The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AY7. Beware of the difference between the letter 'O' and the digit '0'.