Approximations and Mittag-Leffler conditions --- the applications

Abstract

A classic result by Bass says that the class of all projective modules is covering, if and only if it is closed under direct limits. Enochs extended the if-part by showing that every class of modules $\mathcal{C}$, which is precovering and closed under direct limits, is covering, and asked whether the converse is true. We employ the tools developed in [18] and give a positive answer when $\mathcal{C}=\mathcal{A}$, or $\mathcal{C}$ is the class of all locally ${\mathcal{A}}^{\le \omega }$-free modules, where $\mathcal{A}$ is any class of modules fitting in a cotorsion pair $(\mathcal{A},\mathcal{B})$ such that $\mathcal{B}$ is closed under direct limits. This setting includes all cotorsion pairs and classes of locally free modules arising in (infinite-dimensional) tilting theory. We also consider two particular applications: to pure-semisimple rings, and artin algebras of infinite representation type.

16 pages