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In the comments of this question:

What does it mean when the extension of scalars is free?,

it is mentioned that if $ \psi: R \rightarrow S $ is a faithfully flat ring homomorphism between commutative rings, with $R$ Noetherian and $M$ a finitely generated $R$ module then if the extension of scalars $M \otimes_R S$ is a free $S$-module of rank one, $M$ is a (finitely generated) projective $R$-module of rank one. Can anybody provide me with a reference for this fact? I have tried looking in Bourbaki's Commutative Algebra book and have searched on the Stacks project but I could not find it.

Thank you in advance for any help!

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    $\begingroup$ You will find this in two steps in many books (say Matsumura). First, a module which becomes flat after a faithfully flat extension is flat. Second, a finitely generated flat module over a Noetherian ring is projective. $\endgroup$
    – Mohan
    Commented Aug 23, 2023 at 20:33

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