The Stacks project

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

\[ R \longrightarrow \bigoplus \nolimits _{m \geq 1} \bigoplus \nolimits _{(i_1, \ldots , i_ m) \in \mathbf{Z}^{\oplus m}} L_1^{\otimes i_1} \otimes _ R \ldots \otimes _ R L_ m^{\otimes i_ m} \]

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.

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