Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ae<currency> a"mu(0), [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits. In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs' problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].