Remark 35.7.8. Being locally free is a property of quasi-coherent modules which does not descend in the fpqc topology. Namely, suppose that $R$ is a ring and that $M$ is a projective $R$-module which is a countable direct sum $M = \bigoplus L_ n$ of rank 1 locally free modules, but not locally free, see Examples, Lemma 110.33.4. Then $M$ becomes free on making the faithfully flat base change

But we don't know what happens for fppf coverings. In other words, we don't know the answer to the following question: 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? It turns out that if $A$ is Noetherian, then the answer is yes. This follows from the results of [Bass]. But in general we don't know the answer. If you know the answer, or have a reference, please email stacks.project@gmail.com.

## Comments (0)

There are also: