Approximations and Mittag-Leffler conditions --- the applications

TL;DR

This paper investigates conditions under which classes of modules are covering, extending classical results by Bass and Enochs, with applications to pure-semisimple rings and artin algebras.

Contribution

It provides a positive answer to Enochs' question in new settings involving cotorsion pairs and locally free modules, broadening the scope of module covering theory.

Findings

Class of all projective modules is covering iff closed under direct limits.

Extended Enochs' result to classes of modules in cotorsion pairs with certain closure properties.

Applications to pure-semisimple rings and artin algebras of infinite representation type.

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 $\mathcal A ^{\leq \omega}$-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.