 paper, we prove the conjecture for the classes $\mathrm{Filt}(\mathcal S)$ where $\mathcal S$ consists of $\aleph_n$-presented modules for some fixed $n<\omega$. In particular, this applies to the left-hand class of any cotorsion pair generated by a class of $\aleph_n$-presented modules. Moreover, we also show that it is consistent with ZFC that Enochs Conjecture holds for all classes of the form $\mathrm{Filt}(\mathcal{S})$ where $\mathcal{S}$ is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that the Enochs Conjecture holds (possibly under some additional set-theoretic assumption).