The Stacks project

Lemma 17.9.4. Let $X$ be a ringed space. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a homomorphism of $\mathcal{O}_ X$-modules. Let $x \in X$. Assume $\mathcal{F}$ of finite type and the map on stalks $\varphi _ x : \mathcal{G}_ x \to \mathcal{F}_ x$ surjective. Then there exists an open neighbourhood $x \in U \subset X$ such that $\varphi |_ U$ is surjective.

Proof. Choose an open neighbourhood $U \subset X$ of $x$ such that $\mathcal{F}$ is generated by $s_1, \ldots , s_ n \in \mathcal{F}(U)$ over $U$. By assumption of surjectivity of $\varphi _ x$, after shrinking $U$ we may assume that $s_ i = \varphi (t_ i)$ for some $t_ i \in \mathcal{G}(U)$. Then $U$ works. $\square$

Comments (3)

Comment #583 by Anfang on

Typo in the proof. It should be "shrinking U".

Comment #1816 by Keenan Kidwell on

This conceivably could be left implicit, but should be a neighborhood of .

There are also:

  • 2 comment(s) on Section 17.9: Modules of finite type

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Your email address will not be published.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 01B8. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 17.9.4 as pdf