Lemma 115.4.7. Let $R$ be a domain with fraction field $K$. Let $S = R[x_1, \ldots , x_ n]$ be a polynomial ring over $R$. Let $M$ be a finite $S$-module. Assume that $M$ is flat over $R$. If for every subring $R \subset R' \subset K$, $R \not= R'$ the module $M \otimes _ R R'$ is finitely presented over $S \otimes _ R R'$, then $M$ is finitely presented over $S$.
Proof. This lemma is true because $M$ is finitely presented even without the assumption that $M \otimes _ R R'$ is finitely presented for every $R'$ as in the statement of the lemma. This follows from More on Flatness, Proposition 38.13.10. Originally this lemma had an erroneous proof (thanks to Ofer Gabber for finding the gap) and was used in an alternative proof of the proposition cited. To reinstate this lemma, we need a correct argument in case $R$ is a local normal domain using only results from the chapters on commutative algebra; please email stacks.project@gmail.com if you have an argument. $\square$
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