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.

💡 Research Summary

The paper develops a relative version of the classical Mittag‑Leffler condition for modules and shows how this framework unifies several deep results in homological algebra and representation theory. After recalling the original Raynaud‑Gruson theorem, the authors introduce the notion of a “𝒞‑Mittag‑Leffler” module: a module M such that for every directed system (N_i) with transition maps in a fixed class 𝒞, the canonical map
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