The Stacks project

Example 10.91.1. Mittag-Leffler modules.

1. Any finitely presented module is Mittag-Leffler. This follows, for instance, from Proposition 10.88.6 (1). In general, it is true that a finitely generated module is Mittag-Leffler if and only it is finitely presented. This follows from Propositions 10.89.2, 10.89.3, and 10.89.5.

2. A free module is Mittag-Leffler since it satisfies condition (1) of Proposition 10.88.6.

3. By the previous example together with Lemma 10.89.10, projective modules are Mittag-Leffler.