Approximations and Mittag-Leffler conditions the applications

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 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 C = A , or C is the class of all locally A ≤ ω -free modules, where A is any class of modules fitting in a cotorsion pair ( A , B ) such that 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.