Lemma 17.9.4 (01B8)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.9: Modules of finite type
Lemma 17.9.4 (cite)

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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 May 20, 2014 at 23:51

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

Comment #596 by Johan on May 23, 2014 at 20:15

Thanks for this and previous comment! Fixed here.

Comment #1816 by Keenan Kidwell on February 03, 2016 at 22:21

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

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