10.79 Faithfully flat descent for projectivity of modules
In the next few sections we prove, following Raynaud and Gruson [GruRay], that the projectivity of modules descends along faithfully flat ring maps. The idea of the proof is to use dévissage à la Kaplansky [Kaplansky] to reduce to the case of countably generated modules. Given a well-behaved filtration of a module $M$, dévissage allows us to express $M$ as a direct sum of successive quotients of the filtering submodules (see Section 10.83). Using this technique, we prove that a projective module is a direct sum of countably generated modules (Theorem 10.83.5). To prove descent of projectivity for countably generated modules, we introduce a “Mittag-Leffler” condition on modules, prove that a countably generated module is projective if and only if it is flat and Mittag-Leffler (Theorem 10.92.3), and then show that the property of being a Mittag-Leffler module descends (Lemma 10.94.1). Finally, given an arbitrary module $M$ whose base change by a faithfully flat ring map is projective, we filter $M$ by submodules whose successive quotients are countably generated projective modules, and then by dévissage conclude $M$ is a direct sum of projectives, hence projective itself (Theorem 10.94.5).
We note that there is an error in the proof of faithfully flat descent of projectivity in [GruRay]. There, descent of projectivity along faithfully flat ring maps is deduced from descent of projectivity along a more general type of ring map ([Example 3.1.4(1) of Part II, GruRay]). However, the proof of descent along this more general type of map is incorrect. In [G], Gruson explains what went wrong, although he does not provide a fix for the case of interest. Patching this hole in the proof of faithfully flat descent of projectivity comes down to proving that the property of being a Mittag-Leffler module descends along faithfully flat ring maps. We do this in Lemma 10.94.1.
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