The Stacks project

Lemma 28.22.1. Let $j : U \to X$ be a quasi-compact open immersion of schemes.

  1. Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$.

  2. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\mathcal{H}$ of $\mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ as subsheaves of $\mathcal{F}|_ U$.

  3. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\varphi : \mathcal{G} \to \mathcal{F}|_ U$ be a morphism of $\mathcal{O}_ U$-modules. There exists a quasi-coherent sheaf $\mathcal{H}$ of $\mathcal{O}_ X$-modules and a map $\psi : \mathcal{H} \to \mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ and that $\psi |_ U = \varphi $.

Proof. An immersion is separated (see Schemes, Lemma 26.23.8) and $j$ is quasi-compact by assumption. Hence for any quasi-coherent sheaf $\mathcal{G}$ on $U$ the sheaf $j_*\mathcal{G}$ is an extension to $X$. See Schemes, Lemma 26.24.1 and Sheaves, Section 6.31.

Assume $\mathcal{F}$, $\mathcal{G}$ are as in (2). Then $j_*\mathcal{G}$ is a quasi-coherent sheaf on $X$ (see above). It is a subsheaf of $j_*j^*\mathcal{F}$. Hence the kernel

\[ \mathcal{H} = \mathop{\mathrm{Ker}}(\mathcal{F} \oplus j_* \mathcal{G} \longrightarrow j_*j^*\mathcal{F}) \]

is quasi-coherent as well, see Schemes, Section 26.24. It is formal to check that $\mathcal{H} \subset \mathcal{F}$ and that $\mathcal{H}|_ U = \mathcal{G}$ (using the material in Sheaves, Section 6.31 again).

Part (3) is proved in the same manner as (2). Just take $\mathcal{H} = \mathop{\mathrm{Ker}}(\mathcal{F} \oplus j_* \mathcal{G} \to j_*j^*\mathcal{F})$ with its obvious map to $\mathcal{F}$ and its obvious identification with $\mathcal{G}$ over $U$. $\square$


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Your email address will not be published.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 01PE. Beware of the difference between the letter 'O' and the digit '0'.