Mittag-Leffler conditions on modules

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

Tilting and cotilting modules over concealed canonical algebras

We study infinite dimensional tilting modules over a concealed canonical algebra of domestic or tubular type. In the domestic case, such tilting modules are constructed by using the technique of universal localization, and they can be interpreted in terms of Gabriel localizations of the corresponding category of quasi-coherent sheaves over a noncommutative curve of genus zero. In the tubular case, we have to distinguish between tilting modules of rational and irrational slope. For rational slope the situation is analogous to the domestic case. In contrast, for any irrational slope, there is just one tilting module of that slope up to equivalence. We also provide a dual description of infinite dimensional cotilting modules and a classification result for the indecomposable pure-injective modules.Comment: 25 page