Definable Classes and Mittag-Leffler Conditions

We make a systematic approach to (strict) Mittag-Leffler invese system and to dual Mittag-Leffler direct systems. This allows us to prove that a right $R$-module $M$ is Mittag-Leffler with respect to a definable class of left modules $\mathcal{Q}$ if and only if it is strict stationary with respect to the dual definable class of $\mathcal{Q}$.

We also study when classes defined via vanishing either of $\mathrm{Ext}$ functors or $\mathrm{Tor}$ functors are definable, surprisingly enough, Mittag-Leffler conditions apear naturally in this context. For $M$ finitely generated and countably presented, we prove that the functor $\mathrm{Ext}_R(M,-)$ is coherent if and only if so is $\mathrm{Tor}_R(M,-)$ and this happens if and only if $M$ and its first syzygy are finitely presented.

Finally we also show that suitable classes of relative Mittag-Leffler modules give new examples of non deconstructible classes and, over countable rings, new examples of non precovering classes.