Mittag-Leffler conditions on modules Lidia Angeleri HugelDolors Herbera 16D7016E3018E15Mittag-Leffler inverse systemcotorsion pairstilting modulesBaer modulespure semisimple rings We 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. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3325 10.1512/iumj.2008.57.3325 en Indiana Univ. Math. J. 57 (2008) 2459 - 2518 state-of-the-art mathematics http://iumj.org/access/