Let $R$ and $R'$ be two local rings. Let $f:R \rightarrow R'$ be a local ring morphism and $R'$ is flat over $R$. Now, let $M$ be a finitely generated module over $R$. Suppose $ R' \otimes_R M$ is a free module. I want to prove that then $M$ is a free $R$-module. How can I do that?
2026-02-22 23:34:22.1771803262
Free module after tensoring with a flat local ring
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A flat local ring homomorphism is actually faithfully flat.
Then, note that finitely generated projective modules descent along faithfully flat ring maps, see here. Of course, for local rings projective = free.