Approximations and Mittag-Leffler conditions --- the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. 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 [14], and it does not provide for approximations when $R$ has cardinality $\leq \aleph_0$, [6]. We remove the cardinality restriction on $R$ in the latter result. We also prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair $(\mathcal A,\mathcal B)$ is of countable type whenever the class $\mathcal B$ 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 facts above 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 [24]. 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 [20].