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.Comment: 45 page
