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A point worth noting: the proof of fpqc descent for projectivity in Raynaud-Gruson is apparently incorrect (as I learned today from Gabber in connection with something else), but the result is nonetheless true.

Here's the deal. RG deduce the result in 3.1.4(1) of part II of the paper, using 3.1.3 of part II and the fact that faithfully flat ring maps satisfy the property they call (C) there. (Briefly, a ring map satisfies property (C) when flat modules over the base ring which satisfy a certain "Mittag-Leffler" condition after the scalar extension actually satisfy the ML condition before the scalar extension. The content of 3.1.3 in part II is that this condition (C) implies descent of projectivity for flat modules. So the problem is to prove an interesting class of maps satisfies property (C).) But RG's proof of (C) for faithfully flat ring maps in 3.1.4(1) of part II rests on another result (2.5.2 in part II) which Gruson has said is incorrect (in his paper "Dimension homologique...."). That's the problem. Gabber says he does not know a counterexample to this 2.5.2 part II result. (I guess Gruson didn't give one when he said it is false.) Anyway, so to make the proof complete, it is necessary to verify that the ring extension of interest (such as faithfully flat in general, or Zariski-covering in case of the question) satisfies the property which RG call (C). Gabber says that this is an easy exercise adapting the method of proof of 3.1.4 in part I of the paper (which is the case of countably presented modules).

I only ever read part I of the paper, never part II (part I was already exhausting enough, and quite spectacular/useful by itself), so in particular I do not know where an error occurs (if Gruson is right) in the proof of 2.5.2 part II. Maybe someone who has read the argument can identify where an error or gap occurs, and hopefully work out Gabber's exercise. (Bhargav?) If so, please let me know.

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A point worth noting: the proof of fpqc descent for projectivity in Raynaud-Gruson is apparently incorrect (as I learned today from Gabber in connection with something else), but the result is nonetheless true.

Here's the deal. RG deduce the result in 3.1.4(1) of part II of the paper, using 3.1.3 of part II and the fact that faithfully flat ring maps satisfy the property they call (C) there. (Briefly, a ring map satisfies property (C) when flat modules over the base ring which satisfy a certain "Mittag-Leffler" condition after the scalar extension actually satisfy the ML condition before the scalar extension. The content of 3.1.3 in part II is that this condition (C) implies descent of projectivity for flat modules. So the problem is to prove an interesting class of maps satisfies property (C).) But RG's proof of (C) for faithfully flat ring maps in 3.1.4(1) of part II rests on another result (2.5.2 in part II) which Gruson has said is incorrect (in his paper "Dimension homologique...."). That's the problem. Gabber says he does not know a counterexample to this 2.5.2 part II result. (I guess Gruson didn't give one when he said it is false.) Anyway, so to make the proof complete, it is necessary to verify that the ring extension of interest (such as faithfully flat in general, or Zariski-covering in case of the question) satisfies the property which RG call (C). Gabber says that this is an easy exercise adapting the method of proof of 3.1.4 in part I of the paper (which is the case of countably presented modules).

I only ever read part I of the paper, never part II (part I was already exhausting enough, and quite spectacular/useful by itself), so in particular I do not know where an error occurs (if Gruson is right) in the proof of 2.5.2 part II. Maybe someone who has read the argument can identify where an error or gap occurs, and hopefully work out Gabber's exercise. (Bhargav?) If so, please let me know.