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I'm working on exercise II.5.15 in Hartshorne's book. I need to prove the following bit.

Let $ X $ be a noetherian scheme. Let $\mathscr {F }$ be a quasi coherent sheaf on $ X$. Then the subsheaf $ \mathscr{G}\subset\mathscr{F}$ generated by $s\in \mathscr{F}(X)$ is coherent.

One idea I have is to show that $ \mathscr{G}(V)=\mathcal{O}_{X}(V)s$. I'm unable to prove that the gluing axiom of sheaves in this case.

This answer explains the idea of a subsheaf generated by sections. However, the definition is nonconstructive and I don't see a way to use it:

Some question of sheaf generated by sections

As a bonus, can we generalise this question to finitely many sections?

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0) To a section $s\in \mathscr F(X)$ of an $\mathscr O$-module (quasi-coherent or not) we can associate a morphism of $\mathscr O$-modules $h=h_s:\mathscr O\to \mathscr F$ given by $\mathscr O(U)\to \mathscr F(U):f\mapsto f\cdot (s\mid U) $.
The subsheaf of $\mathscr F$ generated by $s$ is then the image $Im(s)=s(\mathscr O)\subset \mathscr F$ of $h$.

1) If now $\mathscr F$ happens to be quasi-coherent then that image is quasi-coherent, like any image of a morphism between quasi-coherent $\mathscr O$-modules.

2) The same technique shows that a section of a coherent sheaf generates a coherent subsheaf as soon as $\mathscr O$ is coherent (which strangely is not always the case if $X$ is not noetherian).

NB
Hartshorne defines "coherent sheaf" in an idiosyncratic way , different from the definition in FAC, EGA or essentially all the literature in algebraic or analytic geometry. This is an unfortunate choice.

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    $\begingroup$ Mumford does it too, so it's okay! $\endgroup$ – Hoot Jun 3 '15 at 23:08
  • $\begingroup$ That's incredibly helpful. It's very simple once you put it this way. $\endgroup$ – PeterM Jun 6 '15 at 10:32
  • $\begingroup$ Dear Peter, I'm glad you find the answer simple and helpful. $\endgroup$ – Georges Elencwajg Jun 6 '15 at 11:31

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