2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128049We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.45 pagesRings and AlgebrasK-Theory and Homology16D70;16L30;18E15text