When do global sections of a $\mathcal{O}_X$-Module $\mathcal{F}$ that generate $\mathcal{F}_x$ also generate $\mathcal{F}$ in a neighborhood of $x$?

Question

Let $\newcommand{\m}{\mathcal}(X, \m{O}_X)$ be a ringed space, $\m{F}$ a $\m{O}_X$-Module and $x \in X$ a point. Now let $s_1, \dots, s_n \in \Gamma(X, \m{F})$ be such that $s_{1, x}, \dots, s_{n, x}$ generate $\m{F}_x$.

Under which conditions (e. g. $X$ (locally Noetherian) Scheme, $\m{F}$ (quasi-)Coherent) doest the above imply that there exists an open neighborhood $U$ of $x$ such that $s_1|_U, \dots, s_n|_U$ generate $\m{F}|_U$ ?

Answer 1

In the book Algebraic Geomtry 1 by Görtz, Wedhorn you'll find the following proposition on page $191$

Let $(X, \mathcal{O}_X)$ be a ringed space and let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite type. Let $x \in X$ be a point and let $s_i \in \Gamma(U,\mathcal{F})$ for $i =1,\dots,n$ be sections over some open neighborhood of $x$ such that the germs $(s_i)_x$ generate the stalk $\mathcal{F}_x$. Then there exists an open neighborhood $V \subseteq U$, such that the $s_i|_V$ generate $\mathcal{F}|_V$.

You also have this proposition on page $196$

Let $X$ be a locally noetherian scheme and let $\mathcal{F}$ be an $\mathcal{O}_{X}$-module. Then the following assertions are equivalent:

• $\mathcal{F}$ is coherent

• $\mathcal{F}$ is of finite presentation

• $\mathcal{F}$ is of finite type and quasi-coherent.

On page $190$ they write if $X$ is a locally noetherian scheme, an $\mathcal{O}_{X}$-module is of finite type if and only if it is of finite presentation. So you might think that we don't need the extra condition in the above proposition. But this statement is false. As mentioned in the errata

In general, there are $\mathcal{O}_X$-modules of finite type that are not of finite presentation even over Noetherian schemes: take a quotient of $\mathcal{O}_X$ by a non-quasi coherent sheaf of ideals. It is true that a quasi-coherent $\mathcal{O}_{X}$-module of finite type over a Noetherian scheme is of finite presentation.