Descent of locally free modules

Locally free modules do not satisfy descent for fpqc coverings. I have an example involving a countable “product” of affine curves, which I will upload to the stacks project soon.

But what about fppf descent? Suppose A —> B is a faithfully flat ring map of finite presentation. Let M be an A-module such that M ⊗_A B is free. Is M a locally free A-module? (By this I mean locally free on the spectrum of A.) It turns out that if A is Noetherian, then the answer is yes. This follows from the results of Bass in his paper on “big” projective modules. But in general I don’t know the answer. If you do know the answer, or have a reference, please email me.

One thought on “Descent of locally free modules

  1. To show fppf descent, it would be enough to show descent along finite fppf morphisms and ├ętale surjective morphisms. Whether one has descent in either of these two cases is not clear to me. One potential problem is the following:

    If M is a locally free A-module of finite rank, then any finite subset Z of Spec(A) is contained in an open subset over which M is free (this follows from the fact that M is free over the semi-local ring given by localizing in Z). What about modules of infinite rank?

    Asking for descent of local freeness along finite flat morphisms is essentially equivalent to the above question.

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