Lemma 31.12.14. Let $X$ be an integral locally Noetherian normal scheme with generic point $\eta$. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Let $T : \mathcal{G}_\eta \to \mathcal{F}_\eta$ be a linear map. Then $T$ extends to a map $\mathcal{G} \to \mathcal{F}^{**}$ of $\mathcal{O}_ X$-modules if and only if

• for every $x \in X$ with $\dim (\mathcal{O}_{X, x}) = 1$ we have

$T\left(\mathop{\mathrm{Im}}(\mathcal{G}_ x \to \mathcal{G}_\eta )\right) \subset \mathop{\mathrm{Im}}(\mathcal{F}_ x \to \mathcal{F}_\eta ).$

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. $\square$

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