Definable Classes and Mittag-Leffler Conditions
Dolors Herbera · Contemporary mathematics - American Mathematical Society · 2014
We make a systematic approach to (strict) Mittag-Leffler inverse system and to dual Mittag-Leffler direct systems. This allows us to prove that a right R R -module M M is Mittag-Leffler with respect to a definable class of left modules Q \mathcal {Q} if and only if it is strict stationary with respect to the dual definable class of Q \mathcal {Q} . We also study when classes defined via vanishing either of E x t \mathrm {Ext} functors or T o r \mathrm {Tor} functors are definable. Surprisingly enough, Mittag-Leffler conditions appear naturally in this context. For M M finitely generated and countably presented, we prove that the functor E x t R 1 ( M , − ) \mathrm {Ext}_R^1(M,-) is coherent if and only if so is T o r 1 R ( M , − ) \mathrm {Tor}_1^R(M,-) , and this happens if and only if M 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, they give new examples of non precovering classes.