Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:

Suppose $A \to B$ is a faithfully flat ring map of finite presentation. Let $M$ be an $A$-module such that $M \otimes_A B$ is free. Is $M$ a locally free $A$-module?

Note that the answer is yes for finite modules. The problem is with "large" modules. Also, if $A$ is Noetherian, then the answer is yes. This follows from results of Bass. But in general, question_bot doesn't know the answer. Do you?

  • $\begingroup$ Since fppf maps admit quasi-finite fppf quasi-sections, by considering henselizations we see that any fppf cover admits a refinement that is finite flat over an etale cover. So it is equivalent to ask the same question when $B$ is either finite free as an $A$-module or an etale cover of $A$. Not sure where to go from there. But please be more forthcoming and mention that "your" question is actually posed in the Stacks Project (rather than have that fact be hidden behind the link). In view of your flurry of recent MO questions, I hope this isn't the beginning of some trend of SP questions... $\endgroup$ – user76758 Jan 21 '14 at 4:13
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    $\begingroup$ I disagree with user76758: if these are good, research-level questions, why does it matter that the questions come from the Stacks Project? $\endgroup$ – Jason Starr Jan 21 '14 at 12:41
  • $\begingroup$ @Jason: I agree, it doesn't matter where questions come from. My "objection" was that since the question is already posed in the Stacks Project, it had looked inappropriate for the OP to write "we ask" rather than "the following question is posed in the Stacks Project". Even if someone wants to go through a document and post many of its open questions on MO, it seems reasonable for this to be done gradually over time, not dumping so many in quick succession. Now that "question_bot" has filled out a profile (not at the time of my initial comment), the context for the flood has became clearer. $\endgroup$ – user76758 Jan 21 '14 at 20:30
  • $\begingroup$ @user76758: Got it, thanks for explaining. $\endgroup$ – Jason Starr Jan 21 '14 at 20:57

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