## 10.80 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 10.84). Using this technique, we prove that a projective module is a direct sum of countably generated modules (Theorem 10.84.5). 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 10.93.3), and then show that the property of being a Mittag-Leffler module descends (Lemma 10.95.1). 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 10.95.6).

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 10.95.1.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (4)

Comment #1088 by Nuno Cardoso on

Comment #1093 by Johan on

Comment #1095 by Nuno Cardoso on

Comment #1129 by Johan on