Alex Perry wrote a paper about faithfully flat descent of projectivity for modules, and submitted it to the stacks project. His writeup largely follows the exposition in Raynaud and Gruson’s paper [RG]. But, as Brian Conrad mentioned here, there is an error in [RG]. Alex avoids this error by proving that the Mittag-Leffler condition satisfies flat descent in the presence of flatness.
The result of the title of this blog post is Theorem Tag 05A9 of the stacks project. To get an (almost) self contained proof of the theorem start reading the introductory Section Tag 058B entitled “Faithfully flat descent for projectivity of modules”.
Edit: You can now find Alex’s write-up at arXiv:1011.0038v1 [math.AC].