The Stacks project

Proof. Because $\mathcal{F}^{**}$ is torsion free and $\mathcal{F}_\eta = \mathcal{F}^{**}_\eta$ an extension, if it exists, is unique. Thus it suffices to prove the lemma over the members of an open covering of $X$, i.e., we may assume $X$ is affine. In this case we are asking the following algebra question: Let $R$ be a Noetherian normal domain with fraction field $K$, let $M$, $N$ be finite $R$-modules, let $T : M \otimes _ R K \to N \otimes _ R K$ be a $K$-linear map. When does $T$ extend to a map $N \to M^{**}$? By More on Algebra, Lemma 15.23.19 this happens if and only if $N_\mathfrak p$ maps into $(M/M_{tors})_\mathfrak p$ for every height $1$ prime $\mathfrak p$ of $R$. This is exactly condition $(*)$ of the lemma. $\square$